ENCYCLOPEDIA OF MATHEMATICS AND ITS ApPLICATIONS

Editorial Board

R. S. Doran, P. Flajolet, M. Ismail, T.-y' Lam, E. Lutwak

Volume 96

Basic Hypergeometric Series Second Edition

This revised and expanded new edition will continue to meet the need for an author­ itative, up-to-date, self contained, and comprehensive account of the rapidly growing field of basic hypergeometric series, or q-series. It contains almost all of the important summation and transformation formulas of basic hypergeometric series one needs to know for work in fields such as combinatorics, number theory, modular forms, quan­ tum groups and algebras, probability and statistics, coherent-state theory, orthogonal polynomials, or approximation theory. Simplicity, clarity, deductive proofs, thought­ fully designed exercises, and useful appendices are among its strengths. The first five chapters cover basic hypergeometric series and integrals, whilst the next five are de­ voted to applications in various areas including Askey-Wilson integrals and orthogonal polynomials, partitions in number theory, multiple series, and generating functions. Chapters 9 to 11 are new for the second edition, the final chapter containing a sim­ plified version of the main elements of the theta and elliptic hypergeometric series as a natural extension of the single-base q-series. Elsewhere some new material and exercises have been added to reflect recent developments, and the bibliography has been revised to maintain its comprehensive nature. ENCYCLOPEDIA OF MATHEMATICS AND ITS ApPLICATIONS

All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://publishing.cambridge.org/stm/mathematics/eom/.

60. Jan Krajicek Bounded Arithmetic, Propositional Logic, and Complex Theory 61. H. Gromer Geometric Applications of Fourier Series and Spherical Harmonics 62. H. O. Fattorini Infinite Dimensional Optimization and Control Theory 63. A. C. Thompson Minkowski Geometry 64. R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices and Applications 65. K. Engel Sperner Theory 66. D. Cvetkovic, P. Rowlinson and S. Simic Eigenspaces of Graphs 67. F. Bergeron, G. Labelle and P. Leroux Combinatorial Species and Tree-Like Structures 68. R. Goodman and N. Wallach Representations of the Classical Groups 69. T. Beth, D. Jungnickel and H. Lenz Design Theory Volume I2 ed. 70. A. Pietsch and J. Wenzel Orthonormal Systems and Banach Space Geometry 71. George E. Andrews, Richard Askey and Ranjan Roy Special Functions 72. R. Ticciati Quantum Field Theory for Mathematicians 76. A. A. Ivanov Geometry of Sporadic Groups I 78. T. Beth, D. Jungnickel and H. Lenz Design Theory Volume II 2 ed. 80. O. Stormark Lie's Structural Approach to PDE Systems 81. C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables 82. J. Mayberry The Foundations of Mathematics in the Theory of Sets 83. C. Foias, R. Temam, O. Manley and R. Martins da Silva Rosa Navier-Stokes Equations and Turbulence 84. B. Polster and G. Steinke Geometries on Surfaces 85. D. Kaminski and R. B. Paris Asymptotics and Mellin-Barnes Integrals 86. Robert J. McEliece The Theory of Information and Coding 2 ed. 87. Bruce A. Magurn An Algebraic Introduction to K-Theory 88. Teo Mora Solving Polynomial Equation Systems I 89. Klaus Bichteler Stochastic Integration with Jumps 90. M. Lothaire Algebraic Combinatorics on Words 91. A. A. Ivanov and S. V. Shpectorov Geometry of Sporadic Groups, 2 92. Peter McMullen and Egon Schulte Abstract Regular Polytopes 93. G. Gierz et al. Continuous Lattices and Domains 94. Steven R. Finch Mathematical Constants 95. Youssef Jabri The Mountain Pass Theorem 96. George Gasper and Mizan Rahman Basic Hypergeometric Series 2 ed. 97. Maria Cristina Pedicchio & Walter Tholen Categorical Foundations ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

BASIC HYPERGEOMETRIC SERIES

Second Edition

GEORGE GASPER Northwestern University, Evanston, Illinois, USA MIZAN RAHMAN Carleton University, Ottawa, Canada

CAMBRIDGE UNIVERSITY PRESS PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom

CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc6n 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org

© Cambridge University Press 1990, 2004

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 1990 Second edition 2004.

Printed in the United Kingdom at the University Press, Cambridge

Typeface Computer Modern 10/12 pt. System 'lEX [TB]

A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Gasper, George. Basic hypergeometric series / George Gasper, Mizan Rahman. - 2nd edn. p. cm. - (Encyclopedia of mathematics and its applications; v. 96) Includes bibliographical references and indexes. ISBN 0 521 83357 4 1. Hypergeometric series. I. Rahman, Mizan. II. Title. III. Series. QA353.H9G37 2004 515'.243--dc22 2004045686

ISBN 0 521 83357 4 hardback

The publisher has used its best endeavors to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate. To Brigitta, Karen, and Kenneth Gasper and Babu, Raja, and to the memory of Parul S. Rahman

Contents

Foreword page xiii Preface xxi Preface to the second edition xxv

1 Basic hypergeometric series 1 1.1 Introduction 1 1.2 Hypergeometric and basic hypergeometric series 1 1.3 The q-binomial theorem 8 1.4 Heine's transformation formulas for 2¢1 series 13 1.5 Heine's q-analogue of Gauss' summation formula 14 1.6 Jacobi's triple product identity, theta functions, and elliptic numbers 15 1. 7 A q-analogue of Saalschiitz's summation formula 17 1.8 The Bailey-Daum summation formula 18 1.9 q-analogues of the Karlsson-Minton summation formulas 18 1.10 The q-gamma and q-beta functions 20 1.11 The q-integral 23 Exercises 24 Notes 34

2 Summation, transformation, and expansion formulas 38 2.1 Well-poised, nearly-poised, and very-well-poised hypergeometric and basic hypergeometric series 38 2.2 A general expansion formula 40 2.3 A summation formula for a terminating very-well-poised 4¢3 series 41 2.4 A summation formula for a terminating very-well-poised 6¢5 series 42 2.5 Watson's transformation formula for a terminating very-well-poised 8¢7 series 42 2.6 Jackson's sum of a terminating very-well-poised balanced 8¢7 series 43 2.7 Some special and limiting cases of Jackson's and Watson's formulas: the Rogers-Ramanujan identities 44 2.8 Bailey's transformation formulas for terminating 5¢4 and 7¢6 series 45 2.9 Bailey's transformation formula for a terminating 1O¢9 series 47

vii viii Contents

2.10 Limiting cases of Bailey's 1O¢9 transformation formula 48 2.11 Bailey's three-term transformation formula for VWP-balanced S¢7 series 53 2.12 Bailey's four-term transformation formula for balanced 1O¢9 series 55 Exercises 58 Notes 67

3 Additional summation, transformation, and expansion formulas 69 3.1 Introduction 69 3.2 Two-term transformation formulas for 3¢2 series 70 3.3 Three-term transformation formulas for 3¢2 series 73 3.4 Transformation formulas for well-poised 3¢2 and very-well-poised S¢4 series with arbitrary arguments 74 3.5 Transformations of series with base q2 to series with base q 77 3.6 Bibasic summation formulas 80 3.7 Bibasic expansion formulas 84 3.8 Quadratic, cubic, and quartic summation and transformation formulas 88 3.9 Multibasic hypergeometric series 95 3.10 Transformations of series with base q to series with base q2 96 Exercises 100 Notes 111

4 Basic contour integrals 113 4.1 Introduction 113 4.2 Watson's contour integral representation for 2¢1 (a, b; c; q, z) series 115 4.3 Analytic continuation of 2¢1(a, b; c; q, z) 117 4.4 q-analogues of Barnes' first and second lemmas 119 4.5 Analytic continuation of r+l ¢r series 120 4.6 Contour integrals representing well-poised series 121 4.7 A contour integral analogue of Bailey's summation formula 123 4.8 Extensions to complex q inside the unit disc 124 4.9 Other types of basic contour integrals 125 4.10 General basic contour integral formulas 126 Contents ix

4.11 Some additional extensions of the beta integral 129 4.12 Sears' transformations of wen-poised series 130 Exercises 132 Notes 135

5 Bilateral basic hypergeometric series 137 5.1 Notations and definitions 137 5.2 Ramanujan's sum for 1 'l/Jl (a; b; q, z) 138 5.3 Bailey's sum of a very-wen-poised 6'l/J6 series 140 5.4 A general transformation formula for an r'l/Jr series 141 5.5 A general transformation formula for a very-wen-poised 2r'l/J2r series 143 5.6 Transformation formulas for very-wen-poised 8'l/J8 and 1O'l/J1O series 145 Exercises 146 Notes 152

6 The Askey-Wilson q-beta integral and some associated formulas 154 6.1 The Askey-Wilson q-extension of the beta integral 154 6.2 Proof of formula (6.1.1) 156 6.3 Integral representations for very-wen-poised 8¢7 series 157 6.4 Integral representations for very-wen-poised 1O¢9 series 159 6.5 A quadratic transformation formula for very-wen-poised balanced 1O¢9 series 162 6.6 The Askey-Wilson integral when max (Ial, Ibl, lei, Idl) ~ 1 163 Exercises 168 Notes 173

7 Applications to orthogonal polynomials 175 7.1 Orthogonality 175 7.2 The finite discrete case: the q-Racah polynomials and some special cases 177 7.3 The infinite discrete case: the little and big q-Jacobi polynomials 181 7.4 An absolutely continuous measure: the continuous q-ultraspherical polynomials 184 7.5 The Askey-Wilson polynomials 188 x Contents

7.6 Connection coefficients 195 7.7 A difference equation and a Rodrigues-type formula for the Askey-Wilson polynomials 197 Exercises 199 Notes 213

8 Further applications 217 8.1 Introduction 217 8.2 A product formula for balanced 4

9 Linear and bilinear generating functions for basic orthogonal polynomials 259 9.1 Introduction 259 9.2 The little q-Jacobi polynomials 260 9.3 A generating function for Askey-Wilson polynomials 262 9.4 A bilinear sum for the Askey-Wilson polynomials I 265 9.5 A bilinear sum for the Askey-Wilson polynomials II 269 9.6 A bilinear sum for the Askey-Wilson polynomials III 270 Exercises 272 Notes 281 Contents xi

10 q-series in two or more variables 282 10.1 Introduction 282 10.2 q-Appell and other basic double hypergeometric series 282 10.3 An integral representation for <[>(1) (qa; qb, qb'; qC; q; x, Y) 284 10.4 Formulas for <[>(2) (qa; qb, qb' ; qC, qC' ; q; x, Y) 286 10.5 Formulas for <[>(3) (qa, qa'; qb, qb'; qC; q; x, Y) 288 10.6 Formulas for a q-analogue of F4 290 10.7 An Askey-Wilson-type integral representation for a q-analogue of Fl 294 Exercises 296 Notes 301

11 Elliptic, modular, and theta hypergeometric series 302 11.1 Introduction 302 11.2 Elliptic and theta hypergeometric series 303 11.3 Additive notations and modular series 312 11.4 Elliptic analogue of Jackson's S¢7 summation formula 321 11.5 Elliptic analogue of Bailey's transformation formula for a terminating 1O¢9 series 323 11.6 Multibasic summation and transformation formulas for theta hypergeometric series 325 11.7 Rosengren's elliptic extension of Milne's fundamental theorem 331 Exercises 336 Notes 349

Appendix I Identities involving q-shifted factorials, q-gamma functions and q-binomial coefficients 351 Appendix II Selected summation formulas 354 Appendix III Selected transformation formulas 359 References 367 Symbol index 415 Author index 418 Subject index 423

Foreword

My education was not much different from that of most mathematicians of my generation. It included courses on modern algebra, real and complex variables, both point set and algebraic topology, some number theory and projective ge­ ometry, and some specialized courses such as one on Riemann surfaces. In none of these courses was a hypergeometric function mentioned, and I am not even sure if the gamma function was mentioned after an advanced calculus course. The only time Bessel functions were mentioned was in an undergradu­ ate course on differential equations, and the only thing done with them was to find a power series solution for the general Bessel equation. It is small wonder that with a similar education almost all mathematicians think of special func­ tions as a dead subject which might have been interesting once. They have no idea why anyone would care about it now. Fortunately there was one part of my education which was different. As a junior in college I read Widder's book The Laplace Transform and the manuscript of its very important sequel, Hirschman and Widder's The Convo­ lution Transform. Then as a senior, 1. 1. Hirschman gave me a copy of a preprint of his on a multiplier theorem for Legendre series and suggested I extend it to ultraspherical series. This forced me to become acquainted with two other very important books, Gabor Szego's great book Orthogonal Polynomials, and the second volume of Higher Transcendental Functions, the monument to Harry Bateman which was written by Arthur Erdelyi and his co-workers W. Magnus, F. Oberhettinger and F. G. Tricomi. From this I began to realize that the many formulas that had been found, usually in the 18th or 19th century, but once in a while in the early 20th century, were useful, and started to learn about their structure. However, I had written my Ph.D. thesis and worked for three more years before I learned that not every fact about special functions I would need had already been found, and it was a couple of more years before I learned that it was essential to understand hypergeometric functions. Like others, I had been put off by all the parameters. If there were so many parameters that it was necessary to put subscripts on them, then there has to be a better way to solve a problem than this. That was my initial reaction to generalized hypergeometric functions, and a very common reaction to judge from the many conversations I have had on these functions in the last twenty years. After learning a little more about hypergeometric functions, I was very surprised to realize that they had occurred regularly in first year calculus. The reason for the subscripts on the parameters is that not all interesting polynomials are of degree one or two. For

Xlll xiv Foreword a generalized hypergeometric function has a series representation

00 (1) n=O with cn+I/cn a rational function of n. These contain almost all the examples of infinite series introduced in calculus where the ratio test works easily. The ratio cn+I/cn can be factored, and it is usually written as

(2) (n + b1) ... (n + bq ) (n + 1) . Introduce the shifted factorial (a)O = 1, (3) (a)n = a(a + 1) ... (a + n - 1), n = 1,2, ...

Then if Co = 1, equation (2) can be solved for Cn as (al)n··· (ap)n xn Cn = - (4) (bdn ... (bq)n n! and (5) is the usual notation. The first important result for a pFq with p > 2, q > 1 is probably Pfaff's sum -n, a, b ] (C - a)n(c - b)n 3F2 [ ; 1 n = 0,1, .... (6) c, a + b + 1 - C - n (c)n(c - a - b)n'

This result from 1797, see Pfaff [1797], contains as a limit when n ----; 00, another important result usually attributed to Gauss [1813],

F [a, b. 1] = r(c)r(c-a-b) Re (c-a-b) > O. (7) 2 1 c' r(c-a)r(c-b)'

The next instance is a very important result of Clausen [1828]:

{ F [ a, b ·x] }2 = F [2a, 2b, a + b ·x] (8) 2 1 a + b + ~ , 3 2 a + b + ~, 2a + 2b ' . Some of the interest in Clausen's formula is that it changes the square of a class of 2Fl'S to a 3F2. In this direction it is also interesting because it was probably the first instance of anyone finding a differential equation satisfied by [Y(X)]2, y(x)z(x) and [Z(X)]2 when y(x) and z(x) satisfy

a(x)y" + b(x)y' + c(x)y = O. (9) Foreword xv

This problem was considered for (9) by Appell, see Watson [1952]' but the essence of his general argument occurs in Clausen's paper. This is a com­ mon phenomenon, which is usually not mentioned when the general method is introduced to students, so they do not learn how often general methods come from specific problems or examples. See D. and G. Chudnovsky [1988] for an instance of the use of Clausen's formula, where a result for a 2Fl is carried to a 3F2 and from that to a very interesting set of expansions of 1[-1. Those identi­ ties were first discovered by Ramanujan. Here is Ramanujan's most impressive example:

There is another important reason why Clausen's formula is important. It leads to a large class of 3F2 's that are nonnegative for the power series variable between -1 and 1. The most famous use of this is in the final step of de Branges' solution of the Bieberbach conjecture, see de Branges [1985]. The integral of the 2Fl or Jacobi polynomial he had is a 3F2' and its positivity is an easy consequence of Clausen's formula, as Gasper had observed ten years earlier. There are other important results which follow from the positivity in Clausen's identity. Once Kummer [1836] wrote his long and important paper on 2Fl 's and IFI 's, this material became well-known. It has been reworked by others. Rie­ mann redid the 2Fl using his idea that the singularities of a function go a long way toward determining the function. He showed that if the differential equa­ tion (9) has regular singularities at three points, and every other point in the extended complex plane is an ordinary point, then the equation is equivalent to the hypergeometric equation

x(l - x)y" + [c - (a + b + l)x]y' - aby = 0, (11) which has regular singular points at x = 0,1,00. Riemann's work was very influential, so much so that much of the mathematical community that consid­ ered hypergeometric functions studied them almost exclusively from the point of view of differential equations. This is clear in Klein's book [1933], and in the work on multiple hypergeometric functions that starts with Appell in 1880 and is summarized in Appell and Kampe de Feriet [1926]. The integral representations associated with the differential equation point of view are similar to Euler's integral representation. This is

F [a, b. x] = r(c) (1(1 _ xt)-atb- 1 (1 _ t)c-b-1 dt (12) 2 1 c' r(b)r(c - b) io '

Ixl < 1, Re c > Re b > 0, and includes related integrals with different contours. The differential equation point of view is very powerful where it works, but it xvi Foreword does not work well for p 2: 3 or q 2: 2 as Kummer discovered. Thus there is a need to develop other methods to study hypergeometric functions. In the late 19th and early 20th century a different type of integral rep­ resentation was introduced. These two different types of integrals are best represented by Euler's beta integral

1 r ta - 1(1 _ t)b-1dt = r(a)r(b) , Re (a, b) > 0 (13) io r(a+b) and Barnes' beta integral

00 -1 1 r(a + it)r(b + it)r(c - it)r(d - it) dt (14) 2n -00 r(a + c)r(a + d)r(b + c)r(b + d) ( ) Re (a, b, c, d) > o. ra+b+c+d

There is no direct connection with differential equations for integrals like (14), so it stands a better chance to work for larger values of p and q. While Euler, Gauss, and Riemann and many other great mathematicians wrote important and influential papers on hypergeometric functions, the devel­ opment of basic hypergeometric functions was much slower. Euler and Gauss did important work on basic hypergeometric functions, but most of Gauss' work was unpublished until after his death and Euler's work was more influ­ ential on the development of number theory and elliptic functions. Basic hypergeometric series are series L Cn with cn+I/cn a rational func­ tion of qn for a fixed parameter q, which is usually taken to satisfy Iql < 1, but at other times is a power of a prime. In this Foreword Iql < 1 will be assumed. Euler summed three basic hypergeometric series. The one which had the largest impact was

00 2) _1)nq(3n 2 -n)/2 = (q; q)oo, (15)

-00 where 00 (a; q)oo = II (1 - aqn). (16) n=O

If (17) then Euler also showed that

1 00 xn ..,.....-....,...--2:-- Ixl

and

(19)

Eventually all of these were contained in the q-binomial theorem

(20)

While (18) is clearly the special case a = 0, and (19) follows easily on replacing x by xa-1 and letting a ----t 00, it is not so clear how to obtain (15) from (20). The easiest way was discovered by Cauchy and many others. Take a = q-2N, shift n by N, rescale and let N ----t 00. The result is called the triple product, and can be written as

00 (x; q)00(qx- 1 ; q)oo(q; q)oo = L (-1tq(~)xn. (21) n=-(XJ

Then q ----t q3 and x = q gives Euler's formula (15). Gauss used a basic hypergeometric series identity in his first proof of the determination of the sign of the Gauss sum, and Jacobi used some to determine the number of ways an integer can be written as the sum of two, four, six and eight squares. However, this particular aspect of Gauss' work on Gauss sums was not very influential, as his hypergeometric series work had been, and Jacobi's work appeared in his work on elliptic functions, so its hypergeometric character was lost in the great interest in the elliptic function work. Thus neither of these led to a serious treatment of basic hypergeometric series. The result that seems to have been the crucial one was a continued fraction of Eisenstein. This along with the one hundredth anniversary of Euler's first work on continued fractions seem to have been the motivating forces behind Heine's introduction of a basic hypergeometric extension of 2Fl (a, b; c; x). He considered

Ixl < 1. (22)

Observe that

so

Heine followed the pattern of Gauss' published paper on hypergeometric series, and so obtained contiguous relations and from them continued fraction xviii Foreword expansions. He also obtained some series transformations, and the sum a b ] (c-a) (c-b ) '" q, q. c-a-b _ q ; q 00 q ; q 00 [ (23) 2'/'1 qc' q, q - ( q,qooqc. ) (c-a-b. ,qoo ) ,

This sum becomes (7) when q ---+ l. As often happens to path breaking work, this work of Heine was to a large extent ignored. When writing the second edition of K ugelfunctionen (Heine [1878]) Heine decided to include some of his work on basic hypergeometric series. This material was printed in smaller type, and it is clear that Heine in­ cluded it because he thought it was important, and he wanted to call attention to it, rather than because he thought it was directly related to spherical har­ monics, the subject of his book. Surprisingly, his inclusion of this material led to some later work, which showed there was a very close connection between Heine's work on basic hypergeometric series and spherical harmonics. The person Heine influenced was L. J. Rogers, who is still best known as the first discoverer of the Rogers-Ramanujan identities. Rogers tried to understand this aspect of Heine's work, and one transformation in particular. Thomae [1879] had observed this transformation of Heine could be written as an extension of Euler's integral representation (12), but Rogers was unaware of this explana- tion, and so discovered a second reason. He was able to modify the transfor­ mation so it became the permutation symmetry in a new series. While doing this he introduced a new set of polynomials which we now call the continuous q-Hermite polynomials. In a very important set of papers which were unjustly neglected for decades, Rogers discovered a more general set of polynomials and found some remarkable identities they satisfy, see Rogers [1893a,b, 1894, 1895]. For example, he found the linearization coefficients of these polynomials which we now call the continuous q-ultraspherical polynomials. These polynomials contain many of the spherical harmonics Heine studied. Contained within this product identity is the special case of the square of one of these polynomials as a double series. As Gasper and Rahman have observed, one of these series can be summed, and the resulting identity is an extension of Clausen's sum in the terminating case. Earlier, others had found a different extension of Clausen's identity to basic hypergeometric series, but the resulting identity was not sat­ isfactory. The identity had the product of two functions, the same functions but one evaluated at x and the other at qx, and so was not a square. Thus the nonnegativity that is so useful in Clausen's formula was not true for the corresponding basic hypergeometric series. Rogers' result for his polynomials led directly to the better result which contains the appropriate nonnegativity. From this example and many others, one sees that orthogonal polynomials provide an alternative approach to the study of hypergeometric and basic hy­ pergeometric functions. Both this approach and that of differential equations are most useful for small values of the degrees of the numerator and denomi­ nator polynomials in the ratio cn+I/cn, but orthogonal polynomials work for a larger class of series, and are much more useful for basic hypergeometric se- Foreword xix ries. However, neither of these approaches is powerful enough to encompass all aspects of these functions. Direct series manipulations are surprisingly useful, when done by a master, or when a computer algebra system is used as an aid. Gasper and Rahman are both experts at symbolic calculations, and I regularly marvel at some of the formulas they have found. As quantum groups become better known, and as Baxter's work spreads to other parts of mathematics as it has started to do, there will be many people trying to learn how to deal with basic hypergeometric series. This book is where I would start. For many years people have asked me what is the best book on special functions. My response was George Gasper's copy of Bailey's book, which was heavily annotated with useful results and remarks. Now others can share the information contained in these margins, and many other very useful results.

Richard Askey University of Wisconsin

Preface

The study of basic hypergeometric series (also called q-hypergeometric series or q-series) essentially started in 1748 when Euler considered the infinite product 00 (q; q)~,I = I1 (1 - qk+ 1)-1 as a generating function for p(n), the number of k= 0 partitions of a positive integer n into positive integers (see §8.1O). But it was not until about a hundred years later that the subject acquired an independent status when Heine converted a simple observation that lim [(1- qa)j(1-q)] = a q---+I into a systematic theory of 2 ¢I basic hypergeometric series parallel to the theory of Gauss' 2FI hypergeometric series. Heine's transformation formulas for 2¢1 series and his q-analogue of Gauss' 2FI (1) summation formula are derived in Chapter 1, along with a q-analogue of the binomial theorem, Jacobi's triple product identity, and some formulas for q-analogues of the exponential, gamma and beta functions. Apart from some important work by J. Thomae and L. J. Rogers the subject remained somewhat dormant during the latter part of the nineteenth century until F. H. Jackson embarked on a lifelong program of developing the theory of basic hypergeometric series in a systematic manner, studying q-differentiation and q-integration and deriving q-analogues of the hyperge­ ometric summation and transformation formulas that were discovered by A. C. Dixon, J. Dougall, L. Saalschiitz, F. J. W. Whipple, and others. His work is so pervasive that it is impossible to cover all of his contributions in a single volume of this size, but we have tried to include many of his important formulas in the first three chapters. In particular, a derivation of his summation formula for an 8¢7 series is given in §2.6. During the 1930's and 1940's many important results on hypergeometric and basic hypergeometric series were de­ rived by W. N. Bailey. Some mathematicians consider Bailey's greatest work to be the Bailey transform (an equivalent form of which is covered in Chap­ ter 2), but equally significant are his nonterminating extensions of Jackson's 8¢7 summation formula and of Watson's transformation formula connecting very-well-poised 8¢7 series with balanced 4¢3 series. Much of the material on summation, transformation and expansion formulas for basic hypergeometric series in Chapter 2 is due to Bailey. D. B. Sears, L. Carlitz, W. Hahn, and L. J. Slater were among the promi­ nent contributors during the 1950's. Sears derived several transformation for­ mulas for 3 ¢2 series, balanced 4¢3 series, and very-well-poised n+ 1 ¢n series. Simple proofs of some of his 3 ¢2 transformation formulas are given in Chap­ ter 3. Three of his very-well-poised transformation formulas are derived in Chapter 4, where we follow G. N. Watson and Slater to develop the theory of

xxi xxii Preface basic hypergeometric series from a contour integral point of view, an idea first introduced by Barnes in 1907. Chapter 5 is devoted to bilateral basic hypergeometric series, where the most fundamental formula is Ramanujan's ] 'l/J] summation formula. Substan­ tial contributions were also made by Bailey, M. Jackson, Slater and others, whose works form the basis of this chapter. During the 1960's R. P. Agarwal and Slater each published a book par­ tially devoted to the theory of basic hypergeometric series, and G. E. Andrews initiated his work in number theory, where he showed how useful the sum­ mation and transformation formulas for basic hypergeometric series are in the theory of partitions. Andrews gave simpler proofs of many old results, wrote review articles pointing out many important applications and, during the mid 1970's, started a period of very fruitful collaboration with R. Askey. Thanks to these two mathematicians, basic hypergeometric series is an active field of re­ search today. Since Askey's primary area of interest is orthogonal polynomials, q-series suddenly provided him and his co-workers with a very rich environment for deriving q-extensions of beta integrals and of the classical orthogonal poly­ nomials of Jacobi, Gegenbauer, Legendre, Laguerre and Hermite. Askey and his students and collaborators who include W. A. AI-Salam, M. E. H. Ismail, T. H. Koornwinder, W. G. Morris, D. Stanton, and J. A. Wilson have produced a substantial amount of interesting work over the past fifteen years. This flurry of activity has been so infectious that many researchers found themselves hope­ lessly trapped by this alluring "q-disease", as it is affectionately called. Our primary motivation for writing this book was to present in one modest volume the significant results of the past two hundred years so that they are readily available to students and researchers, to give a brief introduction to the applications to orthogonal polynomials that were discovered during the current renaissance period of basic hypergeometric series, and to point out important applications to other fields. Most of the material is elementary enough so that persons with a good background in analysis should be able to use this book as a textbook and a reference book. In order to assist the reader in developing a deeper understanding of the formulas and proof techniques and to include additional formulas, we have given a broad range of exercises at the end of each chapter. Additional information is provided in the Notes following the Exercises, particularly in relation to the results and relevant applications contained in the papers and books listed in the References. Although the References may have a bulky appearance, it is just an introduction to the vast literature available. Appendices I, II, and III are for quick reference, so that it is not necessary to page through the book in order to find the most frequently needed identities, summation formulas, and transformation formulas. It can be rather tedious to apply the summation and transformation formulas to the derivation of other formulas. But now that several symbolic computer algebraic systems are available, persons having access to such a system can let it do some of the symbolic manipulations, such as computing the form of Bailey's 1O¢9 transformation formula when its parameters are replaced by products of other parameters. Preface xxiii

Due to space limitations, we were unable to be as comprehensive in our coverage of basic hypergeometric series and their applications as we would have liked. In particular, we could not include a systematic treatment of basic hypergeometric series in two or more variables, covering F. H. Jackson's work on basic Appell series and the works of R. A. Gustafson and S. C. Milne on U(n) multiple series generalizations of basic hypergeometric series referred to in the References. But we do highlight Askey and Wilson's fundamental work on their beautiful q-analogue of the classical beta integral in Chapter 6 and develop its connection with very-well-poised 8(P7 series. Chapter 7 is devoted to applications to orthogonal polynomials, mostly developed by Askey and his collaborators. We conclude the book with some further applications in Chapter 8, where we present part of our work on product and linearization formulas, Poisson kernels, and nonnegativity, and we also manage to point out some elementary facts about applications to the theory of partitions and the representations of integers as sums of squares of integers. The interested reader is referred to the books and papers of Andrews and N. J. Fine for additional applications to partition theory, and recent references are pointed out for applications to affine root systems (Macdonald identities), association schemes, combinatorics, difference equations, Lie algebras and groups, physics (such as representations of quantum groups and R. J. Baxter's work on the hard hexagon model of phase transitions in statistical mechanics), statistics, etc. We use the common numbering system of letting (k.m.n) refer to the n-th numbered display in Section m of Chapter k, and letting (I.n), (Il.n) , and (1II.n) refer to the n-th numbered display in Appendices I, II, and III, respectively. To refer to the papers and books in the References, we place the year of publication in square brackets immediately after the author's name. Thus Bailey [1935] refers to Bailey's 1935 book. Suffixes a, b, ... are used after the years to distinguish different papers by an author that appeared in the same year. Papers that have not yet been published are referred to with the year 2004, even though they might be published later due to the backlogs of journals. Since there are three Agarwals, two Chiharas and three Jacksons listed in the References, to minimize the use of initials we drop the initials of the author whose works are referred to most often. Hence Agarwal, Chihara, and Jackson refer to R. P. Agarwal, T. S. Chihara, and F. H. Jackson, respectively. We would like to thank the publisher for their cooperation and patience during the preparation of this book. Thanks are also due to R. Askey, W. A. Ai-Salam, R. P. Boas, T. S. Chihara, B. Gasper, R. Holt, M. E. H. Ismail, T. Koornwinder, and B. Nassrallah for pointing out typos and suggesting im­ provements in earlier versions of the book. We also wish to express our sincere thanks and appreciation to our TgXtypist, Diane Berezowski, who suffered through many revisions of the book but never lost her patience or sense of humor.

Preface to the second edition

In 1990 it was beyond our wildest imagination that we would be working on a second edition of this book thirteen years later. In this day and age of rapid growth in almost every area of mathematics, in general, and in Orthogonal Polynomials and Special Functions, in particular, it would not be surprising if the book became obsolete by now and gathered dust on the bookshelves. All we hoped for is a second printing. Even that was only a dream since the main competitor of authors and publishers these days are not other books, but the ubiquitous copying machine. But here we are: bringing out a second edition with full support of our publisher. The main source of inspiration, of course, has been the readers and users of this book. The response has been absolutely fantastic right from the first weeks the book appeared in print. The kind of warm reception we enjoyed far exceeded our expectations. Years later many of the leading researchers in the field kept asking us if an updated version would soon be forthcoming. Yes, indeed, an updated and expanded version was becoming necessary dur­ ing the latter part of the 1990's in view of all the explosive growth that the subject was experiencing in many different areas of applications of basic hy­ pergeometric series (also called q-series). However, the most important and significant impetus came from an unexpected source - Statistical Mechan­ ics. In trying to find elliptic (doubly periodic meromorphic) solutions of the so-called Yang-Baxter equation arising out of an eight-vertex model in Statis­ tical Mechanics the researchers found that the solutions are, in fact, a form of hypergeometric series L anzn, where an+ 1/ an is an elliptic function of n, with n regarded as a complex variable. In a span of only five years the study of elliptic hypergeometric series and integrals has become almost a separate area of research on its own, whose leading researchers include, in alphabeti­ cal order, R. J. Baxter, E. Date, J. F. van Diejen, G. Felder, P. J. Forrester, 1. B. Frenkel, M. Jimbo, Y. Kajihara, K. Kajiwara, H. T. Koelink, A. Kuniba, T. Masuda, T. Miwa, Y. van Norden, M. Noumi, Y. Ohta, M. Okado, E. Rains, H. Rosengren, S. N. M. Ruijsenaars, M. Schlosser, V. P. Spiridonov, L. Stevens, V. G. Turaev, A. Varchenko, S. O. Warnaar, Y. Yamada, and A. S. Zhedanov. Even though we had not had any research experience in this exciting new field, it became quite clear to us that a new edition could be justified only if we included a chapter on elliptic hypergeometric series (and modular and theta hypergeometric series), written in an expository manner so that it would be more accessible to non-experts and be consistent with the rest of the book. Chapter 11 is entirely devoted to that topic. Regrettably, we had to be ruthlessly selective about choosing one particular approach from many

xxv xxvi Preface to the second edition possible approaches, all of which are interesting and illuminating on their own. We were guided by the need for brevity and clarity at the same time, as well as consistency of notations. In addition to Chapter 11 we added Chapter 9 on generating and bi­ linear generating functions in view of their central importance in the study of orthogonal polynomials, as well as Chapter 10, covering briefly the huge topic of multivariable q-series, restricted mostly to F. H. Jackson's q-analogues of the four Appell functions Fl , F2 , F3 , F4 , and some of their more recent extensions. In these chapters we have attempted to describe the basic methods and results, but left many important formulas as exercises. Some parts of Chapters 1-8 were updated by the addition of textual material and a number of exercises, which were added at the ends of the sections and exercises in order to retain the same numbering of the equations and exercises as in the first edition. We have corrected a number of minor typos in the first edition, some discovered by ourselves, but most kindly pointed out to us by researchers in the field, whose contributions are gratefully acknowledged. A list of er­ rata, updates of the references, etc., for the first edition and its translation into Russian by N. M. Atakishiyev and S. K. Suslov may be downloaded at arxiv.org/ abs/math.CA/9705224 or at www.math.northwestern.edu/ ~george/ preprints/bhserrata, which is usually the most up-to-date. Analogous to the first edition, papers that have not been published by November of 2003 are referred to with the year 2003, even though they might be published later. The number of people to whom we would like to express our thanks and gratitude is just too large to acknowledge individually. However, we must men­ tion the few whose help has been absolutely vital for the preparation of this edition. They are S. O. Warnaar (who proof-read our files with very detailed comments and suggestions for improvement, not just for Chapter 11, which is his specialty, but also the material in the other chapters), H. Rosengren (whose e-mails gave us the first clue as to how we should present the topic of Chapter 11), M. Schlosser (who led us in the right directions for Chapter 11), S. K. Suslov (who sent a long list of errata, additional references, and sug­ gestions for new exercises), S. C. Milne (who suggested some improvements in Chapters 5,7,8, and 11), V. P. Spiridonov (who suggested some improvements in Chapter 11), and M. E. H. Ismail (whose comments have been very helpful). Thanks are also due to R. Askey for his support and useful comments. Finally, we need to mention the name of a behind-the-scene helper, Brigitta Gasper, who spent many hours proofreading the manuscript. The mention of our ever gracious 'IEXtypist, Diane Berezowski, is certainly a pleasure. She did not have to do the second edition, but she said she enjoys the work and wanted to be a part of it. Where do you find a more committed friend? We owe her immensely. 1

BASIC HYPERGEOMETRlC SERlES

1.1 Introduction

Our main objective in this chapter is to present the definitions and no­ tations for hypergeometric and basic hypergeometric series, and to derive the elementary formulas that form the basis for most of the summation, transfor­ mation and expansion formulas, basic integrals, and applications to orthogonal polynomials and to other fields that follow in the subsequent chapters. We be­ gin by defining Gauss' 2Fl hypergeometric series, the rFs (generalized) hyper­ geometric series, and pointing out some of their most important special cases. Next we define Heine's 2¢1 basic hypergeometric series which contains an addi­ tional parameter q, called the base, and then give the definition and notations for r¢s basic hypergeometric series. Basic hypergeometric series are called q-analogues (basic analogues or q-extensions) of hypergeometric series because an rFs series can be obtained as the q ---+ 1 limit case of an r¢s series. Since the binomial theorem is at the foundation of most of the summation formulas for hypergeometric series, we then derive a q-analogue of it, called the q-binomial theorem, and use it to derive Heine's q-analogues of Euler's trans­ formation formulas, Jacobi's triple product identity, and summation formulas that are q-analogues of those for hypergeometric series due to Chu and Vander­ monde, Gauss, Kummer, Pfaff and Saalschiitz, and to Karlsson and Minton. We also introduce q-analogues of the exponential, gamma and beta functions, as well as the concept of a q-integral that allows us to give a q-analogue of Euler's integral representation of a hypergeometric function. Many additional formulas and q-analogues are given in the exercises at the end of the chapter.

1.2 Hypergeometric and basic hypergeometric series

In 1812, Gauss presented to the Royal Society of Sciences at Gottingen his famous paper (Gauss [1813]) in which he considered the infinite series

ab a(a + l)b(b + 1) 2 a(a + 1)(a + 2)b(b + 1)(b + 2) 3 ( ) 1+-z+ z + z + ... 1.2.1 l·c 1.2·c(c+l) 1.2·3·c(c+l)(c+2) as a function of a, b, c, z, where it is assumed that c -I=- 0, -1, -2, ... , so that no zero factors appear in the denominators of the terms of the series. He showed that the series converges absolutely for Izi < 1, and for Izl = 1 when Re (c - a - b) > 0, gave its (contiguous) recurrence relations, and derived his famous formula (see (1.2.11) below) for the sum of this series when z = 1 and Re(c-a-b»O.

1 2 Basic hypergeometric series

Although Gauss used the notation F(a, b, c, z) for his series, it is now customary to use F(a, b; c; z) or either of the notations

2Fl (a, b; c; z), for this series (and for its sum when it converges), because these notations separate the numerator parameters a, b from the denominator parameter c and the variable z. In view of Gauss' paper, his series is frequently called Gauss'series. However, since the special case a = 1, b = c yields the geometric senes 1+z+z2 +z3 + ... , Gauss' series is also called the (ordinary) hypergeometric series or the Gauss hypergeometric series. Some important functions which can be expressed by means of Gauss' series are (1 + z)a = F( -a, b; b; -z), 10g(1 + z) = zF(l, 1; 2; -z), sin-1 z = zF(1/2, 1/2; 3/2; z2), (1.2.2) tan-1 z = zF(1/2, 1;3/2; _z2), eZ = lim F(a, b; b; z/a), a-HXl where Izl < 1 in the first four formulas. Also expressible by means of Gauss' series are the classical orthogonal polynomials, such as the Tchebichef polyno­ mials of the first and second kinds Tn(x) = F( -n, n; 1/2; (1 - x)/2), (1.2.3) Un(x) = (n + l)F( -n, n + 2; 3/2; (1 - x)/2), (1.2.4) the Legendre polynomials

Pn(x) = F( -n, n + 1; 1; (1 - x)/2), (1.2.5) the Gegenbauer (ultraspherical) polynomials >. (2'\)n en (x) = -,- F( -n, n + 2'\;'\ + 1/2; (1 - x)/2), (1.2.6) n. and the more general Jacobi polynomials

p~a,(3)(x)= (a+,l)n F(-n,n+a+,B+1; a+1;(1-x)/2), (1.2.7) n. where n = 0,1, ... , and (a)n denotes the shifted factorial defined by r(a+n) (a)o=l, (a)n=a(a+1)···(a+n-1)= r(a) , n=1,2, .... (1.2.8)

Before Gauss, Chu [1303] (see Needham [1959, p. 138], Takacs [1973] and Askey [1975, p. 59]) and Vandermonde [1772] had proved the summation for­ mula (c- b)n F( -n, b; c; 1) = (c)n ' n = 0,1, ... , (1.2.9) 1.2 Hypergeometric series 3 which is now called Vandermonde's formula or the Chu- Vandermonde formula, and Euler [1748] had derived several results for hypergeometric series, including his transformation formula F(a, b; c; z) = (1 - z)c-a-b F(c - a, c - b; c; z), Izl < 1. (1.2.10) Formula (1.2.9) is the terminating case a = -n of the summation formula r(c)r(c - a - b) F(a, b; c; 1) = r(c _ a)r(c _ b)' Re(c - a - b) > 0, (1.2.11) which Gauss proved in his paper. Thirty-three years after Gauss' paper, Heine [1846, 1847, 1878] introduced the series (1 - qa)(l - qb) (1 - qa)(l - qa+ 1)(1 - qb)(l - qb+ I) 2 1+ (1 _ q)(l _ qc) z+ (1- q)(l _ q2)(1 _ qC)(l _ qC+ I) Z +- .. , (1.2.12) where it is assumed that q i- 1, c i- 0, -1, -2, ... and the principal value of each power of q is taken. This series converges absolutely for Izl < 1 when Iql < 1 and it tends (at least termwise) to Gauss' series as q ----> 1, because 1 _ qa lim -- = a. (1.2.13) q--+I 1 - q The series in (1.2.12) is usually called Heine's series or, in view of the base q, the basic hypergeometric series or q-hypergeometric series. Analogous to Gauss' notation, Heine used the notation ¢(a, b, c, q, z) for his series. However, since one would like to also be able to consider the case when q to the power a, b, or c is replaced by zero, it is now customary to define the basic hypergeometric series by

¢(a,b;c;q,z) == 2¢I(a,b;c;q,z) == 2¢1 [a~b;q,z]

= f ~a; q~n~b; q~n zn, (1.2.14) n= 0 q; q n c; q n where n=O, (1.2.15) (a; q)n = { ~i _ a)(l - aq) ... (1 _ aqn-I), n= 1,2, ... , is the q-shifted factorial and it is assumed that c i- q-rn for m = 0,1, .... Some other notations that have been used in the literature for the product (a; q)n are (a)q,n' [a]n, and even (a)n when (1.2.8) is not used and the base is not displayed. Another generalization of Gauss' series is the (generalized) hypergeometric series with r numerator parameters aI, ... ,ar and s denominator parameters bl , ... ,bs defined by

rFs(al' a2, ... , ar; bl , ... , bs; z) == rFs [alb' a2,·· ·b' ar ; z] I, ... , s

(1.2.16) 4 Basic hypergeometric series

Some well-known special cases are the exponential function

eZ = oFo(-;-;z), (1.2.17) the trigonometric functions sin z = z oFI (-; 3/2; -z2/4), (1.2.18) cosz = oFI (-; 1/2; _Z2 /4), the Bessel function

Ja(z) = (z/2t oFI (-; a + 1; _z2 /4)/r(a + 1), (1.2.19) where a dash is used to indicate the absence of either numerator (when r = 0) or denominator (when s = 0) parameters. Some other well-known special cases are the Hermite polynomials

Hn(x) = (2x)n 2Fo( -n/2, (1 - n)/2; -; _x-2), (1.2.20) and the Laguerre polynomials

Lna ()x = (a + ,l)n IFI ( -n; a + 1; x ) . (1.2.21 ) n. Generalizing Heine's series, we shall define an r¢s basic hypergeometric series by

A. ( A. r r'f's al,a2,···,ar ; bI,···, bs;q,z ) -= r'f's [aI,b a2, ...b , a ;q,z] 1, ... , s (1.2.22) = f (al;q)n(a2;q)n···(ar ;q)n [(_l t qG)]I+s-r zn n= 0 (q; q)n(bl ; q)n··· (b s; q)n with (~) = n(n - 1)/2, where q -=I 0 when r > s + 1. In (1.2.16) and (1.2.22) it is assumed that the parameters bl , ... , bs are such that the denominator factors in the terms of the series are never zero. Since (-m)n = (q-m; q)n = 0, n = m + 1, m + 2, ... , (1.2.23) an rFs series terminates if one of its numerator parameters is zero or a negative integer, and an r¢s series terminates if one of its numerator parameters is of the form q-m with m = 0,1,2, ... , and q -=I O. Basic analogues of the classical orthogonal polynomials will be considered in Chapter 7 as well as in the exercises at the ends of the chapters. Unless stated otherwise, when dealing with nonterminating basic hyper­ geometric series we shall assume that Iql < 1 and that the parameters and variables are such that the series converges absolutely. Note that if Iql > 1, then we can perform an inversion with respect to the base by setting p = q-I and using the identity

(1.2.24) to convert the series (1.2.22) to a similar series in base p with Ipi < 1 (see Ex. 1.4(i)). The inverted series will have a finite radius of convergence if the original series does. 1.2 Hypergeometric series 5

Observe that if we denote the terms of the series (1.2.16) and (1.2.22) which contain zn by Un and Vn , respectively, then

U n + 1 (al + n)(a2 + n) ... (ar + n) --= z (1.2.25) Un (l+n)(b1 +n)···(bs+n) is a rational function of n, and

(1.2.26) is a rational function of qn. Conversely, if L:: 0 Un and L:: 0 Vn are power series with Uo = Vo = 1 such that un+ dUn is a rational function of nand vn+ l/Vn is a rational function of qn, then these series are of the forms (1.2.16) and (1.2.22), respectively. By the ratio test, the rFs series converges absolutely for all z if r ::; s, and for Izl < 1 if r = s + 1. By an extension of the ratio test (Bromwich [1959, p. 241]), it converges absolutely for Izl = 1 if r = s + 1 and Re [b1 + ... + bs - (a 1 + ... + ar )] > o. If r > s + 1 and z i=- 0 or r = s + 1 and Iz I > 1, then this series diverges, unless it terminates. If 0 < Iql < 1, the r¢s series converges absolutely for all z if r ::; sand for Izl < 1 if r = s + 1. This series also converges absolutely if Iql > 1 and Izl < Ib1b2·· ·bsql/lala2·· ·arl· It diverges for z i=- 0 if 0 < Iql < 1 and r > s + 1, and if Iql > 1 and Izl > Ib1b2·· ·bsql/lala2· ··arl, unless it termi­ nates. As is customary, the rFs and r¢s notations are also used for the sums of these series inside the circle of convergence and for their analytic contin­ uations (called hypergeometric functions and basic hypergeometric functions, respectively) outside the circle of convergence. Observe that the series (1.2.22) has the property that if we replace z by z/ar and let ar ---+ 00, then the resulting series is again of the form (1.2.22) with r replaced by r -1. Because this is not the case for the r¢s series defined (n)]l+s-r without the factors [ (_I)nq 2 in the books of Bailey [1935] and Slater [1966] and we wish to be able to handle such limit cases, we have chosen to use the series defined in (1.2.22). There is no loss in generality since the Bailey and Slater series can be obtained from the r = s + 1 case of (1.2.22) by choosing s sufficiently large and setting some of the parameters equal to zero. An r+ 1 Fr series is called k-balanced if b1 + b2 + ... + br = k + al + a2 + ... + ar+ 1 and z = 1; a I-balanced series is called balanced (or Saalschutzian). Analogously, an r+ 1 ¢r series is called k- balanced if b1 b2 ... br = qk al a2 ... ar+ 1 and z = q, and a I-balanced series is called balanced (or Saalschutzian). We will first encounter balanced series in §1.7, where we derive a summation formula for such a series. For negative subscripts, the shifted factorial and the q-shifted factorials are defined by

1 1 ( a ) -n = -:------:--:------:---:------:- (1.2.27) (a-l)(a-2)···(a-n) (a-n)n 6 Basic hypergeometric series

1 1 ( -qja)nqG) (a;q)-n= (1-aq-l)(1-aq-2) ... (1-aq-n) (aq-n;q)n (qja; q)n ' (1.2.28) where n = 0,1, .... We also define

00 (a; q)oo = II (1- aqk) (1.2.29) k= 0 for Iql < 1. Since the infinite product in (1.2.29) diverges when a -=I- 0 and Iql 2:: 1, whenever (a; q)oo appears in a formula, we shall assume that Iql < 1. The following easily verified identities will be frequently used in this book:

() (a; q)oo a; q n = ( n ) , (1.2.30) aq ; q 00

(a- 1 ql-n; q)n = (a; q)n( _a-1 )nq-G), (1.2.31 )

(a;q)n-k = (a;q)n (_qa-l)kq(~)-nk, (1.2.32) (a- 1ql-n; q)k

(a; q)n+ k = (a; q)n(aqn; q)k, (1.2.33)

(1.2.34)

k) (a;q)n (aq ;q n-k = -( -)-, (1.2.35) a;q k 2k.) _ (a; q)n(aqn; q)k (aq ,q n-k - () , (1.2.36) a; q 2k

-no ) = (q; q)n (_1)k (q ,qk () q,(~)-nk (1.2.37) q; q n-k

-no ) _ (a; q)k(qa-1 ; q)n -nk (aq ,q k - (-1 l-k.) q , (1.2.38) a q ,q n

(a; q)zn = (a; q2)n(aq; q2)n, (1.2.39)

(a2; q2)n = (a; q)n( -a; q)n, (1.2.40) where nand k are integers. A more complete list of useful identities is given in Appendix I at the end of the book. Since products of q-shifted factorials occur so often, to simplify them we shall frequently use the more compact notations (ai, a2,···, am; q)n = (al; q)n(a2; q)n··· (am; q)n, (1.2.41 ) (ai, a2, ... , am; q)oo = (al; q)oo (a2; q)oo ... (am; q)oo. (1.2.42) 1.2 Hypergeometric series 7

The ratio (1 - qa)/(1 - q) considered in (1.2.13) is called a q-number (or basic number) and it is denoted by 1 _ qa [a]q = -1-' q -I- 1. (1.2.43) -q It is also called a q-analogue, q-deformation, q-extension, or a q-generalization of the complex number a. In terms of q-numbers the q-number factorial [n]q! is defined for a nonnegative integer n by n [n]q! = II [k]q, (1.2.44) k= I and the corresponding q-number shifted factorial is defined by n-I [a]q;n = II [a + k]q. (1.2.45) k=O Clearly, 11m· []'n q. -n., , lim [a]q = a, (1.2.46) q-tl q-tl and lim [a]q;n (a)n. (1.2.4 7) q-tl = Corresponding to (1.2.41) we can use the compact notation [aI, a2,···, am]q;n = [adq;n[a2]q;n'" [am]q;n. (1.2.48) Since f [al,a2, ... ,ar]q;n [(_I)nqG)]I+s-rzn n= 0 [n]q![b l , ... , bs]q;n

-_ r'f/sA.. (a1q , qa2 , ... , qa r ,• qb , , ... , qb s ,• q, z (1 - q )I+s-r) , (1.2.49) anyone working with q-numbers and the q-number hypergeometric series on the left-hand side of (1.2.49) can use the formulas for reps series in this book that have no zero parameters by replacing the parameters by q"lj powers and applying (1.2.49). As in Frenkel and Turaev [1995] one can define a trigonometric number [a; u] by [a'u] = sin(7rua) (1.2.50) , sin( 7ru) for noninteger values of u and view [a; u] as a trigonometric deformation of a since lima-to [a; u] = a. The corresponding rts trigonometric hypergeometric series can be defined by rts(al, a2, ... , ar; bl , ... , bs; u, z)

= f [al,a2, ... ,ar;u]n [(_lte11"ia G)]I+s-rzn, (1.2.51 ) n= 0 [n; u]![bl , ... , bs; u]n 8 Basic hypergeometric series where n n-I [n; u]! = II [k; u], [a; u]n = II [a + k; u], (1.2.52) k= I k= 0 and (1.2.53) From eniaa _ e-niaa qa/2 _ q-a/2 _ 1 - qa (l-a)/2 [a· u] = -:-----,---- (1.2.54) , e7ria _ e-7ricr ql/2 _ q-I/2 - 1 - q q , where q = e2nia , it follows that . ] _ (qa;q)n n(l-a)/2-n(n-I)/4 [a, u n - (1 _ q)n q , (1.2.55) and hence rts(al' a2,···, ar; bl ,···, bs; u, z) -_ r'l'srI-. (a1q ,qa2 , ... , qa r •,q b 1 , ... , q bs •,q, cz ) (1.2.56) with (1.2.57) which shows that the rts series is equivalent to the reps series in (1.2.49). Elliptic numbers [a; u, T], which are a one-parameter generalization (de­ formation) of trigonometric numbers, are considered in §1.6, and the corre­ sponding elliptic (and theta) hypergeometric series and their summation and transformation formulas are considered in Chapter 11. We close this section with two identities involving ordinary binomial coef­ ficients, which are particularly useful in handling some powers of q that arise in the derivations of many formulas containing q-series:

(1.2.58)

(1.2.59)

1.3 The q-binomial theorem

One of the most important summation formulas for hypergeometric series is given by the binomial theorem:

2FI (a, C; C; z) = IFo(a;-; z) = f (a),n zn = (1- z)-a, (1.3.1) n=O n. where Izl < 1. We shall show that this formula has the following q-analogue

rI-. (._. ) _ ~ (a;q)n n _ (az;q)oo 1'l'0 a, ,q, z - ~ ( ) z - ( ) , Izl < 1, Iql < 1, (1.3.2) n= 0 q; q n Z; q 00 1.3 The q-binomial theorem 9 which was derived by Cauchy [1843]' Heine [1847] and by other mathemati­ cians. See Askey [1980a], which also cites the books by Rothe [1811] and Schweins [1820]' and the remark on p. 491 of Andrews, Askey, and Roy [1999] concerning the terminating form of the q-binomial theorem in Rothe [1811]. Heine's proof of (1.3.2), which can also be found in the books Heine [1878], Bailey [1935, p. 66] and Slater [1966, p. 92]' is better understood if one first follows Askey's [1980a] approach of evaluating the sum of the binomial series in (1.3.1), and then carries out the analogous steps for the series in (1.3.2). Let us set .. ( ) = ~ (a)n n Ja Z ~ ,z. (1.3.3) n=O n.

Since this series is uniformly convergent in Izl ::; E when 0 < E < 1, we may differentiate it termwise to get

f~(z) = f n(a/n zn-l n= 1 n.

= (a)~+ 1 zn = afa+ 1 (z). (1.3.4) f n. n= 0 Also .. ( ) _.. () _ ~ (a)n - (a + l)n n Ja Z Ja+ 1 Z - ~ , Z n= 1 n. = f (a + l?n-l [a _ (a + n)] zn = _ f n(a + ~)n-l zn n= 1 n. n= 1 n. _ ~(a+1)nn+l_ .. () ~ Z J a+ 1 Z . (1.3.5) -- n.,z -- n= 0

Eliminating fa+ 1 (z) from (1.3.4) and (1.3.5), we obtain the first order differ­ ential equation (1.3.6) subject to the initial condition fa(O) = 1, which follows from the definition (1.3.3) of fa(z). Solving (1.3.6) under this condition immediately gives that fa(z) = (1 - z)-a for Izl < 1. Analogously, let us now set

h ( ) - ~ (a; q)n n aZ-~( )z' Izl < 1, Iql < 1. (1.3.7) n= 0 q; q n

Clearly, hqa (z) ----+ fa(z) as q ----+ 1. Since haq(z) is a q-analogue of fa+ 1 (z), we first compute the difference

h ( ) _ h () = ~ (a; q)n - (aq; q)n n a Z aq Z ~ () Z n= 1 q; q n 10 Basic hypergeometric series

= f (aq; q)n-l [1 - a - (1 - aqn)] zn n= 1 (q; q)n = -a f (1 - qn)(aq; q)n-l zn n= 1 (q; q)n __ ~ (aq; q)n-l n - _ h () - a ~ ( ) z - az aq Z , (1.3.8) n= 1 q; q n-l giving an analogue of (1.3.5). Observing that

!'(z) = lim J(z) - J(qz) (1.3.9) q-+l (1 - q)z for a differentiable function J, we next compute the difference ha(z) - ha(qz) = f (a; q)n (zn _ qnzn) n= 1 (q; q)n

= f (a;q)n zn = f (a;q)n+l zn+l n=l (q;q)n-l n=O (q;q)n = (1 - a)zhaq(z). (1.3.10) Eliminating haq(z) from (1.3.8) and (1.3.10) gives 1- az ha(z) = --ha(qz). (1.3.11) 1-z Iterating this relation n - 1 times and then letting n ----- 00 we obtain h () = (az;q)nh (n) aZ (z;q ) n aqz

= (az; q)oo ha(O) = (az; q)oo, (1.3.12) (z; q)oo (z; q)oo since qn _____ 0 as n _____ 00 and ha(O) = 1 by (1.3.7), which completes the proof of (1.3.2). One consequence of (1.3.2) is the product formula l¢o(a;-;q,z) l¢o(b;-;q,az) = l¢o(ab;-;q,z), (1.3.13) which is a q-analogue of (1 - z)-a(1- z)-b = (1 - z)-a-b. In the special case a = q-n, n = 0,1,2, ... , (1.3.2) gives l¢O(q-n;_;q,z) = (zq-n;q)n = (-ztq-n(n+l)/2(q/Z;q)n, (1.3.14) where, by analytic continuation, z can be any complex number. From now on, unless stated othewise, whenever q-j , q-k, q-m, q-n appear as numerator parameters in basic series it will be assumed that j, k, m, n, respectively, are nonnegative integers. If we set a = 0 in (1.3.2), we get

00 zn 1 l¢O(O;-;q,z) = L -( -. -) = ( . ) ,Izl < 1, (1.3.15) n= 0 q, q n Z, q 00 1.3 The q-binomial theorem 11

which is a q-analogue of the exponential function e Z • Another q-analogue of eZ can be obtained from (1.3.2) by replacing z by -z/a and then letting a ----700 to get 00 qn(n-l)/2 n ocPo(-; -; q, -z) = L (.) z = (-z; q)oo. (1.3.16) n=O q,q n

Observe that if we denote the q-exponential functions in (1.3.15) and (1.3.16) by eq(z) and Eq(z), respectively, then eq(z)Eq( -z) = 1, eq-" (z) = Eq( -qz) by (1.2.24), and

lim eq(z(l - q)) = lim Eq(z(l - q)) = eZ • (1.3.17) q---+l- q---+l- In deriving q-analogues of various formulas we shall sometimes use the observation that

Thus lim (qa z ; q)oo = (1 _ z)-a, Izl < 1, a real. (1.3.19) q---+l- (z; q)oo By analytic continuation this holds for z in the complex plane cut along the positive real axis from 1 to 00, with (1 - z)-a positive when z is real and less than 1. Let Do and \7 be the forward and backward q-difference operators, respec­ tively, defined by

Dof(z) = f(qz) - f(z), \7 f(z) = f(q-l z) - f(z), (1.3.20) where we take 0 < q < 1, without any loss of generality. Then the unique analytic solutions of

\7g(z) - () Do~~) = f(z), f( o) = 1 and \7z - g z , g(O) = 1, (1.3.21 ) are f(z) = eq(z(l - q)) and g(z) = Eq(z(l - q)). (1.3.22)

The symmetric q-difference operator 6q is defined by

(1.3.23)

If we seek an analytic solution of the initial-value problem

6q!(Z) = f(z), f(O) = 1, (1.3.24) uqz in the form 2:: 0 anzn, then we find that 1 - q nl2 (1.3.25) an+ 1 = 1 -qn+ 1 q an, aO = 1, 12 Basic hypergeometric series n = 0,1,2,... Hence, an = (1 - q)nq(n 2-n)/4 /(q; q)n, and we have a third q-exponential function

00 (1 _ )n (n 2-n)/4 00 1 eXPq(z) = L q( .q) zn = L -[ -. -]' zn (1.3.26) n=O q,q n n=O n,u. with q = e21ria . This q-exponential function has the properties

(1.3.27) and it is an entire function of z of order zero with an infinite product represen­ tation in terms of its zeros. See Nelson and Gartley [1994]' and Atakishiyev and Suslov [1992a]. The multi-sheet Riemann surface associated with the q­ logarithm inverse function z = lnq(w) of W = eXPq(z) is considered in Nelson and Gartley [1996]. Ismail and Zhang [1994] found an extension of eXPq(z) in the form

00 m 2 /4 f(z) = L -q-- (aq1 -;=+z,aq1 -;=-Z;q) bm, (1.3.28) m=O (q;q)m m which has the property

8f(z) = f( ) 8x(z) z , 8f(z) = f(z + 1/2) - f(z - 1/2), (1.3.29) where (1.3.30) with C = -abql/4/(1 - q) is the so-called q-quadmtic lattice, and a and b are arbitrary complex parameters such that labl < 1. In the particular case qZ = e-if}, 0 ~ () ~ Jr, X = cos (), the q-exponential function in (1.3.28) becomes the function

00 m 2 /4 "q (1-=.f) 1-= Of}) Eq(x;a,b) = ~ -(-.-)- q-2-ae' ,q-2-ae-';q bm. (1.3.31 ) m=O q,q m m Ismail and Zhang showed that

lim Eq(x; a, b(l - q)) = exp[(l + a2 - 2ax)b], (1.3.32) q--->l and that Eq(x; a, b) is an entire function of x when labl < 1. From (1.3.32) they observed that Eq(x; -i, -it/2) is a q-analogue of ext. It is now standard to use the notation in Suslov [2003] for the slightly modified q-exponential function

2 2) 00 m 2 /4 (a . q 00 " q (1-= of) 1-= of} ) Eq(X; a) = ; 2 ~ --(-iar -iq-2-e', -iq-2-e-' ; q , (qa;q)oom=O(q;q)m m (1.3.33) which, because of the normalizing factor that he introduced, has the nice prop­ erty that Eq(O;a) = 1 (see Suslov [2003, p.17]). 1.4 Heine's transformation formulas for 2

1.4 Heine's transformation formulas for 2

Heine [1847, 1878] showed that (b, az; q)oo ( I ) 2

(cqn; q)oo = f (clb; q)m (bqn)m. (bqn; q)oo m= 0 (q; q)m Hence, for Izl < 1 and Ibl < 1, '" ( b.. ) _ (b; q)oo ~ (a; q)n(cqn; q)oo n 2 '/'1 a, ,c, q, z - ( ) ~ ( ) ( ) z c; q 00 n= 0 q; q n bqn; q 00 = (b; q)oo f (a; q)n zn f (clb; q)m (bqn)m (c;q)oo n=O (q;q)n m=O (q;q)m = (b; q)oo f (clb; q)m bm f (a; q)n (zqmt (c; q)oo m= 0 (q; q)m n= 0 (q; q)n = (b; q)oo f (clb; q)m bm (azqm; q)oo (c; q)oo m= 0 (q; q)m (zqm; q)oo (b, az; q)oo ( I ) = ( ) 2

1.5 Heine's q-analogue of Gauss' summation formula

In order to derive Heine's [1847] q-analogue of Gauss' summation formula (1.2.11) it suffices to set z = e/ab in (1.4.1), assume that Ibl < 1, le/abl < 1, and observe that the series on the right side of

(b, e/b; q)oo ( / ) 2¢1 (a,b;e;q,e /)ab = ( / b ) I¢O e ab;-;q,b e, e a ; q 00 can be summed by (1.3.2) to give

( /) (e/a,e/b;q)oo 2¢1 a,b;c;q,e ab = ( / b ) . (1.5.1) c, e a ; q 00

By analytic continuation, we may drop the assumption that Ibl < 1 and require only that le/abl < 1 for (1.5.1) to be valid. For the terminating case when a = q-n, (1.5.1) reduces to

A. ( -n b nib) (e/b;q)n 2 '/'1 q ,; c; q, eq = ( ) . (1.5.2) c;q n

By inversion or by changing the order of summation it follows from (1.5.2) that

A. ( -n b ) (e/b;q)nbn 2 '/'1 q ,; e; q, q = () . (1.5.3) e;q n

Both (1.5.2) and (1.5.3) are q-analogues of Vandermonde's formula (1.2.9). These formulas can be used to derive other important formulas such as, for example, Jackson's [1910a] transformation formula

A. ( b.. ) = (az; q)oo ~ (a, c/b; q)k (-b )k (~) 2 '/'1 a, ,e, q, z ( ) ~ ( ) z q z; q 00 k= 0 q, e, az; q k (az; q)oo (/ ) = ( ) 2¢2 a,e b;c,az;q,bz . (1.5.4) z;q 00

This formula is a q-analogue of the Pfaff-Kummer transformation formula

2FI (a, b; e; z) = (1 - z)-a 2FI (a, e - b; e; z/(z - 1)). (1.5.5)

To prove (1.5.4), we use (1.5.2) to write 1.6 Jacobi's triple product identity 15 and hence 2¢1 (a, b; c; q, z) = f (a; q)k zk t (q-k, cjb; q)n (bqk) n k= 0 (q; qh n= 0 (q, c; q)n = f f (a; qh(cjb; q)n zk( -btqG) n= 0 k= n (q; q)k-n(q, c; q)n

= ff (a;q)k+n(cjb;q)n(_bz)nZkq(~) n= 0 k= 0 (q; qh(q, c; q)n

= f (a, cjb; q)n (-bz)nqG) f (aqn; q)k zk n=O (q,c;q)n k=O (q;qh = (az; q)oo f (a, cjb; q)n (-bztqG), (z;q)oo n=O (q,c,az;q)n by (1.3.2). Also see Andrews [1973]. If a = q-n, then the series on the right side of (1.5.4) can be reversed (by replacing k by n - k) to yield Sears' [1951c] transformation formula

2¢1 (q-n, b; c; q, z)

= (~jb; r)n (bZ)n 3¢2(q-n, qjz, c-1ql-n; bc-1ql-n, 0; q, q). (1.5.6) c;q n q

1.6 Jacobi's triple product identity, theta functions, and elliptic numbers Jacobi's [1829] well-known triple product identity (see Andrews [1971])

00 (zq!,q!jz,q;q)oo= L (_ltqn2/2 zn, zyfO, (1.6.1) n= -00 can be easily derived by using Heine's summation formula (1.5.1). First, set c = bzq! in (1.5.1) and then let b ---40 and a ---4 00 to obtain

00 (_1)nqn2/2 1 L ( ) zn = (Zq2; q)oo. (1.6.2) n=O q; q n

Similarly, setting c = zq in (1.5.1) and letting a ---4 00 and b ---400 we get

00 2 qn zn 1 (1.6.3) ~ (q,zq;q)n (zq; q)oo . Now use (1.6.2) to find that (zq!, q! jz; q)oo 00 00 (_l)m+ nqCm2+ n2)/2 = L L zm-n m= 0 n= 0 (q; q)m(q; q)n 16 Basic hypergeometric series

00 (_l)n n2/2 00 k2 _ '" q zn '" q qnk - ~ (q;q)n ~ (q,qn+ l;qh

00 (-l)n n2/2 00 k2 + '" q -n '" q nk (1.6.4) ~ (q;q)n Z ~(q,qn+1;qhq·

Formula (1.6.1) then follows from (1.6.3) by observing that

1

An important application of (1.6.1) is that it can be used to express the theta functions (Whittaker and Watson [1965, Chapter 21])

00 19 1(x, q) = 2 L( _1)nq(n+ 1/2)2 sin(2n + l)x, (1.6.5) n= 0 00 192 (x, q) = 2 L q(n+ 1/2)2 cos(2n + l)x, (1.6.6) n= 0 00

193 (x,q) = 1 + 2 L qn2 cos2nx, (1.6.7) n= 1 00 194(x,q) = 1 + 2 L(-ltqn2 cos2nx (1.6.8) n= 1 in terms of infinite products. Just replace q by q2 in (1.6.1) and then set z equal to qe2ix , _qe2ix , _e2ix , e2ix , respectively, to obtain

00 19 1(x, q) = 2q1/4 sinx II (1 - q2n)(1 - 2ln cos 2x + q4n), (1.6.9) n= 1 00 192 (x, q) = 2q1/4 cos x II (1- q2n)(1 + 2q2n cos2x + q4n), (1.6.10) n= 1 00 193 (X, q) = II (1- q2n)(1 + 2q2n-1 cos2x + q4n-2), (1.6.11) n= 1 and 00 194 (X, q) = II (1- q2n)(1_ 2q2n-1 cos2x + q4n-2). (1.6.12) n= 1

It is common to write 19 k (x) for 19 k (x, q), k = 1, ... ,4. Since, from (1.6.9) and (1.6.10),

(1.6.13) one can think of the theta functions 19 1 (x, q) and 192 (x, q) as one-parameter deformations (generalizations) of the trigonometric functions sin x and cos x, 1. 7 q-Saalschutz formula 17 respectively. This led Frenkel and Turaev [1995] to define an elliptic number [a; CT, T] by (1.6.14) where a is a complex number and the modular parameters CT and T are fixed complex numbers such that 1m (T) > ° and CT -I- m + nT for integer val­ ues of m and n, so that the denominator f)l (1[CT, e7riT) in (1.6.14) is never zero. Then, from (1.6.9) it is clear that [a; CT, T] is well-defined, [-a; CT, T] = -[a;CT,TJ, [1;CT,T] = 1, and . sin( 1[CTa) hm [a; CT, T] = . ( ) = [a;CT]. (1.6.15) III T-tOO sIn 1[CT Hence, the elliptic number [a; CT, T] is a one-parameter deformation of the trigonometric number [a; CT] and a two-parameter deformation of the number a. Notice that [a; CT, T] is called an "elliptic number" even though it is not an elliptic (doubly periodic and meromorphic) function of a. However, [a;CT,T] is a quotient of f)l functions and, as is well-known (see Whittaker and Wat­ son [1965, §21.5]), any (doubly periodic meromorphic) elliptic function can be written as a constant multiple of a quotient of products of f)l functions. The corresponding elliptic hypergeometric series are considered in Chapter 11.

1. 7 A q-analogue of Saalschiitz's summation formula

Pfaff [1797] discovered the summation formula (c - a)n(c - b)n 3 F2(a,b,-n;c,l+a+b-c-n;I)= (c)n(c-a-b)n' n=O,I, ... , (1.7.1) which sums a terminating balanced 3F2(1) series with argument 1. It was rediscovered by Saalschutz [1890] and is usually called Saalschiitz formula or the Pfaff-Saalschiitz formula; see Askey [1975]. To derive a q-analogue of (1.7.1), observe that since, by (1.3.2), (abzjc; q)oo = f (abjc; q)k zk (z;q)oo k=O (q;q)k the right side of (1.4.3) equals

f f (abjc; q)k(cja, cjb; q)m (ab)m zk+ m, k= 0 m= 0 (q; qh(q, c; q)m C and hence, equating the coefficients of zn on both sides of (1.4.3) we get t (q-n, cja,cjb; q)j qj = (a,b;q)n . . (q,c,cql-njab;q)j (c,abjc;q)n J= 0 Replacing a, b by cj a, cjb, respectively, this gives the following sum of a termi­ nating balanced 3 (P2 series -n -I I-n ) (cja, cjb; q)n 3(P2 ( a,b,q ;c,abc q ;q,q = ( j b ) , n=O,I, ... , (1. 7.2) c, c a ; q n 18 Basic hypergeometric series which was first derived by Jackson [1910a]. It is easy to see that (1.7.1) follows from (1.7.2) by replacing a, b, c in (1. 7.2) by qa, qb, qC, respectively, and letting q -+ 1. Note that letting a -+ 00 in (1.7.2) gives (1.5.2), while letting a -+ 0 gives (1.5.3).

1.8 The Bailey-Daum summation formula Bailey [1941] and Daum [1942] independently discovered the summation for­ mula ,/, ( b' jb' _ jb) = (-q; q)oc,(aq, aq2 jb2; q2)00 2'f'1 a, ,aq ,q, q ( jb _ jb' ) , (1.8.1) aq , q ,q 00 which is a q-analogue of Kummer's formula r(1 + a - b)r(1 + ta) 2FI(a,b;l+a-b;-I)= ( ) ( I ). (1.8.2) r 1 + a r 1 + 'ia - b Formula (1.8.1) can be easily obtained from (1.4.1) by using the identity (1.2.40) and a limiting form of (1.2.39), namely, (a; q)oo = (a, aq; q2)00, to see that 2¢1 (a, b; aqjb; q, -qjb) (a, -q; q)oo ,/, (jb jb ) ( aq,q,qoo j b _ jb') 2'f'1 q ,-q ;-q;q,a (a,-q;q)oo ~ (q2jb2;q2)n n = (aqjb, -qjb; q)oo ~ (q2; q2)n a (a, -q; q)oo (aq2 jb2; q2)00 by (1.3.2) (aqjb, -qjb; q)oo (a; q2)00 (-q; q)oo (aq, aq2 jb2; q2 )00 (aqjb, -qjb; q)oo

1.9 q-analogues of the Karlsson-Minton summation formulas

Minton [1970] showed that if a is a negative integer and ml, m2, ... , mr are nonnegative integers such that -a 2: ml + ... + mT) then

F [a, b, bl + ml , ... , br + mr . 1] r+ 2 r+ I b + 1, bl , ... , br '

r(b + 1)r(1 - a) (b l - b)ml ... (br - b)m r (1.9.1) r(1 + b - a) (bdml ... (br)m r where, as usual, it is assumed that none of the factors in the denominators of the terms of the series is zero. Karlsson [1971] showed that (1.9.1) also holds when a is not a negative integer provided that the series converges, i.e., if Re( -a) > ml + ... + mr - 1, and he deduced from (1.9.1) that

F [a, l + ml , ... , br + mr . r+1 r b b b ,-1] - 0, Re (-a) > ml + ... + mT) (1.9.2) 1, ... , r 1.9 q-Karlsson-Minton formulas 19

(1.9.3)

These formulas are particularly useful for evaluating sums that appear as so­ lutions to some problems in theoretical physics such as the Racah coefficients. They were also used by Gasper [1981b] to prove the orthogonality on (0,27f) of certain functions that arose in Greiner's [1980] work on spherical harmonics on the Heisenberg group. Here we shall present Gasper's [1981a] derivation of q-analogues of the above formulas. Some of the formulas derived below will be used in Chapter 7 to prove the orthogonality relation for the continuous q-ultraspherical polynomials. Observe that if m and n are nonnegative integers with m ~ n, then

'" (-n -m b ) (brqm; q)n -mn 2'1-'1 q , q ; r; q, q = (b') q r,qn by (1.5.3), and hence '" [al, ... ,ar,brqm. ] r+ 1 'l-'r b b b' q, Z I,···, r-I, r ~ (al, ... ,ar;q)n n~ (q-n,q-m;q)k mn+k = ~ Z ~ q (q,bl , ... ,br-I;q)n (q,br;q)k n= 0 k= 0 = ff (al, ... ,ar;q)n(q-m;qh zn(_l)kqmn+k-nk+(~) (bl, ... ,br-I;q)n(q;q)n-k(q,br;q)k n= 0 k= 0 ~ (q-m,al, ... ,ar;qh( m)k _(k) = ~ -zq q 2 k=O (q,bl, ... ,br;q)k alqk, ... ,arqk m-k] X r¢r-I [ b k b k;q,zq , 1. Iq , ... , r-Iq Izl < (1.9.4) This expansion formula is a q-analogue of a formula in Fox [1927, (1.11)] and independently derived by Minton [1970, (4)]. When r = 2, formulas (1.9.4), (1.5.1) and (1.5.3) yield ¢ [a,b,b1qm. -I I-m] _(q,bq/a;q)oo ¢ ( -m b'b' ) 3 2 bq,bl ,q,a q - (bq,q/a;q)oo 2 1 q "I,q,q = (q,bq/a;q)oo(bl/b;q)mbm ( ) ( ) ( ) , 1.9.5 bq, q / a; q 00 bl ; q m provided that la-1ql-ml < 1. By induction it follows from (1.9.4) and (1.9.5) that if ml,"" mr are nonnegative integers and la-1ql-(m1 + ... + mr) I < 1, then

'" [a,b,blqml, ... ,brqmr. -I 1-(m1+ ... +mr )] r+ 2 'l-'r+ 1 b b b ' q, a q q, 1,···, r = (q, bq/a; q)oo (b l /b; q)ml ... (br/b; q)mr bm1+ ... + mr (1.9.6) (bq, q/a; q)oo (b l ; q)ml ... (br; q)mr 20 Basic hypergeometric series which is a q-analogue of (1.9.1). Formula (1.9.1) can be derived from (1.9.6) by replacing a, b, bl , ... , br by qa, qb, qb1, ... , qbr , respectively, and letting q ----7 1. Setting br = b, mr = 1 and then replacing r by r + 1 in (1.9.6) gives

'" [a,blqml, ... ,brqmr. -I -cm1+ ... +mr)] _ 0 I -I -Cm1+···+mr)1 1 r+ I '/'r b b ' q, a q - , a q < , I, ... , r (1.9.7) while letting b ----700 in the case a = q-Cml+ ... + m r ) of (1.9.6) gives -Cm1+ ... +mr) b qml b qmr ] '" [q ' I , ... , r 1 r+I'/'r b b ;q, I, ... , r (_I)ml+···+mr(. ) = ( ) q(q m)+"'+mrq-cml+"'+mrHml+"'+mr+I)/2, (1.9.8) bl ; q ml ... br ; q mr which are q-analogues of (1.9.2) and (1.9.3). Another q-analogue of (1.9.3) can be derived by letting b ----7 0 in (1.9.6) to obtain

(1.9.9) when la-Iql-Cm1+···+mr)1 < 1. In addition, if a = q-n and n is a nonnegative integer then we can reverse the order of summation of the series in (1.9.6), (1.9.7) and (1.9.9) to obtain

n 2 ml + ... + mn (1.9.10)

-n b ml b mr ] '" [ q , I q , ... , rq. - 0 r+ I '/'r b b ' q, q - , (1.9.11) I, ... , r and the following generalization of (1.9.8)

'" [q-n,blqml, ... ,brqmr. (_I)n(q;q)nq-nCn+I)/2 1] _ (1.9.12) r+ I '/'r b b ' q, - (b) (b) , I,···, r I;qml'" r;qmr where n 2 ml + ... + mn which also follows by letting b ----7 00 in (1.9.10). Note that the b ----7 0 limit case of (1.9.10) is (1.9.11) when n > ml + ... + mn and it is the a = q-Cml+ ... + m r ) special case of (1.9.9) when n = ml + ... +mr .

1.10 The q-gamma and q-beta functions

The q-gamma function

r ( ) = (q; q)oo (1 - )I-x 0 < < 1 q x (x.) q , q , (1.10.1) q , q 00 1.10 The q-gamma and q- beta functions 21 was introduced by Thomae [1869] and later by Jackson [1904e]. Heine [1847] gave an equivalent definition, but without the factor (l-q)l-x. When x = n+ 1 with n a nonnegative integer, this definition reduces to (1.10.2) which clearly approaches n! as q ---+ 1-. Hence f q (n + 1) tends to f( n + 1) = n! as q ---+ 1 -. The definition of f q (x) can be extended to Iq I < 1 by using the principal values of qX and (1 - q)l-x in (1.10.1). To show that lim fq(x) = f(x) (1.10.3) q---+I - we shall give a simple, formal proof due to Gosper; see Andrews [1986]. From (1.10.1),

Hence lim fq(x + 1) = II00 _n_ (+I)X_n_ q---+I- n+x n n= I

= xr(x) = r(x + 1) by Euler's product formula (see Whittaker and Watson [1965, §12.11]) and the well-known functional equation for the gamma function r(x + 1) = xr(x), r(1) = 1. (1.10.4) For a rigorous justification of the above steps see Koornwinder [1990]. From (1.10.1) it is easily seen that, analogous to (1.10.4), fq(x) satisfies the func­ tional equation 1- qX f(x + 1) = -1-f (x), f(l) = 1. (1.10.5) -q Askey [1978] derived analogues of many of the well-known properties of the gamma function, including its log-convexity (see the exercises at the end of this chapter), which show that (1.10.1) is a natural q-analogue of f(x). It is obvious from (1.10.1) that fq(x) has poles at x = 0, -1, -2, .... The residue at x = -n is (1 _q)n+1 x+n lim (x + n) f (x) = --;------:-----:----'---:--'----,------,----:-:- lim x---+-n q (1 - q-n)(1 - ql-n) ... (1 - q-I) x---+-n 1 _ qX+ n (1 _ q)n+ I (1.10.6) 22 Basic hypergeometric series

The q-gamma function has no zeros, so its reciprocal is an entire function with zeros at x = 0, -1, -2, .... Since

1 00 1- qn+ x r (x) = (1- q)x-l II 1 _ qn+ 1 ' (1.10.7) q n= 0 the function 1/r q (x) has zeros at x = -n ± 27rik / log q, where k and n are nonnegative integers. A q-analogue of Legendre's duplication formula

r(2x)r (~) = 22x - 1r(x)r (x +~) (1.10.8) can be easily derived by observing that

rq2(X)rq2(X+t) (q,q2;q2)00 (1_ q2)1-2x r q2 (t) (q2x , q2x+ 1 ; q2 ) 00

= (q; q)oo (1 _ q2)1-2x = (1 + q)I-2x r (2x) (q2x;q)00 q and hence

(1.10.9)

Similarly, it can be shown that the Gauss multiplication formula

r(nx)(27r)(n-l)/2 = nnx-!r(x)r (x + ~) ... r (x + n: 1 ) (1.10.10) has a q-analogue of the form

rq(nx)rr (~) rr (~) ... rr (n: 1)

= (1 + q + ... + qn-l tx-1rr(x)rr (x + ~) ... rr (x + n: 1) (1.10.11) with r = qn; see Jackson [1904e, 1905d]. The q-gamma function for q > 1 is considered in Exercise 1.23. For other interesting properties of the q-gamma function see Askey [1978] and Moak [1980a,b] and Ismail, Lorch and Muldoon [1986]. Since the beta function is defined by

B( ) = r(x)r(y) X,y r(x+y)' (1.10.12) it is natural to define the q-beta function by

(1.10.13) 1.11 The q-integral 23 which tends to B(x, y) as q ----+ 1-. By (1.10.1) and (1.3.2), B (x, y) = (1 _ q) (q, qX+ Y; q)oo q (qX,qY;q)oo = (1 - q) (q; q)oo f (qY; q)n qnx (qY;q)oo n=O (q;q)n

00 ( n+ 1 ) = (1 - q) L (q n+ Y: q) 00 qnx, Re x, Re y > O. (1.10.14) n=O q ,qoo This series expansion will be used in the next section to derive a q-integral representation for Bq(x, y).

1.11 The q-integral Thomae [1869, 1870] and Jackson [1910c, 1951] introduced the q-integral

11 f(t) dqt = (1 - q) ~ f(qn)qn (1.11.1) and Jackson gave the more general definition

a lb f(t) dqt = 1b f(t) dqt - 1 f(t) dqt, (1.11.2) where (1.11.3)

Jackson also defined an integral on (0, (0) by

(1.11.4)

The bilateral q-integral is defined by

100 f(t) dqt = (1 - q) f [f(qn) + f( _qn)] qn. (1.11.5) -00 n= -00 If f is continuous on [0, a], then it is easily seen that

lim r f(t) dqt = fa f(t) dt (1.11.6) q---+l io io and that a similar limit holds for (1.11.4) and (1.11.5) when f is suitably restricted. By (1.11.1), it follows from (1.10.14) that

B q( x, y ) = 1 t x-I ((tq; q)oo ) d qt, R ex> 0 , y r--I- 0 ,- 1 ,- 2 , ... , (1117).. 1o tqY;qoo which clearly approaches the beta function integral

B(x, y) = 11 t x - 1 (1 - t)y-l dt, Re x, Re y > 0, (1.11.8) 24 Basic hypergeometric series as q ----+ 1-. Thomae [1869] rewrote Heine's formula (1.4.1) in the q-integral form

a b. c. ) _ fq(c) b-I (tzqa,tq;q)cx; ( t (1.11.9) 2¢1 q ,q ,q ,q,z - fq(b)fq(c-b) io t (tz,tqc-b;q)cx; dqt, which is a q-analogue of Euler's integral representation

b I b I 2FI (a, b; c; z) = f(b)~~~ _ b) 11 t - (1 - W- - (1 - tz)-a dt, (1.11.10) where I arg(l- z)1 < 7r and Re c> Re b > O. The q-integral notation is, as we shall see later, quite useful in simplifying and manipulating various formulas involving sums of series.

Exercises 1.1 Verify the identities (1.2.30)-(1.2.40), and show that

(i) (aq-n; q)n = (qja; q)n ( _~) n q-G),

(ii) ( -k-n. ) = (qja;q)n+k (_~)n -nk-(~) aq ,q n (j)q a; q k q q , (qa!, -qa!; q)n 1 - aq2n (iii) (a!, -a!; q)n 1 - a '

(iv) (a;q)zn = (a!,-a!,(aq)!,-(aq)!;q)n' (v) (a; q)n(qja; q)-n = (-atq(~), (vi) (q, -q, _q2; q2)cx; = 1.

1.2 The q-binomial coefficient is defined by

[n] (q;q)n k q - (q; q)k(q; q)n-k for k = 0,1, ... , n, and by

(q!3+ I, q<>-!3+ I; q)cx; (q, q<>+ I; q)cx;

for complex a and f3 when Iql < 1. Verify that

(i)

(ii)

(q<>+ I; q)k (iii) (q;q)k ' Exercises 25

(iv) [ -kG L [G + ~ - 1 L(_q-a)kq-(~),

(v) [G: 1 L= [~L qk + [k ~ lL = [~L + [k ~ lL q<>+ I-k,

n k (vi) (Z;q)n=2::[~] (-z)kq(2), k= 0 q when k and n are nonnegative integers. 1.3 (i) Show that the binomial theorem

(a+bt = t (~)akbn-k k= 0 where n = 0,1, ... , has a q-analogue of the form (ab; q)n = t [~] bk(a; q)k(b; q)n-k k= 0 q = t [~] an-k(a; qh(b; q)n-k. k= 0 q (ii) Extend the above formula to the q-multinomial theorem (ala2 ... am+ I; q)n

2:: [ n ] a~l a~l + k2 ... a~: ~2+ ... + krn k l , ... ,km 0:Sklo···,O::;km, q kl +···+krn:Sn

x (al; q)k1 (a2; q)k2 ... (am; q)krn (am+ I; q)n-(k1+ ... + krn ), where m = 1,2, ... , n = 0,1, ... , and

is the q-multinomial coefficient. 1.4 (i) Prove the inversion formula

r¢s [al ' ... ' ar ; q, z] bl , ... ,bs

(Xl (-I -I -I) = '"""' aI' ... ,ar ; q n n=oq~ ( -I 'I,···,s;qb-I b-I -I ) n (ii) By reversing the order of summation, show that

A. [al' ... ' an q-n . ] r+I'Ps b b ,q,z 1, ... , s 26 Basic hypergeometric series

when n = 0,1, .... (iii) Show that

r+ 1 . r+ 1 'Pr" [a1'b ... ' ab' q, q z] 1, ... , r (al, ... , ar+ 1; q)oo t z-1 (qt, b1 t, ... , brt; q)oo d = (l-q)(q,b1 , •.• ,br;q)00}o t (al t, ... ,ar+l t ;q)oo qt, when ° < q < 1, Re z > 0, and the series on the left side does not terminate. 1.5 Show that (e,bqn;q)rn _ (b/e;q)n ~ (q-n,e;q)kqk ( k. ) - ( ) ~ ( 1 / ) cq, q rn· ()b; q rn b; q n k= 0 q, eq -n b; q k

1.6 Prove the summation formulas (i) 2¢I(q-n,ql-n;qb2;q2,q2) = (~:;.q2inq-(~), ,q n .. ) " ( /) (e/a; q)oo (11 1 'PI a; e; q, e a = ( ) , e; q 00 (iii) 2¢o(a, q-n; -; q, qn fa) = a-n,

00 n 2 -n 2 (iv) L -(q)2 = ( ) , n= 0 q; q n q; q 00 (zp-l ;p-l)oo -1 (v) l¢o(a;-;p,z) = ( -1 -1) , Ipl > 1, lazp 1<1, azp ;p 00 ( .) ( ) (a/e, b/e;p-l )00 I I VI 2¢1 a, b; e;p,p = (1/ b/. -1) , p > 1. e, a e,p 00 1.7 Show that, for Izl < 1,

2 ) ( ) (a2qz; q)oo 2¢1 ( a ,aq;a;q,z = l+az ( ) . z;q 00

2 1.8 Show that, when lal < 1 and Ibq/a 1 < 1,

2¢1 (a2, a2/b; b; l, bq/a2) = (a2,q;f)~ [(b/~;q)oo + (-b/~;q)oo] . 2(b, bq/ a ,q)oo (a, q)oo (-a, q)oo (Andrews and Askey [1977])

1.9 Let ¢(a, b, c) denote the series 2¢1 (a, b; e; q, z). Verify Heine's [1847] q-contiguous relations: . -I (l-a)(I-b) (1) ¢(a, b, eq ) - ¢(a, b, c) = ez ( )( ) ¢(aq, bq, cq), q-e l-e (ii) ¢(aq, b, c) - ¢(a, b, c) = az ~ =~ ¢(aq, bq, eq), Exercises 27

... (l-b)(l-c/a) 2 (m) ¢(aq, b, cq) - ¢(a, b, c) = az ( )( ) ¢(aq, bq, cq ), 1-c 1-cq . 1 (1- b/aq) (IV) ¢(aq, bq- ,c) - ¢(a, b, c) = az 1 _ c ¢(aq, b, cq).

1.10 Denoting 2¢1 (a, b; c; q, z), 2¢1 (aq±l, b; c, q, z), 2¢1 (a, bq±l; c; q, z) and 2¢1 (a, b; cq±l; q, z) by ¢, ¢(aq±I), ¢(bq±l) and ¢(cq±I), respectively, show that

(i) b(l - a)¢(aq) - a(l - b)¢(bq) = (b - a)¢, (ii) a(l - b/c)¢(bq-l) - b(l - a/c)¢(aq-l) = (a - b)(l - abz/cq)¢, (iii) q(l - a/c)¢(aq-l) + (1 - a)(l - abz/c)¢(aq) = [1 + q - a - aq/c + a2z(l - b/a)/c]¢, (iv) (1 - c)(q - c) (abz - c)¢(cq-l) + (c - a)(c - b)z¢(cq) = (c - l)[c(q - c) + (ca + cb - ab - abq)z]¢. (Heine [1847]) 1.11 Let g(e;A,/-l,v) = (AeiO,/-lV;q)oo 2¢1(/-le- iO ,ve-iO ;/-lv;q, AeiO ). Prove that g(e; A, /-l, v) is symmetric in A, /-l, v and is even in e. 1.12 Let Vq be the q-derivative operator defined for fixed q by V J(z) = J(z) - J(qz) q (1 - q)z ' and let V~u = Vq(D~-IU) for n = 1,2, .... Show that (i) lim VqJ(z) = dd J(z) if J is differentiable at z, q---+l z .. ) 'T"In rI-. (b ) (a, b; q)n rI-. ( n b n n ) (11 Vq 2'/-'1 a, ;c;q,z = ( ) ( ) 2 '/-'1 aq , q ;cq ;q,z , c; q n 1 - q n ( ... ) 'T"In { (z; q)oo rI-. (b )} m Vq (abz/c;q)oo 2'/-'1 a, ;c;q,z _ (c/a,c/b;q)n (ab)n (zqn;q)oo ¢ ( b' n. n) - (c; q)n(1- q)n C (abz/c; q)oo 2 1 a, ,cq ,q, zq . (iv) Prove the q-Leibniz formula

V;[J(z)g(z)] = ~ [~] qV;-k J(zqk)V;g(z).

1.13 Show that u(z) = 2¢1 (a, b; c; q, z) satisfies (for Izl < 1 and in the formal power series sense) the second order q-differential equation

z(c _ abqz)V2 u + [1 - c + (1 - a)(l - b) - (1 - abq) z] V u q 1-q 1-q q (l-a)(l-b) - (1 _ q)2 u = 0, where Vq is defined as in Ex. 1.12. By replacing a, b, c, respectively, by qa, qb, qC and then letting q ----+ 1- show that the above equation tends to the second order differential equation z(l - z)v" + [c - (a + b + l)z]v' - abv = 0 28 Basic hypergeometric series

for the hypergeometric function v(z) = 2F1 (a, b; c; z), where Izl < 1. (Heine [1847]) 1.14 Let Ixl < 1 and let eq(x) and Eq(x) be as defined in §1.3. Define

. eq(ix) - eq( -ix) 00 (_I)nx2n+ 1 Slllq (x) = 2 . = L (. ) , ~ n= 0 q, q 2n+ 1 ( ) _ eq(ix) +eq(-ix) _ ~ (_I)nx2n COSq x - 2 - ~ (.) . n= 0 q, q 2n Also define S. () Eq(ix) - Eq( -ix) lllq x = 2i ' Show that (i) eq(ix) = cosq(x) + i sinq(x), (ii) Eq(ix) = Cosq(x) + iSinq(x), (iii) sinq(x)Sinq(x) + cosq(x)Cosq(x) = 1, (iv) sinq(x)Cosq(x) - Sinq(x) cosq(x) = 0. For these identities and other identities involving q-analogues of sin x and cosx, see Jackson [1904a] and Hahn [1949c]. 1.15 Prove the transformation formulas

(i) A. [q-n,b. ] _ (bzq-n/c;q)oo A. [q-n,c/b,o. ] 2'/'1 C' q, Z - (bz/c; q)oo 3 '/'2 c, cq/bz ,q, q ,

(ii) A. [q-n,b. ] _ (c/b;q)nbn A. [q-n,b,q/z. /] 2'/'1 C ,q,z - (C;q)n 3'/'1 bq1-n/C ,q,z C ,

q-n,b. ] _ (c/b;q)n [q-n,b,bzq-n/C. ] (iii) 21;1 [ ,q,z - (.) 31;2 b 1-n/ ,q,q . C C, q n q C, ° (See Jackson [1905a, 1927]) 1.16 Show that

1.17 Show that t (a, b; q)k (_ab)n-kq(n-k)(n+ k-1)/2 k=O (q;q)k n (-b)kq(~) (Carlitz [1974]) = (a; q)n+ 1 ~ (q; qh(q; q)n-k(1 _ aqn-k)· Exercises 29

1.18 Show that (i) (c;q)oo 1¢I(a;c;q,z) = (z;q)oo 1¢I(az/c;z;q,c), and deduce that 1¢1(-bq;O;q,-q) = (-bq2;q2)00/(q;q2)00, (ii) (z;q)oo 2¢I(a,0;c;q,z) = (az;q)oo 1¢2(a;c,az;q,cz),

(iii) f ((a;2~)n) qG)(at/z)n 2¢I(q-n,a;ql-n/a;q, qz2/a) n= 0 q, a ,q n = (-zt;q)oo 2¢I(a,a/z2;a2;q,-zt), Iztl < 1.

1.19 Using (1.5.4) show that

A.. [a, q/a. -b] = (ab, bq/a; q2)00 (i) 2'1-'2 -q, b ,q, (b; q)oo ' a2,b2 ] (a2q,b2q;q2)00 (ii) [ 1 1 ; 2¢2 a bq2,-a b q2 q, -q = (q,a2b2q;q2)00 . (Andrews [1973])

1.20 Prove that if Re x > 0 and 0 < q < 1, then

(i)

(ii)

1.21 For 0 < q < 1 and x > 0, show that

d2 2 00 qn+ x dx2 logfq(x) = (logq) ~ (1- qn+ x)2'

which proves that log f q (x) is convex for x > 0 when 0 < q < 1. 1.22 Conversely, prove that if f(x) is a positive function defined on (0, oo)which satisfies 1- qX f(x + 1) = --f(x) for some q, 0 < q < 1, l-q f(l) = 1,

and logf(x) is convex for x > 0, then f(x) = fq(x). This is Askey's [1978] q-analogue of the Bohr-Mollerup [1922] theorem for r(x). For two extensions to the q > 1 case (with f q (x) defined as in the next exercise), see Moak [1980b]. 1.23 For q > 1 the q-gamma function is defined by

f ()= (q-I;q-I)oo( _1)I-X x(x-I)/2 q X (q-x;q-I)oo qq. 30 Basic hypergeometric series

Show that this function also satisfies the functional equation (1.10.5) and that rq(x) ----t r(x) as q ----t 1+. Show that for q > 1 the residue of rq(x) at x = -n is n+ 1) (q _ l)n+ 1q ( 2 (q; q)n logq

1.24 Jackson [1905a,b,el gave the following q-analogues of Bessel functions:

(qv+ 1. q) J~l) (x; q) = ( ') 00 (x/2)V 2¢1 (0, 0; qV+ 1; q, -x2/4), q;q 00 (qv+ 1. q) ( x2qv+ J~2) (x; q) = (q; ;)0000 (x/2t 0¢1 -; qV+ 1; q, --4-1) ,

(qv+ 1. q) J~3)(x;q) = ( \ 00 (x/2t 1¢1(0;qv+1;q,qx2/4), q;q 00 where 0 < q < 1. The above notations for the q-Bessel functions are due to Ismail [1981, 1982, 2003cl. Show that J~2)(x;q) = (-x2/4;q)ooJ~1)(x;q), Ixl < 2, (Hahn [1949c]) and lim J~k)(x(l- q);q) = Jv(x), k = 1,2,3. q--+1

1.25 For the q-Bessel functions defined as in Exercise 1.24 prove that

(i) qV ik) (x· q) = 2(1 - qV) fk) (x· q) _ ik) (x· q) k = 1 2· v+ 1 , X v' v-I" , ,

(ii) J~l) (xq!; q) = qv/2 (J~l) (x; q) + ~J~~ 1 (x; q)) ;

(iii) J~l)(xq!;q) = q-v/2 (J~l)(x;q) - ~J~~l(X;q)).

(iv) qV+ 1 J~~ 1 (xq1/2; q) = 2(1 - qV) J~3) (x; q) - J~~l (x; q). X

1.26 (i) Following Ismail [1982]' let

fv(x) = J~l) (x; q)J£12 (xq!; q) - J£12 (x; q)J~l) (xq!; q). Show that

and deduce that, for non-integral v,

fv(x) = q-v/2(qv, q1-v; q)oo/(q, q, _x2/4; q)oo. (ii) Show that

9v(qX) + (x2 /4 - qV - q-v)9v(X) + 9v(xq-1) = 0 Exercises 31

with gl/(x) = J~3)(xql//2;q2) and deduce that

( 21/ 1-21/. 2) () ( -1 ) () ( -1) _ q , q ,q 00 1/( 1/-1) gl/ X g-I/ xq - g-I/ X gl/ xq - (2 2. 2) q . q ,q ,q 00 (Ismail [2003c]) 1.27 Show that

00

n= -CX)

00

Both of these are q-analogues of the generating function

00 L t n In(x) = ex (t-C 1 )/2.

n= -CX)

1.28 The continuous q-Hermite polynomials are defined in Askey and Ismail [1983] by

H ( I ) - ~ (q; q)n i(n-2k)IJ nxq-L..J( )( ) e , k= 0 q; q k q; q n-k

where x = cos B; see Szego [1926]' Carlitz [1955, 1957a, 1958, 1960] and Rogers [1894, 1917]. Derive the generating function

~ Hn(xlq) n 1 f:'o (q; q)n t = (teiIJ, te-iIJ; q)oo' It I < 1. (Rogers [1894])

1.29 The continuous q-ultraspherical polynomials are defined in Askey and Is­ mail [1983] by

C ( . (31 ) - ~ ((3; q)k((3; q)n-k i(n-2k)IJ n x, q - L..J ( ) ( ) e , k= 0 q; q k q; q n-k

and

00 n ((3teiIJ , (3te-iIJ ; q)oo '" Cn(x; (3lq)t = ( ·IJ -"IJ) < 1. (Rogers [1895]) L..J te' te ' . q ,ItI n=O ' ,00 32 Basic hypergeometric series

1.30 Show that if ml , ... ,mr are nonnegative integers, then

1.31 Let ~b denote the q-difference operator defined for a fixed q by

~bJ(Z) = bJ(qz) - J(z).

Then ~1 is the ~ operator defined in (1.3.20). Show that

~bXn = (bqn - 1 )xn and, if

(al,"" ar; q)n (_1)(1+ s-r)nq(l+ s-r)n(n-l)/2zn, vn ()z = ( ) q,b1 ,···,bs;q n then (~~bdq~b2/q'" ~bs/q)Vn(z) = z(~al ~a2 ... ~aJVn-l (zql+ s-r), n = 1,2, ... Use this to show that the basic hypergeometric series

v(z) = r¢s(al, ... , ar; b1 ,···, bs; q, z) satisfies (in the sense of formal power series) the q-difference equation (~~bl/q~b2/q'" ~bs/q)V(z) = Z(~al ... ~ar)v(zql+ s-r). This is a q-analogue of the formal differential equation for generalized hypergeometric series given, e.g. in Henrici [1974, Theorem (1.5)] and Slater [1966, (2.1.2.1)]. Also see Jackson [191Od, (15)]. 1.32 The little q-Jacobi polynomials are defined by Pn(x; a, b; q) = 2¢1 (q-n, abqn+ 1; aq; q, qx). Show that these polynomials satisfy the orthogonality relation

f ~b~; ql j (aq)j Pn (qj; a, b; q)Pm( qj; a, b; q) j= 0 q, q J 0, if m =I- n, = { (q, bq; q)n(1- abq) (aqt (abl; q)= if m = n. (aq,abq;q)n(l-abq2n+l) (aq;q)= ' Exercises 33

(Andrews and Askey [1977 ]) 1.33 Show for the above little q-Jacobi polynomials that the formula n Pn(x; c, d; q) = L ak,nPk(X; a, b; q) k= 0 holds with __ k (k~l)(q-n,aq,cdqn+l;q)k [qk-n,Cdqn+k+l,aqk+l. ] ak,n - ( 1) q ( q,cq,a bq k+I.),q k 3(Pl cqk+1 , abq2k+2 ,q,q. (Andrews and Askey [1977]) 1.34 (i) If m, ml, m2, ... , mr are arbitrary nonnegative integers and la-Iqm+l-(m1+···+mr)1 < 1, show that a b b qml b qmr ] A.. [ ' ,I , ... , r . a-I m+I-(ml+···+mr) r+ 2 'f'r+ I b 1+ m b b' q, q q ,I, ... , r

= (q, bq/a; q)oo(bq; q)m(bl /b; q)ml ... (br/b; q)mr bm1+ ... + mr-m (bq, q/a; q)oo(q; q)m(bl ; q)ml ... (br; q)mr q-m, b, bq/bl , ... ,bq/br ] [ X r+ 2 ¢r+ I bq / a, bq I -ml /b I,···, bq I-mr /b r ; q, q ; (ii) if ml, m2, ... , mr are nonnegative integers and la-Iql-(m1+ ... + mr) 1< 1, Icql < 1, show that a b b qml b qmr ] A.. [ ' ,I , ... , r . a-II-(m1+···+mr) r+ 2 'f'r+ I b b b ' q, q cq, I,···, r

= (bq/a, cq; q)oo (b l /b; q)ml ... (br/b; q)mr bm1+ ... + mr (bcq, q/ a; q)oo (bl ; q)ml ... (br; q)mr

c-I , b, bq/bl , ... ,bq/br ] [ X r+ 2 ¢r+ I b / b I -ml /b b I -m /b ; q, cq . q a, q I , ... , q r r (Gasper [1981a]) 1.35 Use Ex. 1.2(v) to prove that if x and yare indeterminates such that xy = qyx, q commutes with x and y, and the associative law holds, then

(See Cigler [1979], Feinsilver [1982]' Koornwinder [1989], Potter[1950]' Schiitzenberger [1953], and Yang [1991]). 1.36 Verify that if x and yare indeterminates satisfying the conditions in Ex. 1.35, then (i) eq(y)eq(x) = eq(x + y), eq(x)eq(y) = eq(x + y - yx); (ii) Eq(x)Eq(y) = Eq(x + y), Eq(y)Eq(x) = Eq(x + y + yx). 34 Basic hypergeometric series

(Fairlie and Wu [1997]; Koornwinder [1997], where q-exponentials with q-Heisenberg relations and other relations are also considered.) 1.37 Show that (a2;q2)= { [_qe2iO,_qe-2iO 2 2] Eq(z;a) = ( 2. 2) 2

with Z = cose. 1.38 Extend Jacobi's triple product identity to the transformation formula

1+ f)-ltq(~)(an+bn) = (q,a,b;q)=f (ab/q;q)znqn. n=! n=O (q,a,b,ab;q)n Deduce that = = (.) n 1 2 ~ n 2n 2 ( ) ( 2) ~ -a, q 2nq + ~ a q = q; q = aq; q = ~ ( . ) ( . 2) . n= ! n= 0 q, -aq, q n aq, q n (Warnaar [2003a])

Notes

§§1.1 and 1.2 For additional material on hypergeometric series and orthog­ onal polynomials see, e.g., the books by Erdelyi [1953], Rainville [1960], Szego [1975], Whittaker and Watson [1965], Agarwal [1963], Carlson [1977], T.S. Chi­ hara [1978], Henrici [1974], Luke [1969], Miller [1968], Nikiforov and Uvarov [1988], Vilenkin [1968], and Watson [1952]. Some techniques for using sym­ bolic computer algebraic systems such as Mathematica, Maple, and Macsyma to derive formulas containing hypergeometric and basic hypergeometric se­ ries are discussed in Gasper [1990]. Also see Andrews [1984d, 1986, 1987b], Andrews, Crippa and Simon [1997], Andrews and Knopfmacher [2001], An­ drews, Knopfmacher, Paule and Zimmermann [2001]' Andrews, Paule and Riese [2001a,b], Askey [1989f, 1990], Askey, Koepf and Koornwinder [1999], Baing and Koepf [1999], Garoufalidis [2003], Garoufalidis, Le and Zeilberger [2003], Garvan [1999], Garvan and Gonnet [1992]' Gosper [2001], Gosper and Suslov [2000], Koepf [1998], Koornwinder [1991b, 1993a, 1998], Krattenthaler [1995b], Paule and Riese [1997], Petkovsek, Wilf and Zeilberger [1996], Riese [2003], Sills [2003c], Wilf and Zeilberger [1990], and Zeilberger [1990b]. §§1.3-1.5 The q-binomial theorem was also derived in Jacobi [1846]' along with the q-Vandermonde formula. Bijective proofs of the q-binomial theorem, Heine's 2derivatives, AI-Salam [1966] and Agarwal [1969b]. Toeplitz [1963, pp. 53-55] pointed out that around 1650 Fermat used a q-integral type Riemann sum to evaluate the integral of xk on the interval [0, b]. AI-Salam and Ismail [1994] evaluated a q-beta integral on the unit circle and found corresponding systems of biorthogonal rational functions. Ex.1.2 The q-binomial coefficient [~L' which is also called the Gaus­ sian binomial coefficient, counts the number of k dimensional subspaces of an n dimensional vector space over a field GF(q), q a prime power (Goldman and Rota [1970]), and it is the generating function, in powers of q, for partitions into at most k parts not exceeding n - k (Sylvester [1882]). It arises in such diverse fields as analysis, computer programming, geometry, number theory, physics, and statistics. See, e.g., Aigner [1979], Andrews [1971a, 1976], M. Baker and Coon [1970], Baxter and Pearce [1983], Berman and Fryer [1972], Dowling [1973], Dunkl [1981]' Garvan and Stanton [1990], Handa and Mohanty [1980], Ihrig and Ismail [1981]' Jimbo [1985, 1986], van Kampen [1961]' Kendall and Stuart [1979, §31.25]' Knuth [1971, 1973], P6lya [1970], P6lya and Alexander­ son [1970], Szego [1975, §2.7]' and Zaslavsky [1987]. Sylvester [1878] used the invariant theory that he and Cayley developed to prove that the coefficients of the Gaussian polynomial [~L = '2;ajqj are unimodal. A constructive proof was recently given by O'Hara [1990]. Also see Bressoud [1992] and Zeilberger

[1989a,b, 1990b]. The unimodality of the sequence ([~] q : k = 0,1, ... ,n) is explicitly displayed in Aigner [1979, Proposition 3.13], and Macdonald [1995, Example 4 on p. 137]. Ex.1.3 Cigler [1979] derived an operator form ofthe q-binomial theorem. MacMahon [1916, Arts. 105-107] showed that if a multiset is permuted, then 36 Basic hypergeometric series the generating function for inversions is the q-multinomial coefficient. Also see Carlitz [1963a], Kadell [1985a], and Knuth [1973, p. 33, Ex. 16]. Gasper derived the q-multinomial theorem in part (ii) several years ago by using the q-binomial theorem and mathematical induction. Andrews observed in a 1988 letter that it can also be derived by using the expansion formula for the q-Lauricella function Taylor series" are considered in Jackson [1909b,c] and Wallisser [1985]. A q-Taylor theorem based on the sequence {

Ex.1.32 Masuda et al. [1991] showed that the matrix elements that arise in the representations of certain quantum groups are expressible in terms of little q-J acobi polynomials, and that this and a form of the Peter-Weyl theorem imply the orthogonality relation for these polynomials. Pade approximants for the moment generating function for the little q-Jacobi polynomials are em­ ployed in Andrews, Goulden and D.M. Jackson [1986] to explain and extend Shank's method for accelerating the convergence of sequences. Pade approxi­ mations for some q-hypergeometric functions are considered in Ismail, Perline and Wimp [1992]. 2

SUMMATION, TRANSFORMATION, AND EXPANSION FORMULAS

2.1 Well-poised, nearly-poised, and very-well-poised hypergeometric and basic hypergeometric series

The hypergeometric series

(2.1.1) is called well-poised if its parameters satisfy the relations

1 + al = a2 + b1 = a3 + b2 = ... = ar+1 + bTl (2.1.2) and it is called nearly-poised if all but one of the above pairs of parameters (regarding 1 as the first denominator parameter) have the same sum. The series (2.1.1) is called a nearly-poised series of the first kind if

1 + al i= a2 + b1 = a3 + b2 = ... = ar+l + bTl (2.1.3) and it is called a nearly-poised series of the second kind if

1 + al = a2 + b1 = a3 + b2 = ... = ar + br- 1 i= ar+l + br· (2.1.4) The order of summation of a terminating nearly-poised series can be re­ versed so that the resulting series is either of the first kind or of the second kind. Kummer's summation formula (1.8.2) gives the sum of a well-poised 2Fl series with argument -1. Another example of a summable well-poised series is provided by Dixon's [1903] formula 3F2 [ a, b, c; 1 + a - b, 1 + a - c; 1] r(1 + ~a)r(1 + a - b)r(1 + a - c)r(1 + ~a - b - c) (2.1.5) r(1 + a)r(1 + ~a - b)r(1 + ~a - c)r(1 + a - b - c)'

Re (1 + ~a - b - c) > 0, which reduces to Kummer's formula (1.8.2) by letting c ---+ -00. If the series (2.1.1) is well-poised and a2 = 1 + ~al' then it is called a very-well-poised series. Dougall's [1907] summation formulas F. [ a, 1+ ~a, b, c, d, e, -n . 1] 7 6 ~a, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + n ' (1 + a)n(1 + a - b - c)n(1 + a - b - d)n(1 + a - c - d)n (2.1.6) (1 + a - b)n(1 + a - c)n(1 + a - d)n(1 + a - b - c - d)n

38 2.1 Well-, nearly-, and very-well-poised series 39 when the series is 2-balanced (i.e, 1 + 2a + n = b + c + d + e), and F [ a, 1+ ~a, b, c, d . 1] s 4 ~a, 1 + a - b, 1 + a - c, 1 + a - d' r(1 + a - b)r(1 + a - c)r(1 + a - d)r(1 + a - b - c - d) (2.1. 7) r(1 + a)r(1 + a - b - c)r(1 + a - b - d)r(1 + a - c - d)' when Re (1 + a - b - c - d) > 0, illustrate the importance of very-well-poised hypergeometric series. Note that Dixon's formula (2.1.5) follows from (2.1.7) by setting d = ~a. Analogous to the hypergeometric case, we shall call the basic hypergeo- metric series '" [al ,a2, ... ,ar+l . ] r+ 1 'f'r b b' q, Z (2.1.8) 1, ... , r well-poised if the parameters satisfy the relations

qal = a2 bl = a3 b2 = ... = ar+lbr; (2.1.9)

1 1 very-well-poised if, in addition, a2 = qai, a3 = -qai; a nearly-poised series of the first kind if qal -=I=- a2 bl = a3 b2 = ... = ar+lbn and a nearly-poised series of the second kind if

qal = a2 bl = a3 b2 = ... = arbr- l -=I=- ar+lbr . (2.1.10) In this chapter we shall be primarily concerned with the summation and trans­ formation formulas for very-well-poised basic hypergeometric series. To help simplify some of the displays involving very-well-poised r+lcPr series which arise in the proofs in this and the subsequent chapters we shall frequently replace

by the more compact notation (2.1.11) In the displays of the main formulas, however, we shall continue to use the r+l cPr notation, since in most applications one needs to know the denominator parameters. In all of the very-well-poised 6cPS, 8cP7 and lOcP9 series in Appendix II and Appendix III the parameters aI, a4, ... , ar+l and the argument z satisfy the very-well-poised balancing condition (a4as···ar+l)z= (±(alq)~r-3 (2.1.12) with either the plus or minus sign, where (alq)~ is the principal value of the square root of alq. Thus, we will call a r+l Wr series very-well-poised-balanced (or, for brevity, VWP-balanced) if (2.1.12) holds. A special case of this condi­ tion was stated at the end of §2.4 in the first edition of this book. It follows from (2.1.12) that a VWP-balanced series is balanced (I-balanced) when z = q, 40 Summation, transformation, and expansion formulas a balanced r+l Wr series is VWP-balanced when r is even, but it is not neces­ sarily VWP-balanced when r is odd since then we only know that the squares of both sides of (2.1.12) (with z = q) are equal (see the comments in the first paragraph of §2.6). When r + 1 = 2k is an even integer, which is usually the case, (2.1.12) simplifies to the condition

(a4a5'" a2k)z = (alq)k-2. (2.1.13) Notice that if we set a2k = ±(alq)!, then the 2kW2k-l series reduces to a 2k-l W 2k-2 series that satisfies (2.1.12) with r = 2k - 2 and the corresponding plus or minus sign. Since any well-poised r+l¢r series that satisfies the relations in (2.1.9) can be written as a very-well-poised series in the form

1 1 1 1 r+5 W r+4(al; al, -al, qal, -qal, a2, a3,"" ar+l; q, z), (2.1.14) if follows that the very-well-poised balancing condition for the series in (2.1.14) is consistent with the well-poised balancing condition (ala2'" ar+dz = -(±(alq)!y+l (2.1.15) for a well-poised r+l¢r series to be well-poised-balanced (WP-balanced). In particular, the 2¢1 series in the Bailey-Daum summation formula (1.8.1) is WP-balanced. Clearly, every VWP-balanced basic hypergeometric series is WP-balanced and, by the above observations, every WP-balanced basic hy­ pergeometric series can be rewritten to be a VWP-balanced series of the form in (2.1.14).

2.2 A general expansion formula

Let a, b, c be arbitrary parameters and k be a nonnegative integer. Then, by the q-Saalschiitz formula (1. 7.2)

3¢2 (q-k, aqk, aq/bc; aq/b,aq/c; q,q) (C,ql-k/b;q)k (b,c;q)k (2.2.1) (aq/b, cq-k fa; q)k (aq/b, aq/c; q)k so that ~ (b, c, q-n; qh A ~ (q, aq/b, aq/c; q)k k k=O n k (/b k -k. ) ( -n.) (b) k = L L aq c, aq ,q : q j q. ,q k qj ~ Ak k=Oj=O (q, aq/b, aq/c, q)j(q, q)k aq

_ ~~ (aq/bc, aqi+j,q-i-j ; q)j (q-n;q)i+j . (bc)i+j -~~ ~ - ~H j=Oi=O (q,aq/b,aq/c;q)j(q;q)i+j aq

~ (aq/bc, aqj, q-n; q) . . (j) = ~ J (-I)J q - 2 j=O (q, aq/b, aq/c; q)j 2.3 A summation formula for a terminating 41>3 41

n-j (qj-n, aq2j ;q)i -i. (bC)i+j q J - A+· (2.2.2) X"~ (q,aqj;q)i aq • J' where {Ak} is an arbitrary sequence. This is equivalent to Bailey's [1949] lemma. Choosing (2.2.3)

aq 2j ,a1 q j ,a2q j , ... , arqj ,qj-n . bcz ] X r+ 2 1>r+ 1 [ .. . ., q, ------:::j:l . (2.2.4) b1qJ, b2qJ, ... , brqJ, br+1qJ aqJ This is a q-analogue of Bailey's formula [1935, 4.3(1)]. The most important property of (2.2.4) is that it enables one to reduce the r+41>r+3 series to a sum of r+21>r+1 series. Consequently, if the above r+21>r+1 series is summable for some values of the parameters then (2.2.4) gives a transformation formula for the corresponding r+41>r+3 series in terms of a single series.

2.3 A summation formula for a terminating very-well-poised 41>3 series Setting b = qa!, c = -qa! and ak = bk, k = 1,2, ... , r, br+1 = aqn+l, we obtain from (2.2.4) that 1 1 a, qa 2" , -qa 2" , q-n ] 41>3 [ ! ! n+1 ;q,z a 2 , -a 2 , aq ~ (-q-1,q-n;q)j(a;qhj( ). _(j) = ~ 1 1 qz Jq 2 j=O (q, a2", -a2", aqn+1; q)j

x 21>1 (aq2j,qj-n;aqHn+\q,_zq1-j). (2.3.1)

If we set z = qn so that the above 41>3 series is VWP-balanced, then the 21>1 series on the right of (2.3.1) can be summed by means of the Bailey-Daum summation formula (1.8.1), which gives (_q. q) (aq2j+1 aq2n+2. q2) A, ( 2j j-n. j+n+1. _ 1+n-j ) _ '00 , '00 2'1'1 aq ,q , aq , q, q - ( + ·+1 1+ _.) . aqn J ,-q n J; q 00 (2.3.2) Hence, using the identities (1.2.32), (1.2.39) and (1.2.40), and simplifying, we obtain the transformation formula 1 1 a, qa2", -qa2" , q-n n] 41>3 [ ! ! n+1; q, q a 2 ,-a2 ,aq

(aq, -q; q)n A, (-n -1 -n ) ! !.) 2'1'1 q , -q ; -q ; q, q . (2.3.3) (qa 2 , -qa 2 ,q n 42 Summation, transformation, and expansion formulas

Clearly, both sides of (2.3.3) are equal to 1 when n = O. By (1.5.3) the 2¢1 series on the right of (2.3.3) has the sum(ql-n; q)n (_q-l r j (_q-n; q)n when n = 0,1, .... Since (ql-n; q)n = 0 unless n = 0, it follows that

1 1 a, qa"2 , -qa"2 , q-n n] 4¢3 [ 1 1 n+l; q, q = 6n,o , (2.3.4) a 2 , -a 2 , aq where m=n, (2.3.5) m i- n, is the Kronecker delta function. This summation formula will be used in the next section to obtain the sum of a 6¢5 series.

2.4 A summation formula for a terminating

very-well-poised 6¢5 series

1 1 1 1 +1 Let us now set al = qa"2, a2 = -qa"2, b1 = a"2, b2 = -a"2, br+1 = aqn and ak = bk , for k = 3,4, ... , r. Then (2.2.4) gives

1 1 rf.. [ a, qa"2, -qa"2, b, c, q-n ] 6'i"5 1 1 +1 ; q, z a"2, -a"2, aqjb, aqjc, aqn

= ~ (aqjbc,qa!,-qa!,q-n;q)j(a;qhj (_bacqz)j q_(~) ~j=O ( q,a 12 ,-a 12 ,aq jb ,aq j c,aqn+1.) ,q j aq2j,qJ+la!,_qj+la!,qj-n. bCZ] [ X4¢3 ·1 ·1 . 1 ,q'------:::t=1. (2.4.1) qJ a"2 , _qJ a"2 , aqJ+n+ aqJ

If z = aqn+1 jbc, then we can sum the above 4¢3 series by means of (2.3.4) to obtain the summation formula

rf.. [ a, qa!, -qa!, b, c, q-n . aqn+1 ] 6'i"5 1 1 +1 , q, -- a"2, -a"2, aqjb, aqjc, aqn bc (aqjbc,qa!,-qd,q-n;q)n (a;qhn (_1)nqn(n+l)/2 (q, a!, -a!, aqjb, aqjc, aqn+l; q)n (aq, aqjbc; q)n (2.4.2) (aqjb, aqj c; q)n .

Note that the above 6¢5 series is VWP-balanced and that this summation formula reduces to (2.3.4) when bc = aq. In the next two sections, like climbing the steps of a ladder, we will use (2.4.2) to extend it to a transformation formula and a summation formula for very-well-poised S¢7 series.

2.5 Watson's transformation formula for a terminating very-well-poised S¢7 series We shall now use (2.4.2) to prove Watson's [1929a] transformation formula for a terminating very-well-poised S¢7 series as a multiple of a terminating 2.6 Jackson's sum 43 balanced 4¢3 series: a qa! -qa! b c d e q-n a2q2+n ] 8

A, a,qa.1.12 ,-qa2 , b ,c, d ,e,q-n . a 2 q 2+n] [ 8'1'7 1 1 +1' q, -bd a"2, -a"2, aq/b, aq/c, aq/d, aq/e, aqn c e = t (~q/bc, ;a!, -qa!, d, e, q-n; q)j (a; qhj (_ aqn+1)j q-a) j=O (q, a"2, -a"2, aq/b, aq/c, aq/d, aq/e, aqn+1; q)j de aq2j,qj+1a!,-qj+1a!,dqj,eqj,qj-n aql+n- j ] [ (2.5.2) x 6¢5 . 1 . 1 ·+1 ·+1 .+ +1; q, d ' qJ a"2,-qJ a "2, aqJ /d,aqJ /e,aqJ n e which gives formula (2.5.1) by using (2.4.2) to sum the above 6¢5 series. Clearly, the 8¢7 series in (2.5.1) is VWP-balanced and the transformation formula (2.5.1) reduces to the summation formula (2.4.2) when bc = aq.

2.6 Jackson's sum of a terminating very­ well-poised balanced 8¢7 series

The 8¢7 series in (2.5.1) is balanced when the six parameters a, b, c, d, e and n satisfy the condition (2.6.1) which is the very-well-poised balancing condition. For such a series Jackson [1921] showed that

1 1 a, qa"2 , -qa"2 , b, c, d, e, q-n ] S¢7 [ 1 1 ;q,q a"2, -a"2, aq/b, aq/c, aq/d, aq/e, aqn+1 (aq,aq/bc, aq/bd, aq/cd;q)n (2.6.2) (aq/b, aq/c, aq/d, aq/bcd; q)n ' when n = 0,1,2, .... This formula follows directly from (2.5.1), since the 4¢3 series on the right of (2.5.1) becomes a balanced 3¢2 series when (2.6.1) holds, and therefore can be summed by the q-Saalschiitz formula. Notice that the very-well-poised series in (2.6.2) is also balanced if a2qn+1 = -bcde i=- 0, but then it is not VWP-balanced and it is not summable as a quotient of products of q-shifted factorials. Hence, not every balanced terminating very-well-poised S¢7 series is summable. Formula (2.6.2) is a q-analogue of (2.1.6), as can be seen by replac­ ing a, b, c, d, e by qa, qb, qC, qd, qe, respectively, and then letting q ----> 1. It should be observed that the series S¢7 in (2.6.2) is balanced, while the lim­ iting series 7F6 in (2.1.6) is 2-balanced. The reason for this apparent dis­ crepancy is that the appropriate q-analogue of the term (1 + ~ah/(~ah = (a + 2k)/a in the 7F6 series is not (qa!; qh/(a!; qh = (1 - a! qk)/(l - a!) 44 Summation, transformation, and expansion formulas but (qa!, -qa!; qh/(a!, -a!; q)k, which introduces an additional q-factor in the ratio of the products of the numerator and denominator parameters. Anal­ ogous to (2.1.12), one could call the very-well-poised 7F6 series in (2.1.6) VWP­ balanced if (b + e + d + e - n) + 1 = 2(a + 1), so that then every terminating very-well-poised 7F6(1) series is summable whenever it is VWP-balanced.

2.7 Some special and limiting cases of Jackson's and Watson's formulas: the Rogers-Ramanujan identities Many of the known summation formulas for basic hypergeometric series are special or limiting cases of Jackson's formula (2.6.2). For example, if we take d ---+ 00 in (2.6.2) we get (2.4.2). On the other hand, taking the limit a ---+ 0 after replacing d by aq/d gives the q-Saalschiitz formula (1.7.2). Let us now take the limit n ---+ 00 in (2.6.2) to obtain the following summation formula for a non-terminating VWP-balanced 6¢5 series 1 1 A, [ a, qa'2 , -qa'2 , b, e, d . aq ] 6'/-'5 1 1 , q, -bd a'2, -a'2, aq/b, aq/e, aq/d c (aq,aq/be,aq/bd,aq/ed;q)oo (2.7.1) (aq/b,aq/e,aq/d,aq/bed; q)oo, where, for convergence, it is required that laq/bedl < 1. This formula is dearly a q-analogue of Dougall's 5F4 summation formula (2.1.7), and it reduces to (2.3.4) when be = aq and d = q-n. If cd = aq, then (2.7.1) reduces to the summation formula 4 W3 (a; b; q, l/b) = 0, where Ibl > 1. Setting d = a! in (2.7.1), we get a q-analogue of Dixon's formula (2.1.5) a, -qa'2, b, e qa'2 1 1 1 4¢3 [ 1 ;q,- -a'2, aq/b, aq/e be (aq, aq/be, qa! /b, qa! /e; q)oo (2.7.2) (aq/b, aq/e, qa!, qa! /be; q)oo' provided Iqa! /bel < 1. Watson [1929a] used his transformation formula (2.5.1) to give a simple proof of the famous Rogers-Ramanujan identities (Hardy [1937]):

= (q2,q3,q5;q5)oo, f L (2.7.3) n=O (q; q)n (q; q)oo

qn(n+l) = (q, q4, q5; q5)oo, f (2.7.4) n=O (q; q)n (q; q)oo where Iql < 1. First let b, e, d, e ---+ 00 in (2.5.1) to obtain

~ (a; q)k (1 - aq2k) (q-n; qh q2k 2 (a 2 qn)k 6 (q;q)k (I-a) (aqn+l;q)k n (q-n. q) k = (aq;q)nL 'k (_aqn+l) l(k-l)!2. (2.7.5) k=O (q; qh 2.8 Bailey's transformation formulas 45

Since the series on both sides are finite this limiting procedure is justified as long as the term-by-term limits are assumed to exist. However, our next step is to take the limit n -+ 00 on both sides of (2.7.5), which we can justify by applying the dominated convergence theorem. Thus we have

1 + f (aq; q)k-l(l - aq2k) (_1)ka2k qk(5k-l)/2 k=l (q;q)k

00 k k 2 = (aq; q)oo "a~ -( -. q -) . (2.7.6) k=O q, q k In view of Jacobi's triple product identity (1.6.1) the series on the left side of (2.7.6) can be summed in the cases a = 1 and a = q by observing that

00 (2.7.7) and

00 2:(1- q2k+l)(_1)kqk(5k+3)/2 k=O 00 (2.7.8) n=-oo The identities (2.7.3) and (2.7.4) now follow immediately by using (2.7.6). For an early history of these identities see Hardy [1940, pp. 90-99].

2.8 Bailey's transformation formulas for terminating

5¢4 and 7¢6 series Using Jackson's formula (2.6.2), it can be easily shown that (a, b, c; qh (Abc/a; q)k (q,aq/b, aq/c; q)k (qa2/Abc; qh x t (A;q)j(1- Aq2j)(Ab/a, Ac/a, aq/bc; q)j j=O (q; q)j(l - A)(aq/b, aq/c, Abc/a; q)j

(2.8.1) where A is an arbitrary parameter. If {Ad is an arbitrary sequence, it follows that f (a,b,c;qh A k=O (q, aq/b, aq/c; qh k 46 Summation, transformation, and expansion formulas

= f= (Abc/a; qh Ak t (A; q)j(l - Aq2j) (Ab/a, Ac/a, aq/bc; q)j k=O (qa2/ Abc; qh j=O (q; q)j(l - A)(aq/b, aq/c, Abc/a; q)j x (a;q)k+j(a/A;q)k-j (2::y (Aq; q)k+j(q; q)k-j A = f= (A; q)j(l - Aq2j) (Ab/a, Ac/a, aq/bc; q)j(a; qhj (~)j j=O (q;q)j(1- A)(aq/b,aq/c,qa2/Abc;q)j(Aq;qhj A

00 2· . ,,(aqJ,a/A,AbcqJ/a;qh A ( ) x ~ (q,Aq2Hl,a2qH1/Abc;q)k Hk, 2.8.2 provided the change in order of summation is justified (e.g., if all of the series terminate or are absolutely convergent). It is clear that appropriate choices of A and Ak will lead to transformation formulas for basic hypergeometric series which have at least a partial well­ poised structure. First, let us take Ak = qk(d,q-n;qh/(aq/d,a2q-n/A2;q)k and A = qa2/bcd, so that the inner series on the right of (2.8.2) becomes a balanced and terminating 3¢2. Summing this 3¢2 series and simplifying the coefficients, we obtain Bailey's [1947a,c] formula

A. [ a, b, c, d, q-n ] 5~4 aq/b,aq/c,aq/d,a2q-n/A2;q,q

_ (Aq/a,A2q/a;q)n [A,qA!,-qA!,bA/a,CA/a,dA/a, - ( , '2 / 2 ) 12 ¢11 1 1 Aq, A q a ; q n A2, -A2, aq/b, aq/c, aq/d, a!, -a!, (aq)!, -(aq)!, A2qn+1 la, q-n 1 1 1 1 1 ; q, q , (2.8.3) Aq/a2, -Aq/a2, A(q/a)2, -A(q/a)2, aq-n / A, Aqn+1 where A = qa2/bcd. Note that the 5¢4 series on the left is balanced and nearly-poised of the second kind, while the 12¢11 series on the right is balanced and VWP-balanced. Note also that a terminating nearly-poised series of the second kind can be expressed as a multiple of a nearly-poised series of the first kind by simply reversing the series. By proceeding as in the proof of (2.8.3), one can obtain the following variation of (2.8.3)

q-n, b, c, d, e ] [ 5¢4 q1-n /b, q1-n / C, q1-n / d, eq-2n / A2 ; q, q _ (A2qn+l,Aq/e;q)n ¢ [A,qA!,-qA!,Abqn,ACqn,Adqn, - (A2qn+1/e, Aq; q)n 12 11 A!, -A!, q1-n /b, q1-n /c, q1-n /d, q-n/2, _q-n/2, q(1-n)/2, _q(1-n)/2, e, A2qn+1 / e ] AqHn/2, _ AqHn/2, Aq(1+n)/2, _Aq(1+n)/2, Aq/e, eq-n / A; q, q , (2.8.4) where A = ql-2n /bcd. 2.9 Bailey's transformation formula 47

Next, let us choose Ak = qk(1 - aq2k)(d, q-n; q)k/(1 a)(aq/d, a2q2-n / )...2; q)k and A = qa2/bcd in (2.8.2) so that the inner series on the right side takes the form

aq2j,qJ+1a!,_qJ+1a!,a/A,qj-n [ 1 x 5¢>4 "1 "1 2 "+1 2" +2 2 ; q, q . qJ a '5. , _qJ a '5. , Aq J ,a qJ -n / A

This 5¢>4 series is a special case of the 5¢>4 series on the left side of (2.8.3); in fact, the 12¢>11 series on the right side of (2.8.3) in this case reduces to a termi­ nating 8¢>7 series, which we can sum by Jackson's formula (2.6.2). Carrying out the straightforward calculations, we get Bailey's [1947c] second transformation formula

A, qA! , -qA! ,b.A/ a, CA/ a, dA/ a, (aq)! , -(aq)! , x 12¢>11 [ 1 1 1 1 A'5. ,-A'5., aq/b, aq/c, aq/d, A(q/a) '5. ,-A(q/a)'5., qa!, -qa!, A2 qn-1 la, q-n 1 (2.8.5) A'/.1 a 2 ,-A'/.1 a 2 ,aq2-n/" A,Aq n+1; q, q , where A = qa2 /bcd.

2.9 Bailey's transformation formula for a terminating 1O¢>9 series

One of the most important transformation formulas for basic hypergeometric series is Bailey's [1929] formula transforming a terminating 1O¢>9 series, which is both balanced and VWP-balanced, into a series of the same type: a, qa!, -qa!, b, c, d, e, j, Aaqn+1 /ej, q-n 1 1O¢>9 [ 1 1 1 ;q,q a '5. , -a '5. ,aq/b, aq/c, aq/d, aq/e, aq/ j, ejq-n /A, aqn+ _ (aq, aq/ej, Aq/e, Aq/ j; q)n [A, qA!, -qA!, Ab/a, Ac/a, Ad/a, - (aq/ e, aq/ j, Aq/ej, Aq; q)n 1O¢>9 A! , -A! , aq/b, aq/ c, aq/ d, e,j, Aaqn+1/ej,q-n, ] Aq/e, Aq/ j, ejq-n la, Aqn+1 ; q, q , (2.9.1) where A = qa2 /bcd. To derive this formula, first observe that by (2.6.2) A, qA!, -qA!, Ab/a, Ac/a, Ad/a, aqm, q-m 1 8¢>7 [ 1 1 1 1 ; q, q A'5. ,-A'5., aq/b, aq/c, aq/d, Aq -m la, Aqm+ 48 Summation, transformation, and expansion formulas

(b, c, d, Aq; q)m (2.9.2) (aq/b, aq/ c, aq/ d, a/ A; q)m ' and hence the left side of (2.9.1) equals

~ (a; q)m(l - aq2m)(e, I, Aaqn+l /el, q-n; q)m(a/ A; q)m m m=O~ (q;q)m(l- a)(aq/e,aq/I,elq - n/A,aqn+l;q)m(Aq;q)m q f (A;q)j(l- Aq2j)(Ab/a,Ac/a,Ad/a,aqm,q-m;q)j j x j=O (q; q)j(l - A)(aq/b, aq/c, aq/d, Aql-m la, Aqm+l; q)j q

= t f (a; q)m+j(1- aq2m)(e, I, Aaqn+l /el, q-n; q)m m m=Oj=O (q;q)m-j(l- a)(aq/e,aq/I,elq-n/A,aqn+l;q)m q

x (a/A; ~)m-j(A; :)j(l - Aq2j)(Ab/a, Ac/a, Ad(a; q)j (~)j (Aq, q)m+j(q, q)j(l - A)(aq/b, aq/c, aq/d, q)j A = t ~A; q)j(l - Aq2j)(Ab/a, Ac/a, Ad/a, e, I, Aaqn:~el, q~:l~)j . j=O (q,q)J(l- A)(aq/b,aq/c,aq/d,aq/e,aq/I,elq /A,aq ,q)J aq)j (aq·qh· (2··· . 1 . ) x ( \ (A:) J 8W7 aq J;eqJ,lqJ,a/A,Aaqn+J+ /el,qJ-n;q,q , /\ q, q 2J (2.9.3) where the 8 W 7 series is defined as in §2.1. Summing the above 8 W 7 series by means of (2.6.2) and simplifying the coefficients, we obtain (2.9.1). It is sometimes helpful to rewrite (2.9.1) in a somewhat more symmetrical form:

lOW9 (a; b, c, d, e, I, g, h; q, q) (aq, aq/el,aq/eg, aq/eh, aq/Ig, aq/lh, aq/gh, aq/elgh; q)oo (aq/e,aq/I,aq/g,aq/h,aq/elg, aq/elh, aq/egh,aq/Igh; q)oo

x lO W g (qa 2/bcd; aq/bc, aq/bd, aq/cd, e,J, g, h; q, q) , (2.9.4) where at least one of the parameters e, I, g, h is of form q-n, n = 0,1,2, ... , and (2.9.5)

2.10 Limiting cases of Bailey's lO

a, qa .1.12 , -qa 2 , b, c, d, e, I a 22] q [ 8

A,qA~,-qA~,Ab/a,Ae/a,Ad/a,e,f aq ] XS¢7 [ 1 1 ;q,- , A2, -A2, aq/b, aq/e, aq/d, Aq/e, Aq/ f ef (2.10.1) where A = qa2/bed and

max (Iaq/efl, IAq/efl) < 1. (2.10.2) The convergence of the two series in (2.10.1) is ensured by the inequalities (2.10.2) which, of course, are not required if both series terminate. For exam­ ple, if f = q-n, n = 0,1,2, ... , then (2.10.1) becomes

¢ a, qa .1.12 , -qa 2, b,e, d ,e, q -n . a 2 qn+2] [ S 7 a~, -a~, aq/b, aq/e, aq/d, aq/e, aqn+1 ,q, bcde (aq, Aq/e; q)n (aq/e, Aq; q)n

A, qA~, -qA~, Ab/a, Ae/a, Ad/a, e, q-n. aqn+1 ] x S¢7 [ 1 1 1 ' q, -- . A2, -A2, aq/b, aq/e, aq/d, Aq/e, Aqn+ e (2.10.3)

This identity expresses one terminating VWP-balanced S¢7 series in terms of another. Using (2.5.1) we can now express (2.10.3) as a transformation formula between two terminating balanced 4¢3 series:

rI. [q-n,a,b,e ] 4'1'3 d,e,f ;q,q

_ (e/a, f fa; q)n n rI. [ q-n, a, d/b, d/e . ] - (e, f; q)n a 4'1'3 d, aql-n Ie, aql-n / f' q, q , (2.10.4) where abc = defqn-l. This is a very useful formula which was first derived by Sears [1951c], and hence is called Sears' 4¢3 transformation formula. It is a q-analogue of Whipple's [1926b] formula

F [-n, a, b, e . 1] 4 3 d, e, f' _ (e - a)n (f - a )n F [ -n, a, d - b, d - e . 1] - (e)n(f)n 4 3 d,l+a-e-n,l+a-f-n' , (2.10.5) where a + b + e + 1 = d + e + f + n. Use of (2.5.1) and (2.10.1) also enables us to express a terminating bal­ anced 4¢3 series in terms of a nonterminating S¢7 series. For example, if b, c, or din (2.10.1) is of the form q-n,n = 0,1,2, ... , then the series on the left side of (2.10.1) terminates, but that on the right side does not. In particular, 50 Summation, transformation, and expansion formulas setting d = q-n, and then replacing e and j by d and e, respectively, we obtain

8 W 7 (a; b, e, d, e, q-n; q, a2qn+2 /bede) (aq, aq/de, a2qn+2/bed, a2qn+2/bee; q)oo (aq/d,aq/e, a2qn+2/be, a2qn+2/bede; q)oo

x 8 W 7 (a2qn+1 /be; aqn+1 /b, aqn+1 /e, aq/be, d, e; q, aq/de) , (2.10.6) provided laq/del < 1 to ensure that the nonterminating series on the right side converges. Use of (2.5.1) then leads to the formula ¢ [q-n, a, b, e. ] _ (deqn / a, deqn /b, deqn / e, deqn / abc; q)oo 4 3 d, e, j , q, q - (deqn, deqn / ab, deqn / ae, deqn /be; q)oo

x rI.. [deqn-l, q(deqn-1)!, -q(deqn-1)!, 8'1-'7 1 (deqn-1)2, -(deqn-1)!, a, b, e, dqn, eqn de] ;q,- , (2.10.7) deqn / a, deqn /b, deqn / e, e, d abc provided dej = q1-nabe and Ide/abel < l. As another limiting case of (2.9.1) Bailey [1935, 8.5(3)] found a nontermi­ nating extension of (2.5.1) that expresses a VWP-balanced 8¢7 series in terms of two balanced 4¢3 series. First iterate (2.9.1) to get

10 W g (a; b, e, d, e, j, a3 qn+2 /bedej, q-n; q, q) (aq,aq/de, aq/dj,aq/ej; q)n (aq/d,aq/e, aq/j,aq/dej; q)n

x 10W9 (dejq-n-1/a; aq/be, d, e, j, bdejq-n-1 /a2, edejq-n-1 /a2, q-n; q, q) . (2.10.8)

Clearly, the 10 W g on the left side of (2.10.8) tends to the 8¢7 series on the left side of (2.10.1) as n ---+ 00. However, the terms near both ends of the series on the right side of (2.10.8) are large compared to those in the middle for large n, which prevents us from taking the term-by-term limit directly. To circumvent this difficulty, Bailey chose n to be an odd integer, say n = 2m + 1 (this is not necessary, but it makes the analysis simpler), and divided the series on the right into two halves, each containing m + 1 terms, and then reversed the order of the second series. The procedure is schematized as follows: 2m+1 m 2m+1 L Ak = L Ak + L Ak k=O k=O k=m+1 m m = L Ak + L A2m+1-k' (2.10.9) k=O k=O where {Ad is an arbitrary sequence. Letting m ---+ 00 (and hence n ---+ (0), it follows from (2.10.8) that

rI.. a, qa 12 , -qa 12 , b, e, d, e, j . a 22] q [ 8'1-'7 1 1 ,q,-- a 2 , -a2 , aq/b, aq/e, aq/d, aq/e, aq/ j bedej 2.10 Limiting cases of 1O¢9 transformation formula 51

_ (aq, aq/de, aq/df, aq/ef; q)oo ¢ [ aq/bc, d, e, f . ] - (aq/d, aq/e, aq/ f, aq/def; q)oo 4 3 aq/b, aq/c, def /a' q, q (aq,aq/bc,d,e,f,a2q2/bdef, a2q2/cdef;q)00 +~~--~~~--~--~~~~~~~7-~-- (aq/b,aq/c,aq/d, aq/e,aq/f, a2q2/bcdef,def/aq; q)oo aq/de,aq/df,aq/ef,a2q2/bcdef ] [ (2.10.10) x 4¢3 a2 q2/ bdef, a2 q2/ cdef,aq2/; def q, q , where la2q2/bcdefl < 1, if the 8¢7 series does not terminate. Note that if either b or c is of the form q-n, n = 0,1,2, ... , then the 8¢7 series on the left side terminates but the series on the right side do not necessarily terminate. On the other hand if one of the numerator parameters (except a2q2/bcdef) in either of the two 4¢3 series in (2.10.10) is of the form q-n, then the coefficient of the other 4¢3 series vanishes and we get either (2.5.1) or (2.10.7). If aq/bc, aq/ de, aq/ df or aq/ ef equals 1, then (2.10.10) reduces to the 6¢5 summation formula (2.7.1). If, on the other hand, aq/cd = 1 then the 8¢7 series in (2.10.10) reduces to a 6¢5 which, via (2.7.1), leads to the summation formula (aq,aq/be,aq/bf,aq/ef;q)oo (aq/b,aq/e,aq/f,aq/bef;q)oo _ (aq,c/e,c/f,aq/ef;q)oo ¢ [aq/bc,e,f. ] - (c, aq/e, aq/ f, clef; q)oo 3 2 aq/b, efq/c' q, q (aq,aq/ef,e, f,aq/bc,acq/bef; q)oo +~~-=~--~~--~=-~~~ (aq/e, aq/f,ef/c, c, aq/b, aq/bef;q)oo c/ e, c/ f, aq/bef ] x 3¢2 [ ;q,q . (2.10.11) cq/ef,acq/bef

Solving for the first 3¢2 series on the right and relabelling the parameters we get the following nonterminating extension of the q-Saalschiitz formula

A, [a, b, c ] (q/e, f la, f /b, f /c; q)oo 3'1'2 e, f ; q, q (aq/e, bq/e, cq/e, f;q)oo (q/e, a, b, c, qf Ie; q)oo (e/q, aq/e,bq/e, cq/e, f; q)oo aq/e,bq/e,cq/e ] [ (2.10.12) x 3¢2 q2/ e,q f/ e ; q, q , where ef = abcq. Sears [1951a, (5.2)] derived this formula by a different method. If a, b, or c is of the form q-n, n = 0,1,2, ... , then (2.10.12) reduces to (1.7.2). A special case of (2.10.12) which is worth mentioning is obtained by setting c = 0, f = 0, and then replacing e by c to get ( ) (q/ c, abq/ c; q)oo 2¢1 a, b; c; q, q = ( / b / ) aq c, q c; q 00 (q/c, a, b; q)oo (/ / 2/ ) ( / / b / ) 2¢1 aq c, bq c; q c; q, q . (2.10.13) c q, aq c, q c; q 00 52 Summation, transformation, and expansion formulas

If a or b is of the form q-n, n = 0,1,2, ... , then (2.10.13) reduces to (1.5.3). In general, a 2¢1 (a, b; e; q, q) series does not have a sum which can be written as a ratio of infinite products. However, we can still express (2.10.13) as the summation formula for a single bilateral infinite series in the following way. First, use Heine's transformation formula (1.4.1) to transform both 2¢1 series in (2.10.13): (b, aq; q)oo ( / ) 2¢1 (a, b; e; q, q ) = ( ) 2¢1 e b, q; aq; q, b e, q; q 00

= (b, aq; q)oo f (e/b; q)n bn, (2.10.14) (e,q;q)oo n=O (aq;q)n ,,( / b / . 2/. ) _ (aq/e,bq2/e;q)00 ~ (q/a;q)n (aq)n 2'/'1 aq e, q e, q e, q, q - (2/ .) ~ (b 2/ . ) . q e, q, q 00 n=O q e, q n e (2.10.15) Next, note that

f (q~a; :)n (aq)n = _.:: 1 - bq/e f (l/a;.q)n (aq)n. (2.10.16) n=o(bq/e,q)n e q I-a n=l(bq/e,q)n e

Using (2.10.14) - (2.10.16) and the identity (1.2.28) in (2.10.13) we obtain

(e/b;q)nb (e, q/e, abq/e, q; q)oo , f n = (2.10.17) n=-CX) (aq; q)n (b, aq, aq/e, bq/e; q)oo which is Ramanujan's sum (see Chapter 5 and Andrews and Askey [1978]). However, the conditions under which (2.10.17) is valid, namely, Iql < 1, Ibl < 1 and laq/el < 1, are more restrictive then those for (2.10.13). Note that (2.10.17) tends to Jacobi's triple product identity (1.6.1) when a = 0 and b ---+ O. We shall give an alternative derivation of this important sum in Chapter 5 where bilateral basic series are considered. As was pointed out by AI-Salam and Verma [1982a], both (2.10.10) and (2.10.12) can be conveniently expressed as q-integrals. Thus (2.10.12) is equiv­ alent to l b (qt/a, qt/b, et; q)oo dqt a (dt, et, jt; q)oo

= b(1 _ q) (q, bq/a, alb, e/d, e/e, e/ j; q)oo , (2.10.18) (ad, ae, aj, bd, be, bj; q)oo where e = abdej, while (2.10.10) is equivalent to l b (qt/a, qt/b, et, dt; q)oo dqt a (et,Jt, gt, ht; q)oo = b(1 _ q) (q, bq/ a, alb, ed/ eh, ed/ j h, ed/ gh, be, bd; q)oo (ae, aj, ag, be, bj, bg, bh, bed/h; q)oo x 8 W7 (bed/hq; be, bj, bg, e/h, d/h; q, ah), (2.10.19) 2.11 Bailey's three-term transformation formula for 8¢7 series 53 provided cd = abe j g h and Iah I < 1. By substituting c = abdej into (2.10.18), letting j ----+ 0 and then replac­ ing a, d, e by a, cia, d/b, respectively, we obtain Andrews and Askey's [1981] formula b (qt/a, qt/b; q)oo dqt la (ct/a,dt/b;q)oo b(l - q)(q, alb, bq/a, cd; q)oo (2.10.20) (c, d, bc/a, ad/b; q)oo In view of (1.3.18), (2.10.20) is a q-extension of the beta-type integral

d ( + d)a+b-l j (1 + t/c)a-l(l - t/d)b-l dt = B(a b)":""c_.,..:..-:;---:;-- (2.10.21) -c ' ca-1db- 1 ' which follows from (1.11.8) by a change of variable. The notational compactness of (2.10.18) and (2.10.19) is advantageous in many applications (see, e.g., the next section). In addition, the symmetry of the parameters in the q-integral on the left side of (2.10.19) implies the transformation formula (2.10.1).

2.11 Bailey's three-term transformation formula for VWP-balanced 8¢7 series

The q-integral representation (2.10.19) of an 8¢7 series can be put to advan­ tage in deriving Bailey's [1936] three-term transformation formula for VWP­ balanced 8¢7 series:

A, [ a,qa!,-qa!,b,c,d,e,j . a2q2 ] 8'1-'7 1 1 ,q, -b-d-j a"2, -a"2, aq/b, aq/c, aq/d, aq/e, aq/ j c e (aq,aq/de,aq/dj,aq/ej,eq/c, jq/c, b/a, bej/a; q)oo (aq/d,aq/e,aq/j,aq/dej,q/c,ejq/c, be/a,bj/a; q)oo ej /c, q(ej /c)!, -q(ej /c)!, aq/bc, aq/cd, ej la, e, j. bd] X8¢7 [ 1 1 ,q,- (ej /C)"2, -(ej /c)"2, bej / a, dej la, aq/c, jq/ c, eq/c a b (aq, bq/a, bq/c, bq/d, bq/e, bq/ j, d, e, j; q)oo +-~~~~~~~~~~~~~~~~~~-~ a (aq/b,aq/c,aq/d,aq/e,aq/j,bd/a,be/a,bj/a,dej/a;q)oo (aq/bc, bdej/a2, a2q/bdej;q)oo x --'------'-:------;-:--::--'------:-----::--,,----'--;-----;------'----- (aq/dej, q/c, b2q/a; q)oo b2/a, qba-!, -qba-!, b, bc/a, bd/a, be/a, bj /a. a2q2 ] x 8¢7 [ 1 1 ,q, b d j , ba-"2, -ba-"2, bq/a, bq/c, bq/d, bq/e, bq/ j c e (2.11.1) where Ibd/al < 1 and la2q2/bcdejl < 1. To prove this formula, first observe that by (2.10.19)

8 W 7 (a; b, c, d, e, j; q, a2q2 /bcdej) 54 Summation, transformation, and expansion formulas

aq-dej (aq, d,e,j,aq/be, aq/de, aq/dj,aq/ej;q)oo adejq(l - q) (q, aq/b, aq/e, aq/d, aq/e, aq/ j, dej /aq, aq/dej; q)oo def (t/a, qt/dej, aqt/bdej, aqt/edej; q)oo d x I aq (t/de, t/dj, t/ej, aqt/bedej; q)oo qt. (2.11.2)

Since bdef fa def Idef I j(t) dqt + I j(t) dqt = j(t) dqt, (2.11.3) aq bdef fa aq where j(t) = (t/a, qt/dej, aqt/bdej, aqt/edej; q)oo (2.11.4) (t/de, t/dj, t/ej, aqt/bedej; q)oo ' and bdeffa I aq j(t) dqt bdej(l - q)(q, bdej /a2, a2q/bdej, bq/d, bq/e, bq/ j, bq/a, bq/e; q)oo a(aq/de, aq/dj,aq/ej,bd/a, be/a, bj/a, q/c, b2q/a; q)oo

x 8 W7 (b2/a; b, be/a, bd/a, be/a, bj fa; q, a2q2/bedej) , (2.11.5)

def j(t) dqt = dej(l - q)(q, aq/b, aq/e, eq/e, jq/e, bej la, dej fa; q)oo Ibdeffa (d, e, j, be/a, bj la, aq/be, q/e, ejq/e; q)oo

X 8 W 7 (ej /e; aq/be, aq/ed, ej la, e, j; q, bd/a) , (2.11.6) we immediately get (2.11.1) by using (2.11.5) and (2.11.6) in (2.11.3). The advantage of our use of the q-integral notation can be seen by comparing the above proof with that given in Bailey [1936]. The special case qa2 = bedej is particularly important since the series on the left side of (2.11.1) and the second series on the right become balanced, while the first series on the right becomes a 6¢5 series with sum (aq/ee,aq/ej,ejq/e,q/e;q)oo (aq/e, eq/e, jq/e, aq/eej; q)oo' provided laq/eejl < 1. This gives Bailey's summation formula:

1 1 A, [a,qa"2,-qa"2,b,e,d,e,j ] 8'1-'7 1 1 ;q,q a"2, -a"2 , aq/b, aq/ e, aq/ d, aq/ e, aq/ j b (aq, e, d, e, j, bq/a, bq/e, bq/d, bq/e, bq/ j; q)oo a (aq/b, aq/e, aq/d, aq/e, aq/ j, be/a, bd/a, be/a, bj / a, b2q/a; q)oo b2/a, qba-!, -qba-!, b, be/a, bd/a, be/a, bj /a 1 X8¢7 [ 1 1 ;q,q ba-"2, -ba-"2, bq/a, bq/e, bq/d, bq/e, bq/ j (aq, b/a, aq/ed, aq/ee,aq/ej,aq/de, aq/dj,aq/ej; q)oo (2.11. 7) (aq/e,aq/d,aq/e,aq/j,be/a,bd/a,be/a,bj/a;q)oo ' 2.12 Bailey's four-term transformation formula for balanced 101>9 series 55 where qa2 = bedej, which is a nonterminating extension of Jackson's formula (2.6.2). This can also be written in the following equivalent form b (qt/a, qt/b, t/a!, -t/a!, qt/e, qt/d, qt/e, qt/ j; q)= d I 1 1 qt a (t, bt/a, qt/a'l, -qt/a'l, et/a, dt/a, et/a, jt/a; q)= b(l - q)(q, alb, bq/a, aq/ed, aq/ee, aq/ ej, aq/de, aq/dj, aq/ej; q)= (b, e, d, e, j, be/a, bd/a, be/a, bj fa; q)= (2.11.8)

2.12 Bailey's four-term transformation formula for balanced 10 1>9 series Let us start by replacing a, b, e, d, e, and j in (2.11.8) by A, bqn, Ae/a, Ad/ a, Ae/ a and a/bqn, respectively, to obtain bqn (qt/ A, tql-n /b, t/ A! , -t/ A!, aqt/eA, aqt/dA, aqt/eA, btqn+l / a; q)= d l 1 1 qt A (t, btqn /A, qt/A'l, -qt/A'l, et/a, dt/a, et/a, atq-n /bA; q)= b(l - q)(q, A/b, bq/ A, bq/e, bq/d, bq/e, e, d, e; q)= (a/ A, b, Ae/ a, Ad/a, Ae/ a, alb, be/a, bd/ a, bel a; q)= (b,be/a,bd/a,be/a;q)n (Aq)n (2.12.1) x (bq/a,bq/e,bq/d,bq/e;q)n ~ , where n = 0,1,2,.... Let j, g, h be arbitrary complex numbers such that la3q3/bedejghl < 1. Set p = a3q2/bedejgh, multiply both sides of (2.12.1) by (b2/a; q)n(1- b2q2n /a)(bj la, bg/a, bh/a; q)n (ap)n (q;q)n(1- b2/a) (bq/j, bq/g, bq/h;q)n A and sum over n from 0 to 00. Then the right side of (2.12.1) leads to b(l - q)(q, A/b, bq/ A, bq/e, bq/d, bq/e, e, d, e; q)= (a/ A, b, Ae/ a, Ad/a, Ae/ a, alb, be/a, bd/ a, bel a; q)=

x 1OW9 (b2/a; b,be/a, bd/a, be/a, bf/a, bg/a, bh/a; q,pq) . (2.12.2) The left side leads to two double sums, one from each of the two limits of the q-integral. From the upper limit, bqn, we get b(l - q) (q, bq/ A, abq/ eA, abq/ dA, abq/ eA, b2q/a; q)= (b, a/ A, b2q/ A, be/ a, bd/ a, be/a; q)= x f (b2/a; q)n(1- b2q2n /a)(bj la, bg/a, bh/a; q)n (a:)n n=O (q; q)n (1 - b2/ a) (bq/ j, bq/ g, bq/ h; q)n /\

~ (b2/ A; qhn+j (1- b2q2n+2j / A)(b, be/a, bd/a, be/a; q)n+j(a/ A; q)j j x ~ q j=O (q; q)j(l- b2/A)(bq/A, abq/c>.., abq/dA, abq/eA; q)n+j(b2q/a; qhn+j b(l - q)(q, bq/A, abq/c>.., abq/dA, abq/eA, b2q/a; q)= (b, a/ A, b2q/ A, be/a, bd/ a, bel a; q)=

~ (b2/A;q)m(1- b2q2m/A)(b,be/a, bd/a, be/a, a/A; q)m m x ~ q m=O (q; q)m(1- b2/A)(bq/A, abq/c>.., abq/dA, abq/eA, b2q/a; q)m

X SW7 (b2/a; bj/a, bg/a, bh/a, b2qm/A,q-m; q,pq) . (2.12.3) 56 Summation, transformation, and expansion formulas

Let us now assume that p = 1, that is,

a3 q2 = bedeigh. (2.12.4) Then, by Jackson's formula (2.6.2), the s(P7 series in (2.12.3) has the sum

(b2q/a,aq/ig,aq/ih,aq/gh;q)~ (bq/ i, bq/ g, bq/ h, aq/ i gh; q)~ and the expression in (2.12.3) simplifies to

b(l - q)(q, bq/ A, abq/eA, abq/dA, abq/eA, b2q/a; q)oo (b, a/ A, b2q/ A, be/ a, bd/ a, be/a; q)oo

X lOW9 (b2/A; b,be/a, bd/a, be/a, bi/A, bg/A, bh/A; q,q) , (2.12.5) since, by (2.12.4), aq/ig = bh/A, aq/ih = bg/A, and aq/gh = bf/A. We now turn to the double sum that corresponds to the lower limit, A, in the q-integral (2.12.1). This leads to the series -A(l - q)(q, aq/e, aq/d, aq/e, Aq/b, bAq/a; q)oo (b, Aq, alb, eVa, dVa, eVa; q)oo ~ Wfa; q)n(1- b2q2n /a)(b, bi la, bg/a, bh/a, b/ A; q)n n X ~ q n=O (q; q)n(1- b2 /a)(bq/a, bq/ i, bq/g, bq/h, bAq/a; q)n

X SW7 (A; bqn, eVa, dVa, eA/a, aq-n /b; q, q) -A(l - q)(q, aq/e, aq/d, aq/e, Aq/b, bAq/a; q)oo (b, Aq, a/b, eA/a, dA/a, eA/a; q)oo ~ (A; q)~(1- Aq2~)(b, eVa, dVa, eVa, alb; q)~ ~ x ~ q ~=O (q; q)~(1- A)(Aq/b, aq/e, aq/d, aq/e, bAq/a; q)~

x SW7 (b2/a;bq~,bq-~/A,bi/a,bg/a,bh/a;q,q). (2.12.6) The last s(P7 series in (2.12.6) is balanced and nonterminating, so we may use (2.11.7) to get

s W 7 (b2 fa; bq~, bq-~ /A, bi la, bg/a, bh/a; q, q) (b2q/a, Aq~+l / i, Aq~+l /g, Ag~+l /h, aq/ ig, aq/ ih, aq/gh, aq~ /b; q)oo (bAq~+l la, bq/ i, bq/g, bq/h, a/A, iq~, gq~, hq~; q)oo + aq~ (b2q/a, bq-~ /A, bi la, bg/a, bh/a, aq~+l /b, aq~+l / i; q)oo b (bql-~ la, bAq~+l la, bq/ i, bq/g, bq/h, a/ A, iq~, gq~; q)oo X (aq~+l /g, aq~+l /h, Aq2~+1; q)oo (hq~, aq2~+1; q)oo X SW7 (aq2~; bq~, iq~, gq~, hq~, a/A; q, q). (2.12.7)

Use of this breaks up the double series in (2.12.6) into two parts:

-A(l - q)( q, b2q/a, Aq/b, Aq/ i, Aq/ g, Aq/h, bi / A, bg / A; q)oo (b, i, g, h, bq/ i, bq/ g, bq/h, Aq; q)oo 2.12 Bailey's four-term transformation formula for balanced 101>9 series 57

(bh/A,aq/c,aq/d,aq/e;q)= x-';--c-:-----:---:-=-;'------'--:-:-=-';-----'-----:=-:-:---'---"':---'--'­ (a/A,cA/a,dA/a, eA/a; q)= x lOW9 (A; b, cA/a, dA/a, eA/a, j,g, h; q, q) aA(l - q)(q, b/ A, Aq/b, b2q/a, aq/b, aq/c, aq/d, aq/e, aq/ j, aq/g; q)= (b, a/b,aq, cA/a, dA/a, eA/a, bq/a, bq/j,bq/g,bq/h; q)= (aq/h, bj/a, bg/a, bh/a;q)= x ..:...... ::-'----':,...... ::.,:,----'::--=....:....,,-c---,:-'-.:.....::..:...- (a/ A, j, g, h; q)= ~ (a;q)n(1- aq2n)(b,j,g,h,a/A;q)n n x ~ q n=O (q;q)n(1- a)(aq/b,aq/j,aq/g, aq/h, Aq; q)n x 8Wd A; cA/a, dA/a, eA/a, aqn, q-n; q, q). (2.12.8) Summing the last 81>7 series by (2.6.2) we find that the sum over n in (2.12.8) equals lOW9(a;b,c,d,e,j,g,h;q,q) which is, of course, balanced by virtue of (2.12.4). Equating the expression in (2.12.2) with the sum of those in (2.12.5) and (2.12.8), and simplifying the coefficients, we finally obtain Bailey's [1947b] four-term transformation formula 1 1 A-. [ a,qa'2,-qa'2,b,c,d,e,j,g,h ] 10'1'9 a'2,-a'2,aq1 1 / b,aq / c,aq / d,aq / e,aq / j,aq / g,aq /;h q, q (aq,b/a,c,d,e,j,g,h,bq/c,bq/d;q)= + (b2q/a,a/b,aq/c, aq/d,aq/e,aq/j,aq/g, aq/h, bc/a, bd/a; q)= (bq/ e, bq/ j, bq/g, bq/ h; q)= x~-,;---,--:-=::..,::--:-,~,-,-:-:-:--:-.:....::..:,.- (be/a, bjla, bg/a, bh/a; q)= b2la, qba-~, -qba-~, b, bc/a, bd/a, be/a, bj la, bg/a, bh/a 1 x 101>9 [ 1 1 ;q,q ba- '2, -ba- '2, bq/ a, bq/ c, bq/ d, bq/ e, bq/ j, bq/ g, bq/ h (aq, b/a, Aq/j,Aq/g, Aq/h, bj/A,bg/A,bh/A; q)= (Aq,b/A, aq/j,aq/g, aq/h,bj/a, bg/a, bh/a;q)= A,qA~,-qA~,b,Ac/a,Ad/a,Ae/a,j,g,h 1 x 101>9 [ 1 1 ; q, q A'2, -A'2, Aq/b, aq/c, aq/d, aq/e, Aq/ j, Aq/g, Aq/h (aq,b/a,j,g,h,bq/j,bq/g,bq/h,Ac/a,Ad/a;q)= + (b2q/A,A/b,aq/c,aq/d,aq/e,aq/j,aq/g,aq/h,bc/a,bd/a;q)= (Ae/a,abq/Ac,abq/Ad,abq/Ae;q)= x --'------'-:-::----:---',--::-.,------:-----'-;------:-:,----;-'------:------'--- (be/a, bj la, bg/a, bh/a; q)= b2 /A, qbA-~, -qbA-~, b, bc/a, bd/a, be/a, bj /A, bg/A, bh/A 1 x 101>9 [ 1 1 ; q, q . bA-'2, -bA-'2, bq/A, abq/cA, abq/dA, abq/eA, bq/ j, bq/g, bq/h (2.12.9) In terms of the q-integrals this can be written in a more compact form: b (qt/a, qt/b, ta-~, -ta-~, qt/c, qt/d, qt/e, qt/ j, qt/g, qt/h; q)= d I 1 1 qt a (t, bt/a, qta-'2, -qta-'2, ct/a, dt/a, et/a, jt/a, gt/a, ht/a; q)= a (b/a,aq/b, Ac/a, Ad/a, Ae/a, bj/A, bg/A, bh/A;q)= A (b/A, Aq/b, c, d, e, bj la, bg/a, bh/a; q)= 58 Summation, transformation, and expansion formulas

b 1 1 (qt/ A, qt/b, tA -"2, -tA -"2, aqt/d, aqt/dA, aqt/e>.., qt/ f, qt/ g, qt/h; q)oo d x 1 1 1 qt, A (t, btl A, qtA -"2, -qtA -"2, ct/ a, dt/ a, et/ a, ft/ A, gt/ A, ht/ A; q)oo (2.12.10) where A = qa2/ede and a3q2 = bedefgh.

Exercises 2.1 Show that

3

2.6 Show that q-n b e _q1-n /be ] [ 4

2.7 Derive Jackson's terminating q-analogue of Dixon's sum:

2.8 Show that

q-n, a, b'i,1 -b'i 1 ] 4¢3 [ I-n! I-n! b; q, q q 2 a2, -q 2 a2, 0, n = 2m+ 1, } = { (q, bqja; q2)m n = 2m (bq, qja; q2)m' , where n, m = 0,1, .... (Andrews [1976a])

2.9 Prove that

2.10 The q-Racah polynomials, which were introduced by Askey and Wilson [1979], are defined by q-n abqn+1 q-X cqx-N ] Wn(x;a,b,c,N;q) 4¢3 [ ' b' ;q,q, = aq, cq, q -N

where n = 0,1,2, ... ,N. Show that

(aqjc,bq;q)n n ( -1) Wn ( x; a, b, c, N; q ) = ( b ) c Wn N - x; b, a, c ,N; q . aq, cq;qn

2.11 The Askey- Wilson polynomials are defined in Askey and Wilson [1985] by

Pn(x;a,b,c,d I q) q-n abcdqn-1 aeiO ae-iO ] = a-n(ab,ac,ad;q)n 4¢3 [ ' b 'd' ;q,q , a ,ac,a

where x = cos (). Show that (i) Pn(x; a, b, c, d I q) = Pn(x; b, a, c, d I q), (ii) Pn( -x; a, b, c, d I q) = (-l)npn(x; -a, -b, -c, -d I q).

2.12 Show that lOWg(a;b'i,-b'i,(bq)'i,-(bq)'i,ajb,a1 1 1 1 2 qn +1 jb,q-n;q,q) (aq,a2qjb2;q)n 2 jb' ) ,n = 0,1,2, .... ( aq jb ,a q ,q n 60 Summation, transformation, and expansion formulas

2.13 If A = qa2/bcd and IqA/al < 1, prove that

(i) ¢ [ a,b,c,d . qA2] _ (Aq/a,A2q/a;q)oo 4 3 aq/b,aq/c,aq/d,q, a2 - (Aq,A2q/a2;q)oo

x lOW9 (A;a~,-a~,(aq)~,-(aq)~'Ab/a'Ac/a'Ad/a;q,qA/a),

( •• ) A.. [ a, b, c, d -Aq] 11 4 '/"3 aq/b, aq/ c, aq/ d; q, ------;;- (aq, -q, Aqa-~, -Aqa-~; q)oo (Aq, -Aq/a, qa!, -qa!; q)oo

X 8 W7 (A; a~, -a~, Ab/a, Ac/a, Ad/a; q, -q) .

2.14 (i) Show that

which is a q-analogue of Bailey [1935, 4.5(1.3)].

(ii) Using (i) in the formula (2.8.2) prove the following q-analogue of Bailey [1935,4.5(4)]:

1 A.. [ a, qa'2, b, c, d, q-n ] 6 '/"5 1 ;q,q a'2,aq/b,aq/c,aq/d,a2ql-n/A2 (A/a, A2 la, -Aqa-~; q)n (Aq, A2 /a2, -Aa-~; q)n 1 1 A, qA'2, -qA'2, Ab/a, Ac/a, Ad/a, [ x 12¢U 1 1 A'2, -A'2, aq/b, aq/c, aq/d,

where A = qa2/bcd. This formula is equivalent to Jain's [1982, (4.6)] transformation formula.

2.15 By taking suitable q-integrals of the function

f(t) _ (qt/b, qt/c, aqt/bc, tq2/bcdef; q)oo - (at, qt/bcd, qt/bce, qt/bcf; q)oo ' Exercises 61

prove Bailey's [1936, (4.6)] identity

-1 (aq/d,aq/e,aq/j,q/ad,q/ae,q/aj;q)= a (qa2, ab, ac, bfa, cia; q)= x 8W7 (a 2; ab, ac, ad, ae, aj; q, q2/abcdej) + b-1 (bq/d, bq/e, bq/ j, q/bd, q/be, q/bj; q)= (qb2,ba,bc,a/b,c/b;q)= x 8W7 (b2;ba,bc,bd,be,bj;q,q2/abcdej)

-1 (cq/d,cq/e,cq/j,q/cd,q/ce,q/cj;q)= + c (qc2, ca, cb, a/c, b/c; q)= x 8W7 (c2;ca,cb,cd,ce,cj;q,q2/abcdej) = 0, provided Iq2/abcdejl < 1. 2.16 (i) Let S(A, p" v, p) = (A, q/ A, p" q/p" v, q/v, p, q/ p; q)=. Using Ex. 2.15, prove that S(XA, x/ A, p,v, p,/v) - S(xv, x/v, Ap" p,/ A) p, = >..S(xp" x/p" AV, A/v),

where x, A, p" v are non-zero complex numbers. (Sears [1951c,d], Bailey [1936]) (ii) Deduce that (q!, q!; q)= S(Aq!, Aq~, -p,q!, -p,q~)

= (q! Ap" q! / Ap" q! A/ p" q! p,/ A; q)=

- q! (A + A-I - P, - p,-1 ) (qAp" q/ Ap" qA/ p" qp,/A; q)=. (Ismail and Rahman [2002b]) 2.17 Show that A, qA!, -qA!, a, b, c, -c, Aq/C2 Aq] (i) 8(P7 [ 1 1 2; q, --b A2, -A2, Aq/a, Aq/b, Aq/C, -Aq/C, c a (Aq, C2/A; q)=(aq, bq, c2q/a, c2q/b; q2)= (Aq/a, Aq/b; q)=(q, abq, c2q, c2q/ab; q2)='

where A = -c(ab/q)! and IAq/abl < 1, and

(ii) [-c, q( -c)!, -q( -c)!, a, q/a, c, -d, -q/d ] 8¢7 1 1 ;q,C (-C)2, -( -C)2, -cq/a, -ac, -q, cq/d, cd (-c, -cq; q)=(acd, acq/d, cdq/a, cq2/ad; q2)= Icl < 1. (cd, cq/d, -ac, -cq/a; q)= Verify that (i) is a q-analogue of Watson's summation formula (Bailey [1935,3.3(1)]) while (ii) is a q-analogue of Whipple's formula (Bailey [1935, 3.4(1)]). (See Jain and Verma [1985] and Gasper and Rahman [1986]). 62 Summation, transformation, and expansion formulas

2.18 In the 6(/15 summation formula (2.7.1) let b, c, d ----t 00. Then set a = 1 to prove Euler's [1748] identity

00 1 + L (_l)n { qn(3n-1)/2 + qn(3n+1)/2} = (q; q)oo. n=1 2.19 Show that

10 W g (a; b, c, d, e, f, g, q-n; q, q) (aq,aq/ce,aq/de,aq/ef,aq/eg,b;q)n n = e (aq/c, aq/d,aq/e, aq/f,aq/g, b/e;q)n x lOW9 (eq-n /b; e, aq/bc, aq/bd, aq/bf, aq/bg, eq-n la, q-n; q, q) , where a3qn+2 = bcdefg and n = 0,1,2, ....

2.20 Prove that

10 W g (a; b, c, d, e, f, g, q-n; q, a3qn+3/bcdefg) = (aq,aq/fg;.q)n t (q-n,f,g,aq/de~~)j~j (aq/f,aq/g,q)n j=O (q,aq/d,aq/e,fgq /a,q)j q-j, d, e, aq/bc ] [ x 4¢3 aq / b,aq / c,deq-J0/ a ; q, q , for n = 0,1,2, ....

2.21 Show that lOW9 (a; b, c, d, e, f, g, h; q, a3q3/bcdefgh) = f . (A; q)n(l - Aq2n) (Ab/a, Ac/a, Ad/a, e,j, g, h; q)n\aq; qhn. n=O (q, q)n(1- A)(aq/b, aq/c, aq/d, aq/e, aq/ f, aq/g, aq/h, q)n(Aq, qhn

x (:;;:) n SW7 (aq2n; a/A,eqn, fqn,gqn, hqn; q,a3q3/bcdefgh) ,

where A = qa2/bcd and la3q3/bcdefghl < 1.

2.22 Prove that f . (a; q)n(1- aq2n)(b, c, d, e; q)n . (_ a2q2 ) n qG) n=O (q, q)n(1- a)(aq/b, aq/c, aq/d, aq/e, q)n bcde

= (aq,aq/de;q)oo A, [aq/bc,d,e. aq ] I /d I 1 3'1-'2 /b /' q, ,aq e < . ( aq / d, aq / e; q) 00 aq, aq c de Deduce that f ~a; q)n(1- aq2n)(d, e; q?n (_ adqe)n qG) n=O (q, q)n(1- a) (aq/d, aq/e, q)n (aq, aq/de; q)oo (aq/d, aq/e; q)oo . Exercises 63

2.23 Prove that t (ab, ae, ad; q)j qj j=O (abed,aqz, aq/z; q)j (1 - z/a)(l - abez) = (1 _ bz)(l _ ez) S W7 (abez; ab, ae, be, qz/d, q; q, dz) (ab, ae, ad; q)n+1 (1 - aqn+1 / z) (1 - abezqn+1) (abed, aqz, aq/ z; q)n+1 (1 - a/ z) (1 - abez) x SW7 (abezqn+1; abqn+1, aeqn+\ be, qz/d, q; q, dz) . 2.24 Show that ¢ [ a, b, e, d, e . ] _ ()..q/a, )..q/e, q)..2/a, q/ j; q)oo 5 4 aq/b, aq/e, aq/d, f' q, q - ()..q, aq/ j, eq/ j, aq/ )..j; q)oo

X 12Wll ()..;a!,-a!, (aq)!,-(aq)!,)"b/a,)..e/a,)"d/a,e,aq/j;q,q) (a,e,a/)..,q/j,)..q2/j;q)oo (j/q,)..q,aq/j,aq/)..j,eq/j;q)oo x f f ()..; ~)j(l - )..q2j) ()"b/a, )..e/a, )"d~a; q)j j=Ok=O (q,q)j(l- )..)(aq/b,aq/e, aq/d, q)j x (f/q,aq/j;q)j (aqj+1/j, aq1-j/)..j,eq/j;q)k qHk, ()..q2 / j,)..j fa; q)j (q, q2-J / j, )..q2+J / j; q)k

where ).. = qa2/bed and j = ea2/)..2. Note that this reduces to (2.8.3) when e = q-n, n = 0,1,2, ....

2.25 By interchanging the order of summation in the double sum in Ex. 2.24 and using Bailey's summation formula (2.11.7), prove Jain and Verma's [1982, (7.1)] transformation formula

A-. [ a, b, e, d, e ] 5'1'4 aq/b, aq/e, aq/d, j; q, q + (a,b,e,d,e,q/j,aq2/bj,aq2/ej,aq2/dj;q)oo (aq/b,aq/e,aq/d,j/q,aq/j,bq/j,eq/j,dq/j,eq/j;q)oo

[ eq/ j, aq/ j, bq/ j, eq/ j, dq/ j ] X 5¢4 q2 / j, aq2/bj, aq2/ej, aq2/dj; q, q ()..q/a,)..q/e, q)..2/a, q/j;q)oo ()..q,aq/j,eq/j,aq/)..j;q)oo x 12Wll ()..;a!,-a!,(aq)!,-(aq)!,)"b/a,)..e/a,)"d/a,e,aq/j;q,q) (a,e,)"b/a,)..e/a,)"d/a,q/j,a2q2/)"bj,a2q2/)..ej,a2q2/)"dj,aq3/j2;q)oo +~~~~~~~~~~~~~~~~~~~~~~ (aq/b,aq/e,aq/d,aq/j,bq/j,eq/j,dq/j,eq/j,)..j/aq,a2q3/)..j2;q)oo X 12 Wll (a 2q2 / )..j2; qa~ / )..j, -qa~ / )..j, (qa) ~ / )..j, -(qa) ~ / )..j, )..q/a, aq/ j, bq/ j, eq/ j, dq/ j; q, q), 64 Summation, transformation, and expansion formulas

where the parameters are related in the same way as in Ex. 2.24. Note that this is a nonterminating extension of (2.8.3) and that the first 5¢4 series on the left is a nearly-poised series of the second kind while the second 5¢4 series is a nearly-poised series of the first kind.

2.26 If a = q-n, n = 0,1,2, ... , prove that

¢ [ a, b, c . aqx] _ (ax; q)(X) 3 2 aq/b, aq/c' q, bc - (x; q)(X) a!, -a!, (aq)!, -(aq)!, aq/bc ] x 5¢4 [ ;q,q . aq/b,aq/c,ax,q/x (Sears [1951a, (4.1)], Carlitz [1969a, (2.4)]) 2.27 Show that ~ [q-n,c,ab/c,a1, ... ,ar . ] (c,ab/c;q)n r+3'1'r+2 a b b b ,q, z (b) , , 1,···, r a, ; q n ~ (q-n, cia, c/b; qh k ~ [qk-n, c, a1, ... , ar ] x ~ ( -lb- 1-n ) q r+2'1'r+1 k b b; q, z . k=O q, C, ca 1q; q k cq , 1,···, r

2.28 Show that a,b1, ... ,br,c,q-n ] (aq/cz,q1-n/c;q)n [ r+3¢r+2 /b /b / n+1 ; q, qz (/ 1-n/ ) aq 1, ... , aq To aq C, aq aq c, q cz; q n

X n (q-n/CZ ,l/Z,q-n/a,q-n;q)k (1- q2k-n/CZ ) (aqn+1)k (q, aq/cz, q/CZ, q1-n /C; qh (1 - q-n /cz) c Lk=O a,b1, ... ,br,czq-k,qk-n ] [ X r+3¢r+2 / / k+1/ n+1-k ; q, q . aq b1, ... ,aq br,aq cZ,aq 2.29 Show that

2.30 Iterate (2.12.9) to prove Bailey's [1947b, (8.1)] transformation formula: loVVg(a;b,c,d,e,j,g,h;q,q) (aq,c,d,e,j,g,h,b/a,bq/c,bq/d, bq/e; q)(X) + (qb2/a, bc/a, bd/a, be/a, bj/a, bg/a,bh/a, a/b, aq/c, aq/d,aq/ e;q)(X)

(bq/j,bq/g,bq/h;q)(X) (b 2 / b b / bd/ b / bj/ b / bh/ ) x ( /j / /h) 10 VVg a;, c a, a, e a, a, 9 a, a; q, q aq ,aq g, ag ; q (X) Exercises 65

(aq,b/a,g, bq/g,aq/ch,aq/dh,aq/eh,aq/fh,bch/a, bdh/a;q)oo (bhq/g, bh/a,g/h, aq/h, aq/c, aq/d, aq/e,aq/f,bc/a, bd/a; q)oo (beh/a, bfh/a; q)oo x (be/a,bf/a;q)oo 10VVg(bh/g; b,aq/cg, aq/dg, aq/eg, aq/fg, bh/a, h; q,q)

(aq, b/a, h,bq/h,aq/cg, aq/dg,aq/eg,aq/fg,bcg/a,bdg/a;q)oo +~~~ __ ~~~~~~~~~~~~~~~~~~_ (bgq/h,bg/a, h/g,aq/g, aq/c, aq/d,aq/e, aq/f,bc/a, bd/a; q)oo (beg/a, bfg/a; q)oo x (be/a, bf fa; q)oo lOVVg(b9/h; b, aq/ch, aq/dh, aq/eh, aq/ fh, bg/a, g; q, q),

where a3q2 = bcdefgh. 2.31 By using the q-Dixon formula (2.7.2) prove that the constant term in the Laurent expansion of (XdX2' XdX3; q)al (X2/X3, qX2/X1; q)a2 (qx 3/X1, qX3/X2; q)a3 is

where a1, a2 and a3 are nonnegative integers. (Andrews [1975a]) 2.32 Use (2.10.18), the q-binomial theorem, and the generating function in Ex. 1.29 to derive the formula 2iO 2iO C ( e. (31 ) = 2i sin e ((3, (3, (3e , (3e- ; q)oo n cos, q (2 2 '0 2 '0 ) 1 - q q, (3 ,e ' ,e- , ; q 00

ie ((32;q)n l e - (qteiO,qte-iO;q)oo n x () ('0 _ '0 ) t dqt, 0 < e < 1f. q; q n eie (3te', (3te ' ; q 00 (Rahman and Verma [1986a]) 2.33 (i) Prove that

6+2kVVS+2k(a; b, alb, d, e1, ... ,ek, aqnl +1 /e1, ... , aqnk +1 /ek; q, q1-N /d) = (q, aq, aq/bd, bq/d; q)oo Ilk (aq/bej, bq/ej; q)nj , k ( / / /) ( / /) = 1,2, ... , bq,aq b,aq d,q d;q 00 j=l aq ej,q ej;q nj

where n1, ... , nk are nonnegative integers, N = n1 +-. +nk, and I q1-N /dl < 1 when the series does not terminate. (Gasper [1997]) (ii) Deduce that (be iO , be-iO;p)oo (aqeiO , aqe-iO ; q)oo

00 (k+1) , , 1 - a2q2k (a2q)k (_l)kq 2 (ae'O, ae-'o; q)k(abqk, bq-k /a;p)oo = ~ 1- a2 (q;q)k (q,qa2;q)00 (aqeiO,aqe-iO;q)k when 0 < p < q, or p = q and Ibl < lal. (Ismail and Stanton [2003a]). 66 Summation, transformation, and expansion formulas

2.34 Derive the formulas

where Re "( > Re A > 0 in (i) and (ii), and Re (A, v, "( + J-l- A - v) > 0 in (iii). These formulas are q-integral analogues of Erdelyi's [1939, equations (17), (11), and (20), respectively] fractional integral representations for 2Fl series. (Gasper [2000])

2.35 Derive the following discrete extensions of the formulas in Ex. 2.34:

(Gasper [2000]) 2.36 Extend the formulas in Ex. 2.35 to: (i) 4¢3(a, (3, vqn, q-n; ,,(, 8, p; q, q) Notes 67

= b/)..,vqh;q)n)..n b,)..vqh;q)n x ~ ()..vh,)..,8/1,j0!/3,/1pI0!/3, vqn,q-n;qh()..vqh;qhk (8(J)k ~ (q, vqh, p, 8, )..q1-n h, )..vqn+1 h; q)k()..vh; qhk /1V k=O x 4cP3(/110!, /11/3, )..vqk h, q-k;).., 8/110!/3, /1pIO!/3; q, q) x 4cP3(0!/31/1, /11).., vqk+n, qk-n; "(I).., 8qk, pqk; q, q), (ii) 4cP3(0!, /3, vqn, q-n; "(, 8, p; q, q) = b/)..,vqh;q)n)..n b,)..vqh;q)n ~ ()..vl,,(,)..,pI0!,81(J, vqn,q-n;q)k()..vql,,(;q)2k X (q, vqh, 8, P(J IO!, )..q1-n h, v)..qn+1 h; q)k()..vh; qhk k=O x 4cP3(0!1(J, /3, )..vqk h, q-k;).., 81(J, p; q, q) X 4cP3((J, (31).., vqnH, qk-n, "(I).., 8qk, (Jpqk IO!; q, q), (iii)

A, [q-n,V,a1,a2, ... ,ar . ] r+2 'f's+l b b b ' q, z "(, 1, 2,···, s

[I/1 v,"( I v,q k-n 1-k ] [ q -k ,/1,a1,···,ar ] X3cP2 "(/1I)..V,q1-nlv;q,q I).. r+2cPs+1 )..,b1,b2, ... ,bs ;q,z ,

where O!/3vq = 8"(p in (i) and (ii). (Gasper [2000])

Notes

§2.5 Some applications of Watson's transformation formula (2.5.1) to mock theta functions are presented in Watson [1936, 1937]. §2.7 For additional proofs of the Rogers-Ramanujan identities, identities of Rogers-Ramanujan type, applications to combinatorics, Lie algebras, statis­ tical mechanics, etc., see Adiga et al. [1985], Alladi [1997], Alladi, Andrews and Berkovich [2003], Alladi and Berkovich [2002]' Andrews [1970b, 1974b,c, 1976, 1976b, 1979b, 1981a, 1984b,d, 1986, 1987a,b, 1997, 2001], Andrews, Askey, Berndt et al. [1988], Andrews and Baxter [1986, 1987], Andrews, Baxter, Bres­ soud et al. [1987], Andrews, Baxter, and Forrester [1984]' Andrews, Schilling and Warnaar [1999], Bailey [1947a, 1949, 1951], Baxter [1980-1988], Baxter and Andrews [1986], Baxter and Pearce [1983, 1984], Berkovich and McCoy [1998], Berkovich, McCoy and Schilling [1998], Berkovich and Paule [2001a,b], Berkovich and Warnaar [2003], Berndt [1985-2001]' Borwein and Borwein [1988], Bressoud [1980a, 1981a,b, 1983a], Bressoud, Ismail and Stanton [2000], Burge [1993], Dobbie [1962]' Dyson [1988], Fine [1988], Foda and Quano [1995], Garrett, Ismail and Stanton [1999], Garsia and Milne [1981]' Garvan [1988], 68 Summation, transformation, and expansion formulas

Lepowsky [1982]' Lepowsky and Wilson [1982]' McCoy [1999], Misra [1988], Paule [1985a], Ramanujan [1919]' Rogers and Ramanujan [1919]' Schilling and Warnaar [2000], Schur [1917]' Sills [2003a,b,c], Slater [1952a], Stanton [2001a,b], Warnaar [1999-2002a, 2003a-2003e], and Watson [1931]. §2.9 Agarwal [1953e] showed that Bailey's transformation (2.9.1) gives a transformation formula for truncated 8¢7 series, where the sum of the first N terms of an infinite series is called a truncated series. §2.10 Many additional transformation formulas for hypergeometric series are derived in Whipple [1926a,b]. §2.11 Additional transformation formulas for 8¢7 series are derived in Agarwal [1953c]. Ex.2.6 Also see the summation formulas for very-well-poised series in Joshi and Verma [1979]. Ex. 2.31 This exercise is the n = 3 case of the Zeilberger and Bres­ soud [1985] theorem that if Xl, ... , X n , q are commuting indeterminates and al, ... ,an are nonnegative integers, then the constant term in the Laurent expansion of II (xi/Xj; q)ai (qXj/Xi; q)aj l~i

(q; q)al +·+an

(q; q)al ... (q; q)a n • This was called the Andrews' q-Dyson conjecture because Andrews [1975a] had conjectured it as a q-analogue of a previously proved conjecture of Dyson [1962] that the constant term in the Laurent expansion of II (1 _Xi) ai l~i#-j~n Xj is equal to the multinomial coefficient (al+···+an)! al!'" an! The n = 4 case of the Andrews' q-Dyson conjecture was proved indepen­ dently by Kadell [1985b]. Additional constant term results are derived in Baker and Forrester [1998, 1999], Bressoud [1989], Bressoud and Goulden [1985], Cherednik [1995], Cooper [1997a,b], Evans, Ismail and Stanton [1982]' Forrester [1990], Kadell [1994, 1997, 2000], Kaneko [1996-2001], Macdonald [1972-1998b], Morris [1982]' Opdam [1989], Stanton [1986b, 1989], Stembridge [1988], and Zeilberger [1987-1994]. 3

ADDITIONAL SUMMATION, TRANSFORMATION, AND EXPANSION FORMULAS

3.1 Introduction In this chapter we shall use the summation and transformation formulas of Chapters 1 and 2 to deduce additional transformation formulas for basic hy­ pergeometric series which are useful in many applications. In §3.2 and §3.3 we shall obtain q-analogues of some of Thomae's [1879] 3F2 transformation formulas, typical among which are

p. [-n, a, b .1] = (d - b)n p. [ -n, c - a, b .1] (311) 3 2 C, d' (d)n 3 2 c,1 + b - d - n' , .. n = 0,1,2, ... ,

a,b,c ] = f(d)f(e)f(s) D [d-a,e-a,s.l] (312) 3F2 [ d,e;1 f(a)f(s+b)f(s+c) 3r 2 s+b,s+c' , .. s = d + e - a - b - c, and

p. [a, b, c. 1] f(1 - a)r(d)r(e)f(c - b) 3 2 d, e ' f(d - b)f(e - b)f(1 + b - a)f(c)

D [b, b - d + 1, b - e + 1 1] ·d (b) (3 1 3) x 3 r 2 1 + b _ c, 1 + b _ a; + 1 em ; c, .. where the symbol "idem (b; c)" after an expression means that the preceding expression is repeated with band c interchanged. The main topic of this chapter, however, will be the q-analogues of a large class of transformations known as quadratic transformations. Two functions J(z) and g(w) are said to satisfy a quadratic transJormation if z and w iden­ tically satisfy a quadratic equation and J(z) = g(w). Among the important examples of quadratic transformation formulas are

a (a a+ 1 4z ) (l+z) 2 Fl(a,b;l+a-b;-z) = 2Fl 2'-2--b;l+a-b;(I+z)2 , (3.1.4)

a (a a+ 1 1 Z2) (1 - z) 2Fl(a, b; 2b; 2z) = 2Fl -, --; b + -; ( )2' (3.1.5) 2 2 2 l-z 1 Z2) (1 - z)a 2Fl (2a, a b; 2a 2b; z) = 2Fl ( a, b; a b -2; (3.1.6) + + + + 4(z-l) ,

69 70 Additional summation, transformation, and expansion formulas

1 1 2FI(2a, 2b; a + b + "2; z) = 2FI(a, b; a + b + "2; 4z(1 - z)), (3.1. 7)

(1 - z)a 3F2 [1 + a _ab~i c+ a _ c; z]

= Po 4z ] [~a,~(a+1),1+a-b-c._ (3.1.8) 3 2 1 + a _ b, 1 + a - c ' (1 - z)2 '

(l-z)a+1 [ a,l+~a,b,c ] -'------'----- 4 F3 I ;z 1 + z "2a, 1 + a - b, 1 + a - c _ [ ~ (a + 1), 1 + ~ a, 1 + a - b - c. 4Z] - 3 F 2 ,-. (3.1.9) l+a-b,l+a-c (l-z)2 The above definition of a quadratic transformation cannot be directly applied to basic hypergeometric series. For example, the a = q-n case of the identity in Ex. 2.26 is a q-analogue of the a = -n case of (3.1.8), but it does not fit into the above definition of a quadratic transformation. So we shall just say that a transformation between basic hypergeometric series is "quadratic" if it is a q-analogue of a quadratic transformation for hypergeometric series. It will be seen that one important feature of the quadratic transformations derived for basic hypergeometric series in the following sections is that the series obtained from an r

3.2 Two-term transformation formulas for 3

In general, a convergent 32 series 71 with n = 0,1,2, .... Keeping n fixed and choosing special or limiting values of one of the other parameters leads to transformation formulas for terminating 31>2 series. Let us consider this class of formulas first.

Case (i) Letting c ----t °in (3.2.1) we get

A, [q-n,a,b. ] _ (e/a;q)n n A, [q-n,a,d/b. bq] (3.2.2) 3'1-'2 d , e ,q, q - (.)e, q n a 3'1-'2 d , aq 1-n/'e q, e . Note that the series on the left is of type I and that on the right is of type II. Formula (3.2.2) is a q-analogue of (3.1.1).

Case (ii) Letting a ----t °in (3.2.1) gives

A, [q-n,b,c. ] _ (de/bc;q)n (bC)n A, [q-n,d/b,d/C. ] 3'1-'2 d ' q, q - (.) d 3'1-'2 d d /b ' q, q. (3.2.3) ,e e, q n , e c

If we let c ----t °in (3.2.3) we obtain

21>1 (q-n, d/b; d; q, bq/e) = (-ltq-G)(e;q)ne-n 31>2 [q-;:~,O;q,q], which may be written in the form (Ex. 1.15(i))

( .. )_ (abz/c;q)oo [a,c/b,o. ] 21>1 a, b, c, q, z - (bz/c; q)oo 31>2 c, cq/bz' q, q , (3.2.4) where a = q-n, n = 0, 1,2, ....

Case (iii) Let c ----t 00 in (3.2.1). This gives Sears' [1951c, (4.5)] formula

A, [q-n,a,b. deqn ] _ (e/a;q)n A, [q-n,a,d/b. ] 3'1-'2 d ,q, b - (.) 3'1-'2 d 1-n/' q, q . (3.2.5) , e a e, q n , aq e Note that there is no essential difference between (3.2.2) and (3.2.5) since one can be obtained from the other by a change of parameters.

Case (iv) Replacing a by aqn in (3.2.1) and simplifying, we get q-n , aqn, b, c ] [ 41>3 d ,e, a bcq /de ;q,q

= (aq/e, de/bc; q)n (bC)n 1> [q-n,aqn,d/b,d/C. ] (e, abcq/de; q)n d 4 3 d, de/bc, aq/e ,q, q .

Set d = 'xc and then let c ----t 00. In the resulting formula we replace ,x, e and abq/'xe by c, d and e, respectively, to get

A, [ q-n , aqn, b . de] 3'1-'2 d ,q'--b ,e a

= (aq/d, aq/e; q)n (de)n 31>2 [q-n,aqn,abq/de;q,fJ.] (3.2.6) (d, e; q)n aq aq/d, aq/e b 72 Additional summation, transformation, and expansion formulas which is a transformation formula between two terminating 3¢2 series of type II. Let us now consider the class of transformation formulas that connect two nonterminating 3¢2 series.

Case (v) In (3.2.1) let us take n ----; 00. A straightforward term-by-term limiting process gives the formula a,b,e. ~] _ (e/a,de/be;q)oo [a,d/b,d/e.~] 3¢2 [ ,q, b - ( d / b ) 3¢2 /' q, . (3.2.7) d, e ace, e a e; q 00 d, de be a Apart from the general requirement that no zero shall appear in the denomina­ tors of the two 3¢2 series, the parameters must be restricted by the convergence conditions: Ide/abel < 1 and le/al < 1. This formula is a q-analogue of the Kummer-Thomae-Whipple formula F [a,b,e. 1] = r(e)r(d+e-a-b-e) F [a,d-b,d-e .1] ( ) 3 2 d,e' r(e-a)r(d+e-b-e) 3 2 d,d+e-b-e' , 3.2.8 where Re (e - a) > 0 and Re (d + e - a - b - c) > o.

Case (vi) Iterating (3.2.1) once gives q-n, a, b, e ] [ 4¢3 d ,e, a beq I-n/d e ;q,q _ (b, del ab, de/be; q)n [q-n, d/b, e/b, del abc. ] (3.2.9) - (d, e, de/abc; q)n 4¢3 de/ab, de/be, ql-n /b' q, q .

Let us assume that max (Ibl, Ide/abel) < 1. Then, taking the limit n ----; 00, we obtain Hall's [1936] formula

A. [ a, b, e . de ] 3~2 d,e ,q, abc

= (b,de/ab,de/be;q)oo ¢ [d/b,e/b,de/abe. b] (3.2.10) (d, e, de/abc; q)oo 3 2 de/ab, de/be ,q, . Note that this is a q-analogue of formula (3.1.2). Before leaving this section it is worth mentioning that by taking the limit n ----; 00 in Watson's formula (2.5.1), we get another transformation formula: ¢ [aq/be, d, e. aq] _ (aq/d, aq/e; q)oo 3 2 aq/b, aq/e' q, de - (aq, aq/de; q)oo ~ (a;q)k(l- aq2k)(b,e,d,e;qh (k) ( a2q2 )k x ~ q 2 --- (q;q)k(1 -a)(aq/b,aq/e,aq/d,aq/e;q)k bede' k=O (3.2.11) provided laq/del < 1. This is a q-analogue of the formula 3 F2 [ 1 + a - b - e, d, e .1] = r(1 + a)r(1 + a - d - e) 1 + a - b, 1 + a - e' r(1 + a - d)r(1 + a - e)

a,I+~a,b,e,d,e ] x 6F5 [ 1 ; -1 , (3.2.12) "2a,1 + a - b, 1 + a - e, 1 + a - d, 1 + a - e 3.3 Three-term transformation formulas for 3¢2 series 73 where Re (1 + a - d - e) > 0; see Bailey [1935, 4.4(2)].

3.3 Three-term transformation formulas for 3¢2 series

In (2.10.10) let us replace a,b,e,d,e,j by AqN,BqN,C,D,E,FqN, respec­ tively, and then let N ----; 00. In the resulting formula replace C, D, E, Aq/B and Aq/F by a, b, e, d and e, respectively, to obtain

A. [a, b, e. ~] _ (e/b, e/e; q)oo A. [d/a, b, e. ] 3'1'2 d ,q, b - ( /b ) 3'1'2 / ,q, q , e a e e, e e; q 00 d, beq e

(d/ a, b, e, de/be; q)oo [e/b, e/ e, de/abe ] (3.3.1) + (d, e, be/e, de/abe; q)oo 3¢2 de/be, eq/be ; q, q , where Ide/abel < 1, and be/e is not an integer power of q. This expresses a 3¢2 series of type II in terms of a 3¢2 series of type 1. As a special case of (3.3.1), let a = q-n with n = 0,1,2, .... Then

A. [q-n, b, e . deqn ] 3'1'2 d, e ' q, ----,;;;-

_ (e/b, e/e; q)oo A. [b, e, dqn . ] 3'1'2 / ' q, q - (e, e /b e; q) 00 beq e, d

(b, e; q)oo (de/be; q)n ¢ [e/b, e/e, deqn /be. ] (3.3.2) + (e, be/e; q)oo (d; q)n 3 2 eq/be, de/be ,q, q . Setting n = 0 in (3.3.2) gives the summation formula (2.10.13). We shall now obtain a transformation formula involving three 3¢2 series of type II. We start by replacing a, b, e, d, e, j in (2.11.1) by AqN, BqN, C, D, E, FqN, respectively, and then taking the limit N ----; 00. In the resulting formula we replace C, D, E, Aq/ B, Aq/F by a, b, e, d, e, respectively, and obtain

[a, b, e de] 3¢2 d,e ;q, abe

_ (e/b, e/e, eq/ a, q/ d; q)oo ¢ [e, d/ a, eq/ e . bq] - (e, eq/d, q/a, e/be; q)oo 3 2 eq/a, beq/e' q, d

(q/ d, eq/ d, b, e, d/ a, de/beq, beq2 / de; q)oo ¢ [aq/ d, bq/ d, eq/ d . de ] - (d/q, e, bq/d, eq/d, q/a, e/be, beq/e; q)oo 3 2 q2/d, eq/d ,q, abe ' (3.3.3) provided Ibq/dl < 1, Ide/abel < 1 and none of the denominator parameters on either side produces a zero factor. If Iql < Ide/abel < 1, then

A. [a, b, e . de ] 3'1'2 d,e ,q, abe

_ (e/b, e/e, q/d, bq/a, eq/a, abeq/de; q)oo ¢ [q/a, d/a, e/a. abeq] - (e, e/be, q/a, bq/d, eq/d, beq/e; q)oo 3 2 bq/a, eq/a ,q, de 74 Additional summation, transformation, and expansion formulas

(b, c, ql d, dl a, eql d, delbcq, bcq2 I de; q)oo rP [aql d, bql d, cql d . de ] - (e, elbc, qla, bqld, cqld, bcqle, dlq; q)oo 3 2 q2ld, eqld ,q, abc ' (3.3.4) by observing that from (3.2.7) C,dla,cqle. bq] [ 3rP2 cq I a, bcq I e ,q'-d _ (abcql de, bql a; q)oo rP [ql a, dl a, el a . abcq] - (bcqle,bqld;q)oo 3 2 bqla,cqla ,q, de .

If we set e = '\c in (3.3.3), let c ----+ 0 and then replace d and ,\ by c and abzlc, respectively, where Izl < 1, Ibqlcl < 1, then we obtain (abzlc, qlc; q)oo ( I I I I ) 2rPl ( a, b; c; q, z ) = ( I I ) 2rPl C a, cq abz; cq az; q, bq c az c, q a; q 00 (b, qlc, cia, azlq, q2 laz; q)oo ,/, ( I b I 21 ) ( ) 2'1'1 aq c, q c; q c; q, z. 3.3.5 (c I q, bq I c, q I a, az I c, cq I az; q) 00 Sears' [1951c, p. 173] four-term transformation formulas involving 3rP2 se­ ries of types I and II can also be derived by a combination of the formulas obtained in this and the previous section. Some of these transformation for­ mulas also arise as special cases of the more general formulas that we shall obtain in the next chapter by using contour integrals.

3.4 Transformation formulas for well-poised 3rP2 and very-well-poised 5rP4 series with arbitrary arguments

Gasper and Rahman [1986] found the following formula connecting a well­ poised 3rP2 series with two balanced 5rP4 series: a, b, c aqx] [ 3rP2 aq I b, aq I c ;q, -b- c

_ (ax; q)oo ,/, [a!, -a!, (aq)!, -(aq)!, aqlbc. ] - ( ) 5'1'4 ,q, q x; q 00 aqlb, aqlc, ax, qlx (a,aqlbc,aqxlb,aqxlc;q)oo + (aqlb, aqlc, aqxlbc, X-I; q)oo

,/, [xa!,-xa!,x(aq)!,-x(aq)!,aqxlbc ] x 5'1'4 ; q, q . (3.4.1) aqxlb,aqxlc,xq,ax2 Convergence of the 3rP2 series on the left requires that laqxlbcl < 1. It is also essential to assume that x does not equal q±j, j = 0,1,2, ... , because of the factors (x; q)oo and (x-\ q)oo appearing in the denominators on the right side of (3.4.1). Note that if either a or aqlbc is 1 or a negative integer power of q, 3.4 Transformation formulas of well-poised 3¢>2 and 5¢>4 series 75 then the coefficient ofthe second 5¢>4 series on the right vanishes, so that (3.4.1) reduces to the Sears-Carlitz formula (Ex. 2.26). An important application of (3.4.1) is given in §8.8. To prove (3.4.1) we replace d by dqn in (2.8.3) and then let n ----7 00. This gives

¢> [ a, b, e . d] _ (bed/aq; q)oo 3 2 aq/b, aq/e' q, ~ - (bed/qa2; q)oo x lim 12W11 (a2q1 - n/bed; a!, -a!, (aq)!, -(aq)!,aql-n/be, n---+oo aql-n /bd, aql-n /ed, a3q3-n /b2e2d2, q-n; q, q) . (3.4.2)

To take the limit on the right side of (3.4.2) it suffices to proceed as in (2.10.9) to obtain

'" [d,-d,(aq)!,-(aq)!,aq/be ] = 5'1'4 ;q,q aq/b,aq/e,bed/aq,a2q2/bed

bed (bed/ a2, bd/ a, ed/ a, aq/be, a; q)oo qa2 (d/a,aq/b,aq/e,bed/a,a2q2/bed;q)oo

d/a, bed/qa~, -bed/qa~ , bed/q! a~, -bed/q! a~ ] x 5¢>4 [ ; q, q . (3.4.3) bd/a,ed/a,bed/a2,b2e2d2/qa2 Using this in (3.4.2) and replacing d by qxa2/be, we get (3.4.1). If we now replace d by dqn in (2.8.5) and then let n ----7 00, we obtain the transformation formula

'" [ a,qa!,-qa!,b,e. x(aq)!] 5'1'4 1 1 ,q, -b- a"2, -a"2, aq/b, aq/e e (1 - x 2)(xq(aq)!; q)oo (x(aq)-!;q)oo

(aq)!, -(aq)!, qa!, -qd, aq/be ] x 5¢>4 [ 1 1 ;q,q aq/b,aq/e,xq(aq)"2,q(aq)2/x

(aq,aq/be,x(aq)!/b,x(aq)!/e;q)oo + 1 1 (aq/b, aq/e, x(aq) "2 /be, (aq)2 Ix; q)oo

X, -x, xq!, -xq! , x(aq)! /be ] x 5¢>4 [ 1 1 1;q, q . (3.4.4) x(aq)"2/b,x(aq)"2/e,x(q/a)"2,qx2 In terms of q-integrals formulas (3.4.1) and (3.4.4) are equivalent to

a,b,e aqx] [ 3¢>2 aq/b, aq/e; q, bc 76 Additional summation, transformation, and expansion formulas

(a, aq/bc; q)oo s(l - q) (q, aq/b, aq/ c, q/ x, X; q)oo

8 (qu/xs, qui s, aqu/bs, aqu/cs, axu/ s; q)oo d x 1 1 1 1 1 qU, 8X (ua"2 / s, -ua"2 / s, u(aq) "2 / s, -u(aq) "2 / s, aqu/bcs; q)oo (3.4.5) and

a, qa"2,1 -qa"2, 1 b, c. x(aq) "21 ] 5¢4 [ 1 1 , q, -b- a"2, -a"2, aq/b, aq/c c

(1 - x 2) (aq, aq/bc; q)oo s(l - q)(q, aq/b, aq/c, x(aq)-~, q(aq)~ Ix; q)oo

8 (uq(aq) ~ / sx, qui s, aqu/bs, aqu/cs, uxq(aq) ~ / S; q)oo d x 1 1 1 1 1 qU, x(aq)-~ (u(aq) "2 / s, -u(aq) "2 / s, uqa"2 / s, -uqa"2 / s, aqu/bsc; q)oo (3.4.6)

respectively, where s -=I=- 0 is an arbitrary parameter. If we now set c = (aq)~ in (3.4.5), replace x by x/b(aq)~, and use (2.10.19), then we get

2 (xq/b, aqx2 /b2; q)oo 2¢1(a, b;aq/b; q,qx/b ) = ( /b 2/F) aqx ,qx ,q 00

ax/b, q(ax/b)~, -q(ax/b)~, x, a!, -a~, (aq)~, -(aq)~ qx] x 8¢7 [ 1 1 1 1 1 1; q, 2 (ax/b) "2 ,-(ax/b)"2, aq/b, xqa"2 /b, -xqa"2 /b, x(aq) "2 /b, -x(aq) "2 /b b (3.4.7) 2 provided Iqx/b 1 < 1 when the two series do not terminate. Similarly, setting c = (aq)~ and replacing x by x/bq in (3.4.6) we obtain ¢ [a,qa~,-qa!,b . ....::...] _ (ax2/b2,x/qb;q)00 4 3 a~, -a~, aq/b ,q, qb2 - (aqx/b, x 2 /qb2; q)oo

ax/b, q(ax/b)~, -q(ax/b)~, (aq)~, -(aq)~, qa~, -qa~, x x ] X 8¢7 [ 1 1 1 1 1 1 ; q, -2 ' (ax/b) "2 ,-(ax/b)"2, x(aq) "2 /b, -x(aq)"2 /b, xa"2 /b, -xa"2 /b, aq/b qb (3.4.8)

provided Ix/qb2 1 < 1 when the series do not terminate. 3.5 Quadratic transformations 77

3.5 Transformations of series with base q2 to series with base q

If in Sears' summation formula (2.10.12) we set b = -c, e = -q, replace a by aqr, r = 0, 1, 2, ... , multiply both sides by (x2,y2;q2)r b2rqr (-q; q)r(x2y2b2; q2)r and then sum over r from 0 to 00, we get

(-1,-q,ab,-ab,b2;q)00 A,. [ a2,x2,y2 .2 b2] 3'1-'2 2b2 2 2b2 , q , q ( a, b , -a, - b , b , -b) ; q 00 a, x y

00 j 2 2 ] _ (_q ' ab2. , q) 00 (a ",b -b· q) J" j [-2q,x,y. 2 2 - ( b -b·) b2.) "q 3¢2 x 2y2b2 q2-2j jb2 , q , q a" ,qooj=oL (_q, q,a ,qJ '

j 2 + (-q,-ab2;q)00 ~ (-a,-b,b;q)j j A,. [ q-2 ,x ,y2 . 2 2] ·) ~ ( b2. ) q 3'1-'2 22b2 2-2j jb2 ,q,q ( -a, -b , b , q 00" q, -q, -a , q J" x y , q J=O (3.5.1) 2 assuming that Iqb 1 < 1 when the series on the left is nonterminating. Since the two 3¢2 series on the right side can be summed by the q-Saalschiitz formula (1.7.2) with the base q replaced by q2, it follows from (3.5.1) that

_ (-a, ab2; q)oo A,. [a, bx, -bx, by, -by. ] - (-1, b2; q)oo 5'1-'4 -q, ab2, bxy, -bxy' q, q

(a, -ab2; q)oo A,. [-a, -bx, bx, -by, by. ] (3.5.2) + (-1,b2;q)00 5'1-'4 -q,-ab2,-bxy,bxy,q,q .

Note that one of the terms on the right side of (3.5.2) drops out when a = ±q-n, n = 0,1,2, .... Setting y = ab and using (2.10.10) gives

( 2 2 2 2 4 2 2) (b2,a2b2x2;q2)00(ab2,b2x2;q)00 2¢1 a , x ; a x b ; q , qb = ( 2b2 b2 2. 2) (b2 b2 2. ) a , x,q 00 ,a x,q 00 X 8 W 7(ab2x 2 jq; a, x, -x, bx, -bx; q, ab2), (3.5.3)

2 2 where Iqb 1 < 1 and lab 1 < 1 when the series do not terminate. Byapply­ ing Heine's transformation formula (1.4.1) twice to the 2¢1 series above and replacing b by q! jb we find that

2¢1 (a2, b2; a2q2 jb2; q2, x 2q2 jb4)

(qa2x2jb2,q2a2x2jb4;q2)00(aqjb2,qx2jb2;q)00 (qx2jb2,q2x2jb4;q2)00(qa2jb2,aqx2jb2;q)00 78 Additional summation, transformation, and expansion formulas

(3.5.4) provided laq/b2 1 < 1 and Ixq/b2 1 < 1 when the series do not terminate. This formula was derived by Gasper and Rahman [1986], and a terminating version of it was given earlier by Verma [1980]. Application of the transformation formula (2.10.1) to the 8(P7 series on the right of (3.5.4) yields an equivalent formula

(qa2x 2/b 2, q2a2x 2/b 4; q2)oo (-xq~ /b, -axq~ /b3; q)oo (qx2 /b2, q2x 2/b 4 ; q2)oo( -axq~ /b, -a2xq~ /b3; q)oo

-xa2q~ /b3, q( -xa2q~ /b3) ~ , -q( -xa2q~ /b3) ~, a, [ x 8eP7 2 1 3 1 2 1 3 1 3 3 (-xa q2/b )2, -(-xa q2/b )2, -axq2/b,

aq~ /b, -aq2/b,1 -aq/b,2 -xq2III /b . xq2 (3.5.5) -aqx/b2, aqx/b2, axq~/b, qa2/b2 ,q'--b- ,

2 where Ixq/b 1 < 1 and Ixq~ /bl < 1 when the series do not terminate. It is clear that formula (3.5.5) is a q-analogue of the quadratic transformation formula 1 -4x 2Fl (a, b; 1 + a - b; x 2) = (1 - x) -2a 2Fl (a, a + - - b; 2a + 1 - 2b; ( )2). 2 I-x (3.5.6)

We shall now prove the following transformation formula due to Jain and Verma [1982]:

eP [ a, q2a~, -q2a~, b, e, eq, d, dq, e, eq 2 a3q3 ] 10 9 a~, -a~, aq2/b, aq2/e, aq/e, aq2/d, aq/d, aq2/e, aq/e; q , be2d2e2

(aq, aq/ed, aq/ee, aq/de; q)oo [ (aq/b) ~, -(aq/b) ~,e, d, e 1 = (aq/e, aq/d, aq/e, aq/ede; q)oo 5eP4 ( )1 ( )1 /b d / ; q, q aq 2, - aq 2, aq ,e e a + (aq2, a3q3 /e2d2e2; q2)oo(e, d, e, a2q2 /bede; q)oo (aq2/b,a3q3/be2d2e2;q2)oo(aq/e,aq/d,aq/e,ede/aq;q)oo (a3q3/be2d2e2)~,-(a3q3/be2d2e2)~,aq/ed,aq/ee,aq/de 1 x 5eP4 [ ; q, q , (a3q3/e2d2e2)~,-(a3q3/e2d2e2)~,a2q2/bede,aq2/ede (3.5.7) 3.5 Quadratic transformations 79 with the usual understanding that if the 104>9 series on the left does not ter­ minate then the convergence condition la3q3/bc2d2e21 < 1 must be assumed to hold. First we rewrite (2.10.12) in the form

(aq4n+l, aq/cd, aq/ce, aq/de; q)oo 00 (cq2n, dq2n, eq2n; q)r r (cq2n, dq2n, eq2n, aql-2n / cde; q)oo ~ (q, aq4n+l, cdeq2n / a; q)r q

+ (a 2q2n+2/cde; q)oo 00 (aq/cd, aq/ce, aq/de; q)r qr (cdeq2n-l fa; q)oo ~ (q, a2q2n+2 /cde, aq2-2n /cde; q)r

(aq/c, aq/d, aq/e; q)oo (c, d, e; qhn (3.5.8) (c, d, e; q)oo (aq/c, aq/d, aq/e; qhn' where n is a nonnegative integer. Using (1.2.39) and (1.2.40), multiplying both sides of (3.5.8) by (a,b;q2)n(1-aq4n) ( a3q3 )n (q2, aq2/b;q2)n(1-a) bc2d2e2 ' and summing over n from 0 to 00, we get loVVg(a;b,c,cq,d,dq,e,eq;q2,a3q3/bc2d2e2) (aq, aq/cd, aq/ce, aq/de; q)oo f (a, b; q2)n(1- aq4n) - (aq/c, aq/d, aq/e, aq/cde; q)oo n=O (q2, aq2/b; q2)n(1- a)

( d ) 3 ) n [ cq2n, dq2n, eq2n ] c, ,e; q 2n n(2n-l) ( aq x (cde/a; qhn(aq; q)4n q b 34>2 aq4n+1, cdeq2n /a; q, q

(c,d, e,a2q2/cde; q)oo ~ (a,b;q2)n(1 _aq4n) + (aq/ c, aq/ d, aq/ e, cde/ aq; q)oo t:o (q2, aq2/b; q2)n (1- a)

(cde/aq; qhn ( a3q3 ) n [aq/cd, aq/ce, aq/de ] x (a 2q2/cde;q)2n bc2d2e2 34>2 a2q2n+2/cde,aq2-2n/cde;q,q . (3.5.9) The first double series on the right side of (3.5.9) easily transforms to

f (c,d,e;q)m qm 6 VV5 (a;b,ql-m,q-m;q2,aq2m+l/b) , m=O (q, aq, cde/a; q)m which, by (2.4.2), equals c, d, e, (aq/b)!, -(aq/b)! 1 54>4 [ 1 1 ; q, q . aq/b, cde/a, (aq)2, -(aq)2 Similarly we can express the second double series on the right side of (3.5.9) as a single balanced 54>4 series. Combining the two we get (3.5.7). The special case of (3.5.7) that results from setting e = (aq)! is particu­ larly interesting because both 54>4 series on the right side become balanced 44>3 80 Additional summation, transformation, and expansion formulas

series which, via (2.10.10), combine into a single 8cP7 series with base q. Thus we have the formula

a, q2a~, -q2a~, b, e, eq, d, dq 2 a2q2 ] [ 8cP7 1 1 2 2 2 ; q , b 2d2 a>, -a>, aq /b, aq /e, aq/e, aq /d, aq/d e

(aq, aq/be, aq/ed, -aq/ed, aq/db~, -aq/db~; q)oo (aq/b, aq/ e, aq/d, -aq/d, aq/edb~, -aq/edb~; q)oo

-aid, q( -a/d)~, -q( -a/d)~, e, b~, -b~, (aq)~ /d, -(aq)~ /d aq ] x 8cP7 [ 1 1 1 1 1 1 ; q, -b ' (-a/d) >, -( -a/d» , -aq/ed, -aq/db> , aq/db> , -(aq) > , (aq) > e (3.5.10) 2 2 2 where la q2/be d 1 < 1 and laq/bel < 1 when the series do not terminate.

3.6 Bibasic summation formulas

Our main objective in this section is to derive summation formulas containing two independent bases. Let us start by observing that when d = a/be Jackson's 8cP7 summation formula (2.6.2) reduces to the following sum of a truncated series

1 - aq2k (a, b, e, a/bc; q).k = (aq, bq, eq, aq/be;?)n , t l (3.6.1) 1 - a (q, aq/b, aq/e, beq, qh (q, aq/b, aq/e, beq, q)n k=O

where n = 0,1, .... Notice that this series telescopes, for if we set 0"-1 = 0 and (aq,bq,eq,aq/be;q)k O"k = (3.6.2) (q,aq/b,aq/e,beq;q)k

for k = 0,1, ... , and apply the difference operator ~ defined by ~Uk = Uk - Uk-l to O"k, then we get

A (1 - aq2k)(a, b, c, a/bc; qh k 1...l.00k = q (3.6.3) (1 - a)(q, aq/b, aq/e, beq; qh ' which gives (3.6.1), since n L~Uk = Un - U-l (3.6.4) k=O for any sequence {Uk}. These observations and the bibasic extension (ap,bp;p)k(eq,aq/be;q)k Tk = (3.6.5) (q,aq/b; q)k(ap/c, bep;p)k of O"k were used in Gasper [1989a] to show that

A (1 - apkqk)(1 - bpkq-k) (a, b;p)k(e, a/bc; q)k k I...l.~= q (3.6.6) (1 - a)(1 - b) (q, aq/b; qh(ap/c, bep; ph ' 3.6 Bibasic summation formulas 81 which, by (3.6.4), gave the indefinite bibasic summation formula t (1 - apkqk)(l - bpkq-k) (a, b;ph(e, a/be; qh qk (1 - a)(l - b) (q, aq/b; q)k(ap/e, bep;p)k k=O (ap, bP;P)n(eq, aq/be; q)n (3.6.7) (q,aq/b;q)n(ap/c,bep;P)n for n = 0,1, .... Notice that the part of the series on the left side of (3.6.7) containing the q-shifted factorials is split-poised in the sense that aq = b(aq/b) and e(ap/e) = (a/be) (bep) = ap, while the expression on the right side is balanced and well-poised since (ap) (bp) (eq) (aq/be) = q( aq/b) (ap/ e) (bep) and (ap)q = (bp)(aq/b) = (eq)(ap/e) = (aq/be)(bep). The b ----; °case of (3.6.7) ..[!--. 1 - apkqk (a;p)de; q)k -k (ap;P)n(eq; q)n -n ~ -----:----,-----:---,,-:-----,-----:-----,----:--e = e (3.6.8) k=O 1- a (q;q)k(ap/e;p)k (q;q)n(ap/e;P)n is due to Gosper. To derive a useful extension of (3.6.7), Gasper and Rahman [1990] set (ap,bp;p)k(eq, ad2q/be; q)k Sk = (dq, adq /)b; q k ( adp / e, bep / d;p )k (3.6.9) for k = 0, ±1, ±2, ... , and observed that /}.Sk = Sk - Sk-l (ap, bp;ph-l (eq, ad2q/be; qh-l (dq,adq/b; q)k(adp/e, bep/d;p)k x {(I - apk)(l - bpk)(l - eqk)(l - ad2qk /be) -(1 - dqk)(l - adqk /b)(l - adpk /e)(l - bepk /d)} d(l - e/d)(l - ad/be)(l - adpkqk)(l - bpk /dqk) (1 - a)(l - b)(l - e)(l - ad2/be) (a, b;p)k(e,ad2/be; q)kqk x ( dq, adq /)b; q k ( adp / e, bep /)d; p k (3.6.10) Since (3.6.4) extends to n L /}.Uk = Un - U-'m-l, (3.6.11) k=-'m where we employed the standard convention of defining a'm + a'm+l + ... + an, m:::::n, n { L ak = 0, m = n+ 1,

k='m -(an+l + an+2 + ... + a'm-d, m ~ n+2, (3.6.12) 82 Additional summation, transformation, and expansion formulas for n, m = 0, ±1, ±2, ... , it follows from (3.6.10) that (3.6.7) extends to the indefinite bibasic summation formula ~ (l-adpkqk)(l-bpk/dqk) (a,b;ph(e,ad2 /be;q)k k (1 - ad)(l - bid) (dq, adq/b; q)dadp/e, bep/d;p)k q k=-rn~ (1 - a)(l - b)(l - e)(l - ad2 /be) d(l- ad)(l - b/d)(l - e/d)(l - ad/be) 2 { (ap, bp;p)n(eq, ad q/be; q)n (e/ad, d/be;p)rn+l(l/d, b/ad; q)rn+l} x (dq, adq/b; q)n(adp/e, bep/d; P)n - (l/e, be/ad2 ; q)rn+l (l/a, l/b; P)rn+l (3.6.13) for n, m = 0, ±1, ±2, ... , by applying the identity (1.2.28). Observe that (3.6.7) is the case d = 1 of (3.6.13) and that the right side of (3.6.9) is balanced and well-poised since (ap) (bp) (eq)( ad2 q/be) = (dq) (adq/b) (adp/ c) (bep/ d) and (ap)(dq) = (bp)(adq/b) = (eq)(adp/e) = (ad 2 q/be)(bep/d). It is these observations and the factorization that occurred in (3.6.10) which motivated the choice of Sk in (3.6.9). If Ipl < 1 and Iql < 1, then by letting n or m tend to infinity in (3.6.13) we find that (3.6.13) also holds with n or m replaced by 00. In particular, this yields the following evaluation of a bilateral bibasic series ~ (1 - adpkqk) (1 - bpk / dqk) (a, b; ph (e, ad2 /be; q)k k (1 - ad)(l - bid) (dq, adq/b; q)k(adp/e, bep/d;ph q k=-oo~ (1 - a)(l - b)(l - e)(l - ad2 /be) d(l - ad)(l - b/d)(l - e/d)(l - ad/be) 2 { (ap, bp; p)oo (eq, ad q/be; q)oo (e/ad, d/be;p)oo(1/d, b/ad; q)oo} x (dq, adq/b; q)oo(adp/e, bep/d; p)oo - (l/e, be/ad2 ; q)oo(1/a, l/b; p)oo ' (3.6.14) where Ipi < 1 and Iql < 1. In §3.8 we shall use the m = °case of (3.6.13) in the form ~ (1 - adpkqk)(l- bpk /dqk) (a, b;p)k(e, ad2 /be; q)k k (1- ad)(l- bid) (dq, adq/b; qh(adp/e, bep/d;ph q k=O~ (1 - a)(l - b)(l - e)(l - ad2 /be) d(l - ad)(l - b/d)(l - e/d)(l - ad/be) (ap,bp;p)n(eq,ad2q/be;q)n X~'-----::---;-:,------'--,-----;-----::----,------'::-----;-:'------:-- (dq, adq/b; q)n(adp/e, bep/d; P)n (1 - d)(l - ad/b)(l - ad/c) (1 - be/d) (3.6.15) d(l - ad)(l - b/d)(l - e/d)(l - ad/be)· There is no loss in generality since, by setting k = j - m in (3.6.13), it is seen that (3.6.13) is equivalent to (3.6.15) with n, a, b, e, d replaced by n + 3.6 Bibasic summation formulas 83 m, ap-m , bp-m, cq-m, dq-m, respectively. We shall also use the special case c = q-n of (3.6.15) in the form

t (1 - adpkqk)(l - bpk /dqk) (a, b;p)k(q-n, ad2qn /b; qh k (1 - ad)(l - bid) (dq, adq/b; qh(adpqn, bp/dqn;ph q k=O (1 - d)(l - ad/b)(l - adqn)(l - dqn /b) (3.6.16) (1 - ad)(l - d/b)(l - dqn)(l - adqn /b)' where n = 0,1, .... The d ----+ 1 limit case of (3.6.16) ~ (1 - apkqk)(l - bpkq-k) (a, b;ph(q-n, aqn /b; qh k _ 6 (3.6.17) ~ (1- a)(l- b) (q, aq/b; q)dapqn, bpq-n;p)k q - n,O, k=O where 6n ,m is the Kronecker delta function and n = 0,1, ... , was derived inde­ pendently by Bressoud [1988], Gasper [1989a], and Krattenthaler [1996]. If we replace n, a, band k in (3.6.17) by n - m, apmqm, bpmq-m and j - m, respectively, we obtain the orthogonality relation n L anjbjm = 6n,m (3.6.18) j=m with (-l)n+j(l- apJqj)(l- bpJq-j)(apqn,bpq-n;P)n_l anj = (q; q)n_j(apqn, bpq-n;p)j(bql-2n/a; q)n-j , (3.6.19)

(3.6.20)

This shows that the triangular matrix A = (anj) is inverse to the triangular matrix B = (bjm). Since inverse matrices commute, by computing the jkth term of BA, we obtain the orthogonality relation . k t (1 - apkqk)(l - bpkq-k) (apk+lqk+n, bpk+lq-k-n;p)j_k_l n=O (q; q)n(q; q)j_k_n(aq2k+n /b; q)j-k-l

x (1 - ~q2k+2n) (_ltqn(j-k-l)+e-~-n) = 6j,k, (3.6.21) which, by replacing j, n, a, b by n + k, k, ap-k-lq-k, bp-k-lqk, respectively, yields the bibasic summation formula

1- ~) (1 - ~) t (aqk, bq-k;P)n_l(l - aq2k /b) (_l)kq(~) = 6n 0 ( p p k=O (q; q)k(q; q)n_k(aqk /b; q)n+l ' (3.6.22) for n = 0,1,... . The b ----+ ° limit case of (3.6.22) was derived in AI-Salam and Verma [1984] by using the fact that the nth q-difference of a polynomial in q of degree less than n is equal to zero. For applications to q-analogues of Lagrange inversion, see Gessel and Stanton [1983, 1986] and Gasper [1989a]. Formulas (3.6.17) and (3.6.22) will be used in §3.7 to derive some useful general expansion formulas. 84 Additional summation, transformation, and expansion formulas

3.7 Bibasic expansion formulas

One of the most important general expansion formulas for hypergeometric series is the Fields and Wimp [1961] expansion

(3.7.1) where we employed the contracted notation of representing al, ... , ar by aR, (al)n'" (ar)n by (aR)n, and n + al,"" n + ar by n + aR. In (3.7.1), as elsewhere, either the parameters and variables are assumed to be such that the (multiple) series converge absolutely or the series are considered to be formal power series in the variables x and w. Special cases of (3.7.1) were employed, e.g., in Gasper [1975a] to prove the nonnegativity of certain sums (kernels) of Jacobi polynomials and to give additional proofs of the Askey and Gasper [1976] inequalities that de Branges [1985] used at the last step in his proof of the Bieberbach conjecture. Verma [1972] showed that (3.7.1) is a special case of the expansion

OO A B (xw)n _ Loo (_x)n Loo (a)n+d,8)n+k B k L n n - n+k X n-O_ n! n-O_ n!b+n)n k=O k!b+2n+1h

X ~ (-n)j(n+'Y)jA.wj ( ) ~ () () J 3.7.2 j=O j! a j ,8 j and derived the q-analogue

fAnBn (xw)n n=O (q; q)n _L oo (-x)n C2) Loo (a,,8; q)n+k B k - q n+k X n=O (q,'Yqn;q)n k=O (q,'Yq2n+l;qh

X t (q-n,'Yqn;q)j Aj(wq)j. (3.7.3) j=O (q,a,,8;q)j

To derive a bibasic extension of (3.7.3) we first observe that, by (3.6.17),

(3.7.4) 3.7 Bibasic expansion formulas 85 for m = 0,1, .... Hence, if Cr,m are complex numbers such that Cr,o = 1 for r = 0,1, ... , then

B r 00 1_"W-1q2r+2m ("(a-1q2r;q)m("(pqr,apq-r;P)r rX = ~ 1 _ _W- 1q2r (q; q)m("(pqr+m, apq-r-m;P)r

-mrB C r+ms: X q r+m r,mx Um,O

00 00 (1 _1'pnqn)(l _ apnq-n) (1 _1'a-1q2n+2k) = {;~ (q; q)k(q; q)n("(pqn+k, apq-n-k;P)n

x ("(a-1qn+r+1; q)n+k-r-1 ("(pqr, apq-r;P)n_1 (q-n; q)r

x (_l)n B n+k C r,n+k-r xn+kqn(1+r-n-kl+G) (3.7.5) by setting j = n - rand m = n + k - r. Then by multiplying both sides of (3.7.5) by Arwr /(q; q)r and summing from r = °to 00 we obtain Gasper's [1989a] bibasic expansion formula f AnBn (xw)n = f (l-1'pnqn)(l - apnq-n) (-xtqn+G) n=O (q; q)n n=O (q; q)n

00 1 - 1'a-1q2n+2k k Bn+kx x'"'~ (q; q)k("(pqn+k, apq-n-k;P)n

n (-n ) ( -1 n+j+1 ) X L q ; q j 1'a q ; q n+k-j-1 j=O (q; q)j

x (1'pqj, apq-j; P)n-1AjCj,n+k-jwj qn(j-n-k), (3.7.6) where Cj,o = 1, for j = 0,1, .... Note that if p = q and Cj,m == 1, then (3.7.6) reduces to an expansion which is equivalent to fAnBn (xw)n = f (a, 1'qn+1/a , a, (3; q)n (::.r n=O (q;q)n n=O (q,1'qn;q)n a

00 ("(q2n/a,qn+l"h/a,-qn+1yh/a,1/a,aqn,(3qn;q)kB k n+kX xLk=O (q, qn V1' /a, -qnV1' /a, 1'q2n+1; q)k

n (-n n) X '"' q, 1'q ; q j A- ( wq)j . (3.7.7) ~ (q,1'qn+1/a ,q1-n/a ,a,(3;q)j J

Verma's expansion (3.7.3) is the a ----7 00 limit case of (3.7.7). For basic hyper­ geometric series, (3.7.7) gives the following q-extension of (3.7.1) 86 Additional summation, transformation, and expansion formulas

rf, [q-j,,,(qj,aR,jM' ] (3.7.8) x r+rn+2'f's+k+2 "(qj+l jer, ql-j jer, bs' eK ' q, wq , where we used a contracted notation analogous to that used in (3.7.1). Note that by letting er ----t 00 in (3.7.8) and setting m = 2, II = h = 0 we get the expansion

r+t'f's+urf, [aR,cb d T ; q, xw] s' u

X r+2¢s+k [q-Jb',"(qj,aR;q,wq] , (3.7.9) s,eK which is equivalent to Verma's [1966] q-extension of the Fields and Wimp expansion (3.7.1). Other types of expansions are given in Fields and Ismail [1975]. AI-Salam and Verma [1984] used the b ----t 0 limit case of the summation formula (3.6.22) to show that Euler's transformation formula

00 00 k Lanbnxn = L(-l)k~! j(k) (x)b.kao, (3.7.10) n=O k=O where and

has the bibasic extension

00 00 k (1 n n) n A L AnBn(xw)n = L(apq\ph-lXk L ( .) ap ~a v:.. )n n=O k=O n=O q, q k-n pq ,p n 3.7 Bibasic expansion formulas 87

oo(kk) . X " ap q ; p j B. (_)j ~ () J+k x q m . (3.7.11) j=O q; q j The p = q case of (3.7.11) is due to Jackson [1910a]. In order to employ (3.6.22) to extend (3.7.10), replace n in (3.6.22) by j, multiply both sides by Bn+jxn+j (a/b)jqj2, sum from j = 0 to 00, change the order of summation and then replace k by k - nand j by j + k - n to obtain

(3.7.12) Next we replace a by apn+lqn, b by bpn+lq-n, multiply both sides by Anwn and then sum from n = 0 to 00 to get

00 00 ( k b -k ) "A B ( )n =" apq ,pq ;p k-l k ~ n n XW ~ (a k/b') X n=O k=O q, q k k (1 - apnqn) (1 - bpnq-n)(aqk /b; q)n n w X n=OL ()q; q k-n (apq k ,bpq-k) ;p n An

X 00 (apkqk,bpkq-k;p)j (_'!.-q2n)j+k-n ~ (q; q)j(aq2k+l /b; q)j b

x Bj+k( _x)jq(k-n)(j+k-n-l)+(~)+e+k2n+l). (3.7.13)

This formula tends directly to (3.7.11) as b ---+ O. By replacing An, B n , x, w by suitable multiples, we may change (3.7.13) to an equivalent form which tends to (3.7.11) as b ---+ 00. In addition, by replacing An, B n , x, w by Anq2(;) , B nq-2(;) , bx/a, aw/b, respectively, we can write (3.7.13) in the simpler looking equivalent form

00 00 ( k b -k ) k+l "A B n n _" apq ,pq ;p k-l (_ )k ( 2 ) ~ n nX W - ~ ( k/b) x q n=O k=O q, aq ; q k

X k (1 - apnqn) (1 - bpnq-n)(q-k, aqk /b; q)n Anwn ~ (apqk, bpq-k;P)n

00 (apkqk bpkq-k. p) ,j B· j j x"~ ( '2k+l/b') . J+k X q . (3.7.14) j=O q, aq ,q J

As in the derivation of (3.7.6), one may extend (3.7.14) by replacing Bj+k by Bj+kCn,j+k-n with Cn,o = 1 for n = 0,1, .... Multivariable expansions, which are really special cases of (3.7.6) and (3.7.14), may be obtained by 88 Additional summation, transformation, and expansion formulas replacing An and Bn in (3.7.6) and (3.7.14) by multiple power series, see, e.g. Gasper [1989a], Ex. 3.22 and, in the hypergeometric limit case, Luke [1969]. For a multivariable special case ofthe AI-Salam and Verma expansion (3.7.11), see Srivastava [1984].

3.8 Quadratic, cubic, and quartic summation and transformation formulas

By setting P = qj or q = p1, j = 2,3, ... , in the bibasic summation formulas of §3.7 and using summation and transformation formulas for basic hypergeomet­ ric series, one can derive families of quadratic, cubic, etc. summation, transfor­ mation and expansion formulas. To illustrate this we shall derive a quadratic transformation formula containing five arbitrary parameters by starting with the q = p2 case of (3.6.16) n (1 - adp3k) (1 - b/dpk) (a, b;ph(p-2n, ad2p2n /b; p2h 2k ~ (1 - ad)(l - bid) (dp2, adp2 /b;p2)dadp2n+l, bpl-2n /d;p)k p (1 - d)(l - ad/b)(l - adp2n)(1 - dp2n /b) (3.8.1) (1 - ad)(l - d/b)(l - dp2n)(1 - adp2n /b)' where n = 0,1, .... Change p to q and d to c in (3.8.1), multiply both sides by (ac2/b; q2)n(c/b; q)zn C (q2; q2)n(acq; q)zn n and sum over n to get

(3.8.2)

Setting c _ (1 - ac2q4n /b)(d, e, j; q2)n(a2q3/deJ)n n - (1 - ac2/b)(ac 2q2/bd, ac2q2 /be, ac2q2 /bj; q2)n' it follows from (3.8.2) that lOVVg(ac2/b;ac/b,c,cq/b,cq2/b,d,e,j;q2,a2c2q3/dej) 3.8 Quadratic, cubic, and quartic formulas 89

x (d,e,J;q2)k(a2c3q4jbde!)k q(~) (ac2q2jbd,ac2q2jbe, ac2q2jbJ; q2)k

x 8 W7(ac2q4k jb; cl jb, cqk+l jb, dq2k, eq2k, Jq2\ q2, a2c2q3 jde!). (3.8.3) If we now assume that a2c2q = deJ, (3.8.4) then we can apply (2.11.7) to get 8W7(ac2q4kjb; cqkjb, cqk+ljb, dq2k, eq2k, Jq2k; q2,q2)

(ac2q4k+2jb, bJjac2q2k,abq2k+l, acqk+2jd;q2)oo (acq3k+2, acq3k+l, ac2q2k+2 jbd, ac2q2k+2 jbe; q2)oo

X (acqk+2je,acqk+ljd,acqk+lje,ac2q2jbde;q2)oo (beJ j ac2, bdJ j ac2, J j acqk, J j acqk-l; q2)oo

+ bJq-2k(ac2q4k+2 jb, cqk jb, cqk+l jb, dq2k, eq2k; q2)oo ac2(ac2q2k+2jbJ,acq3k+2,acq3k+l,ac2q2k+2jbd;q2)oo

X (Jq2 je, Jq2 j d, bJq2-2k jac2, bJqk+l jc, bJqk+2 jc; q2)oo (ac2q2k+2jbe,beJjac2, bdJjac2, bJ2 q2jac2, Jjacqk, Jjacqk-l ;q2)oo

X 8 W7(bJ2 jac2; Jq2k, beJ jac2, bdJ jac2, J jacl, J jacl-\ q2, q2), (3.8.5) which, combined with (3.8.3), gives

lOW9 (ac2jb;c,d,e,J,acjb,cqjb,cq2jb;q2,q2)

(ac2q2jb,ac2q2jbde,abq,bJjac2;q2)oo(acqjd,acqje;q)oo (ac2q2 jbd, ac2q2 jbe, bdJ jac2, beJ jac2; q2)oo(acq, J jac; q)oo

~ (1 - acq3k)(1 - bjcqk)(a, b, cjb; q)k(d, e, J; q2h 2k x ~ (1 - ac)(l - bjc) (cq2, acq2 jb, abq; q2)k(acqjd, acqje, acqj J; q)k q + bJ(ac2q2 jb, d, e, Jq2 jd, Jq2 je, bJq2 jac2; q2)oo ac2(ac2q2jbJ,ac2q2jbe,ac2q2jbd, bdJjac2, beJjac2,bJ2q2jac 2;q2)oo (bJqjc, cjb; q)oo x --';------'------::-;-'---c-- (acq, J j ac; q)oo

~ (1 - acq3k)(1 - bjcqk) (a, b; qh(ac2 jbJ, J; q2)k 2k X ~ (1 - ac)(l - bjc) (cq2, acq2 jb; q2h(bJqjc, acqj J; q)k q

x 8 W7(bJ2 jac2; Jq2k, bdJ jac2, beflac2, J jacqk, J jacqk-\ q2, q2). (3.8.6) 90 Additional summation, transformation, and expansion formulas

The last sum over k in (3.8.6) is

~ (bP lac2,j, bdflac 2, bef lac2, f lac, fqlac; q2)j(1 - bpq4 j lac2) q2j f;:o (q2, bfq2 lac2,jq2 Id, fq2 Ie, bfq2/c, bfqlc; q2)j(1 - bP lac2)

~ (1 - acq3k)(1 - blcqk) (a, b; q)dac2 Ibfq2j ,jq2j; q2)k 2k X ~ (1 - ac)(l - blc) (cq2, acq2/b; q2)k(bfq2J+1 Ic, acql-2j If; q)k q

= ~ (bP lac2, f, bd!lac2, bef lac2, !lac, fqlac; q2)j(1 - bpq4j lac2) q2j f;:o (q2, bfq2 lac2, fq2ld, fq2 Ie, bfq2/c, bfqlc; q2)j(1 - bP lac2) (1 - c)(l - aclb)(l - acl fq2j)(1 - bfq2j Ic) x .. c(l - ac)(l- blc)(l - fq2) Ic)(l - aclbfq2J)

(a, b; q)00(Jq2j , ac2 Ibfq2j; q2)00 { I} x (c,aclb;q2)00(aclfq2j,bfq2jlc;q)00-

(1 - c)(l - aclb)(l - f lac)(l - bf Ic) (1 - clb)(l - ac)(l - f Ic)(l - bf lac)

x 10 W g(bf2 lac2; f, bdf lac2, beflac 2, bf lac, f Ic, fql ac, fq2 lac; q2, q2) + f ac(l- ac)(l - clb)(l - f Ic)(l- bf lac) (a, b; q)oo(J, ac2/bf; q2)00 x-;----c;o---'---'--;;-'-:-:'-'-~_'__;_____'___c~'-:'--;---;-­ (cq2, acq2/b; q2)00(acql f, bfqlc; q)oo

00 (bP lac2, bdflac 2, bef lac2, f Ic, bf lac; q2)j (1 - bpq4j lac2) ( fq2)j j2 X 2: -- q j=O (q2, fq2 Id,jq2 Ie, bfq2 lac, fq2/c; q2)j (1 - bP lac2) ab (3.8.7)

by the n = 00 case of (3.6.15). Thus,

2 2 2 2 bf(l - c)(l - aclb) lOW g(ac Ib;c,d,e,f,aclb,cqlb,cq Ib;q ,q ) - ac2(1- !lc)(l- bflac)

x (ac2q2/b, d, e, fq2ld, fq2 Ie, bfq2 lac2; q2)00 (bf2q2/ac2, ac2q2/be,ac2q2/bd,bdflac2, beflac2,ac2q2/bf; q2)00

(bf Ic, cqlb; q)oo W. (bP. f bdf bef f bf fq fq2. 2 2) X (ac,fqlac;q)oo 10 9 ac2' 'ac2'ac2'~'ac'ac'ac2,q ,q 3.8 Quadratic, cubic, and quartic formulas 91

x (1, ac2/bf, ac2q2 /b, d, e, f q2/d, f q2/e, bfq2 /ac2; q2)00 (cq2, acq2/b, ac2q2/bf,ac2q2/be, ac2q2/bd, bdf/ac2, bef/ac2 ,bf2q2/ac2;q2)00

x ~ (bP /ac2, bdflac2, bef /ac2 , flc, bflac; q2)j ~ (q2, f q2/d, f q2/e, bfq2/ac, f q2/c; q2)j

(1- bpq4j /ac2) ( fq2)j j2 x -- q (3.8.8) (1 - bj2 /ac2) ab when (3.8.4) holds. Now observe that since f (a, qva, -qva, c, d, e, f; q)j (_ a2q2 )j q(~) j=O (q, va, -va, aq/c, aq/d, aq/e, aq/ f; q)j cdef

_ (aq,aq/ef;q)oo ¢ [aq/cd,e,f. aq ] (3.8.9) - (aq/e,aq/f;q)oo 3 2 aq/c,aq/d,q, ef by the n ~ 00 limit case of (2.5.1), the sum over j in (3.8.8) equals (q2,bPq2/ac2;q2)00 ¢ [f/c,bf/ac,ac2q2/bde. 2 2] (3.8.10) (bfq2/ac, f q2/C; q2)00 3 2 f q2/d, f q2/e ,q ,q .

Hence, by setting e = a2c2q/df in (3.8.8) we obtain the Gasper and Rahman [1990] quadratic transformation formula lOVVg(ac2/b;f,ac/b,c,cq/b,cq2/b,d,a2c2q/df;q2,q2)

(ac2q2/b,bf/ac2,ac/b,c,cq/b,cq2/b,bfq2/ac;q2)00 +~~~~~~--~~--~~~~~~~~~~ (bf2q2/ac2, ac2/bf,ac2q2/bd,dfq/ab,bdf/ac2,abq/d,cq2; q2 )00

x (1q2/c, bf /c, bfq/c, fq2 / d, dPq/a2c2, d, a2c2q/df; q2)00 (acq2/b,f/c, bf/ac, aC,acq, fq/ac, f q2/ac;q2)00

(a, b, cq/b; q)oo (ac, ac/ f, fq/ac; q)oo x (f,d,a2c2q/df,bf/ac2,ac2q2/b,fq2/d,df2q/a2c2,q2;q2)00 (bf/ac, f/c, cq2, acq2/b, ac2q2/bd, dfq/ab, bdf/ac2, abq/d;q2 )00

flC,bf/ac,fq/ab [ 22] x 3¢2 f q2/d, dpq/a2c2 ; q ,q

(acq/d, df lac; q)00(ac2q2 /b, abq, bf /ac2, fq/ab; q2)00 (acq, f lac; q)00(ac2q2 /bd, dfq/ab, bdf /ac2, abq/d; q2)00 f 1 - acq3k (a,b,cq/b;q)k(d,f,a2c2q/df;q2)kqk x 1 _ ac (cq2, acq2/b, abq; q2h(acq/d, acq/ f, df lac; q)k· (3.8.11) k=O 92 Additional summation, transformation, and expansion formulas

Note that the first two terms on the left side of (3.8.11) containing the 10 Wg series can be transformed to another pair of 10 W g series by applying the four-term transformation formula (2.12.9). Since the 31>2 series in (3.8.11) is balanced it can be summed by (1.7.2) whenever it terminates. When c = 1 formula (3.8.11) reduces to the quadratic summation formula ~ 1 - aq3k (a, b, q/b; q)k(d, j, a2q/dj; q2)k k ~ 1 - a (q2, aq2/b, abq; q2h(aq/d, aq/ j, dj fa; q)k q k=O (aq, j/a, b, q/b; q)=(d,a2q/dj,jq2/d, dj2q/a2;q2)= +~~~~--=-~~~~~~~~~~~~-=-- (a/j,jq/a, aq/d,dj/a; q)=(aq2/b,abq, jq/ab,bj/a; q2)=

j, bj la, jq/ab 2 2] X 31>2 [ jq2/d,dj2q/a2 ;q ,q

(aq, j/a; q)=(aq2/bd, abq/d, bdj/a, djq/ab; q2)= (3.8.12) (aq/d, dj fa; q)=(aq2 /b, abq, bj la, jq/ab; q2)= . By multiplying both sides of (3.8.11) by (J lac; q)= and then setting j = ac we obtain Rahman's [1993] quadratic transformation formula ~ (a; q2h(1 - aq3k)(d, aq/d; q2)k(b, c, aq/bc; q)k k ~ (q;q)k(1 -a)(aq/d,d;q)k(aq2/b,aq2/c,bcq;q2)k q k=O _ (aq2,bq,cq, aq2/bc;q2)= [b,c,aq/bc. 1> 2 2] (3.8.13) - (q, aq2/b, aq2/c, bcq; q2)= 3 2 dq, aq2/d ,q ,q , provided d or aq/d is not of the form q-2n, n a nonnegative integer. Also, the case d = q-2n of (3.8.11) gives n 1 - acq3k (a, b, cq/b; q)k(J, a2c2q2n+1 / j, q-2n; q2)k k ~ 1 - ac (cq2, acq2/b, abq; q2h(acq/ j, Jlacq2n, acq2n+1; q)k q (acq;q)2n(ac2q2/bj,abq/j;q2)n (acq/j;q)2n(abq,ac2q2/b;q2)n

x 10 W g (ac2/b; j, ac/b, c, cq/b, cq2/b, a2c2q2n+l / j, q-2n; q2, q2) (3.8.14) and the case b = cq2n+1 gives 2n 1 - acq3k (d, j, a2c2q/dj; q2h(a, cq2n+l, q-2n; qh k ~ 1 - ac (acq/d, acq/ j, dJlac; q)k(cq2, aql-2n, acq2n+2; q2h q (acq2,dq/ac, jq/ac, acq2/dj; q2)n (q/ac,acq2/d,acq2/j,djq/ac;q2)n x 10 W g (acq-2n-\ c, d, j, a2c2q/dj, aq-2n-\ ql-2n, q-2n; q2, q2) (3.8.15) for n = 0, 1, .... Similarly, the special case n (l-acq4k)(I-b/cq2k) (a,b;qh(q-3n,ac2q3n/b;q3h 3k ~ (1 - ac)(1 - b/c) (cq3, acq3/b; q3h(acq3n+1, b/cq3n-1; qh q 3.8 Quadratic, cubic, and quartic formulas 93

(1 - e)(1 - ae/b)(1 - aeq3n) (1 - eq3n /b) = 0 1 2 (3.8.16) (1 - ae)(1 - e/b)(1 - eq3n)(1 - aeq3n /b)' n ", ... , of (3.6.16) is used in Gasper and Rahman [1990] to show that Gosper's sum (see Gessel and Stanton [1982]) F. [ a, a + 1/2, b, 1 - b, e, (2a + 1)/3 - e, a/2 + 1 .1] 7 6 1/2, (2a - b + 3)/3, (2a + b + 2)/3, 3e, 2a + 1 - 3e, a/2'

2 r(e+!)r(e+~)r(~)r(~) V3 r (2at2) r (2at3) r (3c+;+l) r (3c+i- b)

x r (2+2~-3C) r (3+2~-3C) sin ~(b + 1) (3.8.17) r (2+2atb-3c) r (3+2a 3b-3c) has a q-analogue of the form

~ 1 _ aeq4k (a,q/a;q)k(ae;q)2k(d,aeq/d;q3)k qk 1 -ae (eq3,a2eq2;q3)k(q;q)2k(aeq/d,d;q)k k=O~ (aeq2,aeq3,d/ae,dq/ae,adq,aq,q2/a,dq2/a;q3)00 (q,q2,dq,dq2,a2eq2,eq3,dq/a2e,d/e;q3)00 d(a,q/a,aeq;q)00(q3,d,aeq/d,d2q2/ae;q3)00 +~~~--~7-~~~~~--~~~-­ ae(q,d,aeq/d;q)00(eq3,a2eq2,d/e,dq/a2e;q3)00 die, dq/a2e 3 3] x 2

(a,b,eq/b;q)00(q3,d,ae2q3/b,a2be/d,d2q3/a2be,bd/ae2;q3) 00 (ae,dq/ae,ae/d;q)00(eq3,aeq3/b,d/e,bd/ae,edq3/ab2,ab2/e;q3)00

(ab, dq/ ab; q)oo (bd/ ae2, ae2q3/b; q3)00 (aeq, d/ ae; q)oo (ab2/ e, edq3/ ab2; q3)00 oo 1 - aeq4k (a, b; q)k(eq/b; qhk(d, a2be/d; q3)k k X L q (3.8.19) k=O 1 -ae (eq3,aeq3/b;q3)k(ab;q)2k(aeq/d,dq/ab;q)k and some other cubic transformation formulas. 94 Additional summation, transformation, and expansion formulas

The special case

n (1-acq5k)(1-bjcq3k) (a,b;qh(q-4n,ac2q4njb;q4)k 4k {; (1 - ac) (1 - bj c) (eq4, aeq4 jb; q4 h (aeq4n+1, bj eq4n-1; q)k q (1 - e)(l - aejb)(l - aeq4n) (1 - eq4n jb) (1 _ ae)(l _ ejb)(l _ eq4n)(1 _ aeq4n jb)' n = 0,1,2, ... , (3.8.20) of (3.6.16) was used in Gasper and Rahman [1990] to derive the quartic trans­ formation formula

+ (1 - e)(l - acjb) (a 2b3 jcq2, eqjb; q)00(ae2q4 jb, ab3 jc2q2; q4)00 (1 -a2b2jeq2)(1 -ab3jeq2)(ab2jeq,ae;q)00(a3b5je2,e2q2jab3;q4)00

+ ab2(a, b, eqjb; q)00(ae2q4 jb, ab3 je2q2, a2b2 jq2; q4)00 eq2(1 -a2b2jeq2)(ae,ab2jeq2,eq3jab2;q)00(ab3jeq2,cq4,aeq4jb;q4)00

a2b2jeq2. 4 ab3q2 ] x 11>1 [a2b2q2 je' q , -e-

(ab; q)oo (abj q; q2)00 (ae2q4 jb, ab3 j C2q2; q4)00 (aeq, ab2 jeq2; q)oo

X 00 1 - aeq5k (a, b; q)k(eqjb, eq2 jb, eq3 jb; q3h(a2b2 jq2; q4)k k {; 1 - ac (eq4, aeq4 jb; q4)k (abq, ab, abj q; q2h (eq3 j ab2; qh q . (3.8.21 )

When b = q2 j a the sum of the two 10 W g series in (3.8.21) reduces to a sum of two 8W7 series, which can be summed by (2.11.7) to obtain the quartic summation formula

~ 1 - aeq5k (a, q2 ja; q)k(aejq, ae, aeq; q3h(q2; q4h l ~ 1 - ae (eq4,a2eq2;q4)k(q3,q2,q;q2)k(aejq;q)k q2 (a, q2 j a, acq; q)oo (q2; q4)00

(3.8.22) (q,q3;q2)00(eq4,q2je,a2eq2,q4ja2e;q4)00 . Additional quadratic, cubic and quartic summations and transformation for­ mulas are given in the exercises. 3.9 Multibasic hypergeometric series 95

3.9 Multibasic hypergeometric series

In view of the observation in §1.2 that a series 2::::"=0 Vn is a basic hypergeo­ metric series in base q if Vo = 1 and Vn+Ii Vn is a rational function of qn, a series 2::::"=0 Vn will be called a bibasic hypergeometric series in bases p and q if vn+ I / Vn is a rational function of pn and qn, and p and q are independent. More generally, we shall call a series 2::::"=0 Vn a multibasic (or m-basic) hypergeo­ metric series in bases ql, ... , qm if vn+Ii Vn is a rational function of q'l, ... , q;;:', and ql,"" qm are independent. Similarly a bilateral series 2::::"=-00 Vn will be called a bilateral multibasic (or m-basic) hypergeometric series in bases ql, ... , qm if vn+ I / Vn is a rational function of q'l, ... , q;;:', and ql,· .. , qm are independent. Multibasic series are sometimes called polybasic series. Since a multibasic series in bases ql,"" qm may contain products and quotients of q-shifted factorials (a; q)n with q replaced by q~l ... q~rn where kl , ... , k m are arbitrary integers, the form of such a series could be quite complicated. Therefore, in working with multibasic series either the series are displayed explicitly or notations are employed which apply only to the series under consideration. For example, to shorten the displays of many of the formulas derived in §3.8 we employ the notation

if. [al,"" ar : CI,I,···, cI,rl : ... : Cm,l,"" cm,rrn . . ] 'l' b b d d d d ,q,ql,···,qm,z (3.9.1) 1,0", s: 1,1,···,1,81:"': m,l,···, m,srn to represent the (m + I)-basic hypergeometric series 00 ( ) [ ] l+s-r L al,···,ar;q n zn (_I)nqG) q, bl , ... , b ; q)n n=O ( s j x (Cj,I, ... ,Cj,rj;.qj)n [(_I)n G)]sj-r IT q (3.9.2) j=l (dj,l, ... ,dj,sj,qj)n J

The notation in (3.9.1) may be abbreviated by using the vector notations: a = (al, ... ,ar ), b = (bl, ... ,bs), Cj = (Cj,I, ... ,Cj,rj)' d j (dj,l, ... ,dj,sj) and letting

if. [ a : CI : ... : Cm ] 'l' b' d' . d ; q, ql, ... , qm; Z (3.9.3) . I····· m denote the series in (3.9.2). If in (2.2.2) we set

Ak = (a; q)k zk (Cj,l, ... , Cj,rj; qj)k [( , IT _1)kq}~)]Sj-rj (3.9.4) (q-n;q)k j=l (dj,l, ... ,dj,sj;qj)k then we obtain the expansion

[ a, b, C : CI,I,···, cI,rl : ... : Cm,l,···, cm,rrn ; q, ql, ... , qm; z] aq/b, aq/ C : dl,l, ... , dl,Sl : ... : dm,l, ... , dm,srn

= f= (aq/bc;q)n(a;qhn (_ bCZ)n q-G) n=O (q,aq/b,aq/c;q)n aq 96 Additional summation, transformation, and expansion formulas

: C1,lqr,··· ,c1,rlqr: ... : : d1,lqr, .. ·, d1,Sl qr : ... :

Cm 1qmn , ... , Cm r qmn bcz -n-1p ] d , n d , '" n"'q qq'1, .. ·, m, - q n m,lqm"'" m,s",qm a (3.9.5) where m Pn = II q;(Sj-rj ), (3.9.6) j=l provided at least one Cj,rj is a negative integer power of qj so that the series terminate. Note that this formula is valid even if a = q-k, k = 0,1, ... , in which case the upper limit of the sum on the right side can be replaced by [k/2]' where [k/2] denotes the greatest integer less than or equal to k/2. Similarly, use of the expansion formula (2.8.2) leads to the formula

aq2n , X: c1,lqr,···, cI,rl qr : ... : cm,lq;::', ... , cm,r",q;::' ] x

3.10 Transformations of series with base q to series with base q2

Following Nassrallah and Rahman [1981]' we shall derive some quadratic trans­ formation formulas for basic hypergeometric series by using the following spe­ cial cases of (3.9.5) and (3.9.7):

a 2 , -aq, 2 b2 , C 2. . al,"" an q -n 2 ] [

2 2 2. 2) ( 2. 2) ( 2 = '"'n ( a q 2/b2 c ,-aq ,q j a ,q 2j al,···, aT) q -no,q ) j -2(~) . ( _ ~b2 )j ~. ( q 2 ,-a, a 2 q2/b2 ,a2 q 2/ c,2. q2) J·(b 1,···, bT) br+l, .). q J q a 2 q 2 J=O a2q4j , _ aq2j+2 : al qj, ... , arqj, qj-n X

a2q4j , _ aq2j+2, a2/ A: alqj, ... , arqj, qj-n x

_ (w/a,-aq;q)n ¢ [ a,aq,a2q2/w2,a2q2/b2c2,q-2n . 2 2] - (w,-q;q)n 5 4 a2q2/b2,a2q2/c2,aql-n/w,aq2-n/w,q ,q . (3.10.3) Note that the above 5¢4 series is balanced and that the [ '" " "" . 10 9 a~, -a~, aq/b, aq/x, -aq/x, aq/y, -aq/y, _ aqn+l, aqn+l ,q, - bx2y2 _ (a 2q2, a2q2 /X2y2; q2)n ¢ [ q-2n, x 2, y2, -aq/b, - aq2/b . 2 2] - (a2q2/x 2, a2q2 /y2; q2)n 5 4 x2y2q-2n /a2, a2q2/b2, -aq, -aq2' q ,q . (3.10.4) 98 Additional summation, transformation, and expansion formulas

For a nonterminating extension of (3.10.4) see Jain and Verma [1982]. Since the 5¢>4 series on the right side of (3.10.3) is balanced, it can be summed by (1.7.2) whenever it reduces to a 3¢>2 series. Thus, we obtain the summation formulas:

a2,aq2,_aq2: -aq/w,q-n w qn - 1 ]

(-aq, aq2 /w, w/aq; q)n (3.10.5) (-q, aq/w, w; q)n a2, _ aq2, b2 : _aqn /b2, q-n ]

(3.10.6)

(3.10.7) and

(3.10.8)

These are q-analogues of formulas 4.5(1.1) - 4.5(1.4) in Bailey [1935]. Since the series on the left sides of (3.10.5) and (3.10.6) can also be written as VWP­ balanced 8¢>7 series in base q, which are transformable to balanced 4¢>3 series by Watson's formula (2.5.1), formulas (3.10.5) and (3.10.6) are equivalent to the summation formulas a, qa!, -w/qa!, q-n 1 4¢>3 [ .1 .1 I-n ; q, q a 2 ,w,-a2 q (w/aq, -a!, aq2 /w; q)n (3.10.9) (w, -a-!, aq/w; q)n and

1 2 1 (1 + l/b)(l + a"2 qn /b)(a/b , qa"2 /b, l/b; q)n (3.10.10) (1 + qn /b)(l + a! /b)(aq/b, a! /b, 1/b2; q)n ' respectively, which are closer to what one would expect q-analogues of formulas 4.5(1.1) and 4.5(1.2) in Bailey [1935] to look like. 3.10 Transformations of series 99

It is also of interest to note that if we set c2 = aq in (3.10.3), rewrite the il> series on the left side as an 8¢7 series in base q and transform it to a balanced 4¢3 series, we obtain a, b, -w /b, q-n ] [ 4¢3 aq /b , -bq -n ,w ; q, q _ (w/a,-aq/b;q)n ¢ [ a,a2q2/w2,aq/b2,q-2n . 2 2] ( ) - (w,-q/b;q)n 4 3 a2q2/b2,aql-n/w,aq2-n/w,q,q , 3.10.11 which is a q-analogue of the c = (1 + a)/2 case of Bailey [1935, 4.5(1)]. Using (2.10.4), the left side of (3.10.11) can be transformed to give q-n, a, aq/b2, -aq/w ] [ 4¢3 aq /b , -aq /b ,aqI-n/ w ; q, q

(3.10.12)

This formula was first proved by Singh [1959] and more recently by Askey and Wilson [1985]. The latter authors also wrote it in the form a2,b2,c,d ] [ 4¢3 a b!q2, -ab! q2,-c d; q, q 2b22d2 ] [ a, , c , 2 2 4¢3 2b2 d d; q , q , (3.10.13) = qa , -c , -qc provided that both series terminate. Now that we have the summation formulas (3.10.5)-(3.10.8), we can use them to produce additional transformation formulas. Set r = 3 and al = -a2 = qa!, a3 = -aq/w, bl = -b2 = a!, b4 = _aqn+1 in (3.10.1). The il> series on the right side can now be summed by (3.10.5) and this leads to the following q-analogue of Bailey [1935, 4.5(2)]

(-aq, aq2 /w, w/aq; q)n (-q, aq/w, w; q)n aq,aq2,a2q2/b2c2,a2q2/w2,q-2n [ 2 2] X5¢4 a2q2/b2,a2q2/c2,aq2-n/w,aq3-n/w;q,q . (3.10.14)

Let us now turn to applications of (3.10.2). If we set r = 1, al = _Aqn+l/a, bl = a2q-n /A, b2 = _aqn+l, z = q in (3.10.2), where A = (qa2/bcd)2, then the il> series on the right side reduces to a balanced very-well-poised 8¢7 series in base q which can be summed by Jackson's formula (2.6.2). Thus, we obtain the transformation formula a2,_aq2,b2,c2,d2: _Aqn+l/a,q-n 1 il> [ ;q2,q;q -a, a2q2 /b2, a2q2 / c2, a2q2 / d2 : a2q-n / A, _ aqn+1 100 Additional summation, transformation, and expansion formulas

(-aq, >..qj a; q)n(>..q2 j a2; q2)n (-q, >..qja2; q)n(>..q2; q2)n

This is a q-analogue of Bailey [1935, 4.5(3)]. Similarly, setting r = 3 and choosing the parameters so that the series on the right side of (3.10.2) can be summed by (3.10.7), we get the following q-analogue of Bailey [1935,4.5(4)]

a2 aq2 _aq2 b2 c2 d2 . ->..qnja q-n ] [ '" , ,., 2 a, -a, a2q2 jb2, a2q2 j c2, a2q2 j d2 : a2ql-n j >.., _ aqn+l ; q ,q; q

Exercises

3.1 Deduce from (3.10.13) that

provided that both series terminate. Show that this formula is a q-analogue of Gauss' quadratic transformation formula (3.1.7) when the series terminate.

3.2 Using the sum

(b2. q2) (n) -+. (q-n ql-n. qb2. q2 q2) = ' n q- 2 2 '/'1 , , " (b 2 . ) , ,q n prove that

(Z; q)oo -+. ( b2 2 2) I I (i) = ( .) 2'/'1 a, aq; q ; q ,Z , Z < 1. az, q 00 Exercises 101

b22;q,azq,z<1.2 222 ]11 a Z Show that (iii)

Formulas (i) and (ii), due to Jain [1981]' are q-analogues of (3.1.5) and (3.1.6), respectively. T. Koornwinder in a letter (1990) suggested part (iii) . 3.3 Show that a, q/a, Z ] 3rP2 [ ;q,q c,-q

_ (-1, -qz/c; q)oo A,. [cia, ac/q, Z2. 2 2] - (/-qc,-z,qoo .) 3'1'2 c,O2 ,q ,q , when the series terminate. This is a q-analogue of 2Fl (a, 1 - a; c; z) = (1 - zt-1 2Fl ((c - a)/2, (c + a - 1)/2; c; 4z(1 - z)) , when the series terminate. 3.4 Show that ~ (q-n, b, -b; qh nk-(~) (b2 ) q k=O~ q,; q k

vanishes if n is an odd integer. Evaluate the sum when n is even. Hence, or otherwise, show that

Deduce that -n I-n ] [ q ,q ,a,aq 2 2 4rP3 b2 d d ; q ,q q , , q

_ (d/a;q)n n A,. [q-n,a,b,-b. _Cfc] - (d.),q n a 4'1'2 b2 ,aq I-n/d ' q, d . (Jain [1981]) 102 Additional summation, transformation, and expansion formulas

3.5 By using Sears' summation formula (2.10.12) show that f (a, aq/e; q)r Ar r=O (q, abq/e, acq/e; q)r (aq/e,bq/e,cq/e,abcq/e;q)oo (q/e,abq/e,acq/e,bcq/e;q)oo

~ (a, b, c; qhqk ~ (aqk; q)r A x t='o (q, e, abcq/e; q)k ~ (q, abcqk+l Ie; q)r r

+ (a, b, c, abcq2 /e2; q)oo (e/q,abq/e, acq/e, bcq/e; q)oo

~ (aq/e, bq/e, cq/e; q)k k ~ (aqk+ 1 Ie; q)r A x t='o (q, q2/e, abcq2 /e2; q)k q ~ (q, abcqk+2 /e2; q)r r, where a, b, c, e are arbitrary parameters such that e -I=- q, and {Ar} is an arbitrary sequence such that the infinite series on both sides converge absolutely. 3.6 Prove that b, (q/e,a,b,c,dq/e;q)oo 3(P2 [a, C] d, e; q, q + (e/q, aq/e, bq/e, cq/e, d; q)oo aq/e, bq/e, cq/e ] 3¢2 [ 2/ q, q x q e, d q / e ; _ (q/e, abq/e, acq/e, d/a; q)oo ¢ [a, aq/e, abcq/de. ~] - (d, aq/e, bq/e, cq/e; q)oo 3 2 abq/e, acq/e ,q, a ' where e -I=- q and Id/al < l. 3.7 Prove that a,aq/e,e/bc bcq] 3¢2 [ ;q,-- abq/e,acq/e e (aq, aq/e, bq/e, cq/e, -q; q)oo (q/e, abq/e, acq/e, bcq/e, -bcq/e; q)oo

00 (a,b,c;q)k (ab2c2qk+2/e2;q2)00 k x ~ (q, aq, e; q)k(aqk+2; q2)00 q, provided Ibcq/el < l. 3.8 Assuming that Ixl < 1 and alb -I=- qj, j = 0, ±1, ±2, ... , prove that

2¢1 (a, b; c; q, x) _ (b, cia, ax; q)oo '" [a, c/b, 0 . ] - (b/ ) 3'1-'2 / ' q, q a, c, x; q 00 aq b, ax (a,c/b,bx;q)oo '" [b,c/a,o ] + ( / ) 3'1-'2 ; q, q . a b, c, x; q 00 bq/a, bx Exercises 103

Show that this is a q-analogue of the formula 2F1(a, b; C; x)

r(c)r(b-a) -a -1 =r(b)r(c_a)(l-x) 2F1(a,c-b;a-b+1;(1-x) )

r(c)r(a - b) -b -1 +r(a)r(c-b)(l-x) 2F1(b,c-a;b-a+1;(1-x) ).

3.9 Show that a,>..q,b >..2] [ 3¢2 >..,q>..2/b;q, ab2

1 - >.. >../b(l - >../a) (>..2 /b2, q>..2/ab; q)oo + 2 -'------'c--c--'---=--,"--:----,,----''''------I>.. 2/ ab 1 < l. (1 - >")(1 + >../b) (q>..2/b, >..2 /ab2; q)oo'

3.10 Show that

8 W7 ( ->..; q>,,~, -q>,,~, a, b, -b; q, >..jab2) 2 (>..ja, >"q, ->..q, >../b , >..q/ab, ->..q/ab;.q)oo , l>..jab21 < l. (>.., >"q/a, ->..q/a, >"q/b, ->..q/b, >../ab ,q)oo Show that this is a q-analogue of the formula F [a, 1+ >"/2, b . 1] 3 2 >..j2, 1 + >.. - b' r(>../2)r (1 + A;a) r(l + >.. - b)r(>.. - a - 2b) Re(>..-a-2b) >0. r(l + >..j2)r (A;a) r(l + >.. - a - b)r(>.. - 2b)'

3.11 Derive Jackson's [1941] product formula

2(Pl (a2, b2; qa2b2; q2, z) 2¢1 (a 2, b2; qa2b2; q2, qz) a2, b2, ab, -ab ] [ Izl < 1, Iql < 1, = 4¢3 a 2b2 ,ab! q2, -ab! q2 ; q, z , and show that it has Clausen's [1828] formula

1 )] 2 [2a, 2b, a + b ] [ 2F1 ( a, b; a + b + -; z = 3F2 1 ; Z , Izl < 1, 2 2a + 2b, a + b + "2 as a limit case. Additional q-analogues of Clausen's formula are given in §8.8. 3.12 Prove that q-2n _q2-n b2 c2 . d _q1-n/w wq2-n] IJ? [ ' "., . 2 • _q-n,q2-2n/b2,q2-2n/c2: _q1-n/d,w,q ,q, b2c2d = (-l,w/d;q)ndn (-d,w; q)n d2,q2-2n/b2c2,q2-2n/w2,q-n,q1-n 2 2] [ X 5¢4 q2-2n/b 2,q 2-2n/ c,2 d q1-n/ w, d q2-n/ w ; q ,q . 104 Additional summation, transformation, and expansion formulas

3.13 If A = a4q2 /b2c2d2, show that a2 _q2a b2 c2 d2 . -Aqn/a q-n ] [ ' ",., 2 2 _a,a2q2/b2,a2q2/c2,a2q2/d2: a2ql-n/A,_aqn+l;q ,q;q (-aq, Aja; q)n(Aja2, Aq2/a; q2)n (-q, A/a2; q)n(Aq2, A/a; q2)n X lOVVg(A; a,aq, Ab2/a2, Ac2/a2, Ad2/a2,A2q2n/a2,q-2n;q2,a2q3 /A) and a2 q2a -q2a b2 c2 d2 . _Aqn-l/a q-n ] [ "",. , 2 2 a, -a, a2q2 /b2, a2q2 / c2, a2q2 / d2 : a2q2-n / A, _aqn+l ; q , q; q (-aq,Ajaq;q)n(A/a2q2;q2)n(1- Aq2n-l/a) n = q (-q, A/qa2; q)n(Aq2; q2)n(1- Aq-l/a)

x lQVVg (A; aq, aq2, Ab2/a 2, AC2/a 2, Ad2/a 2, A2q2n-2 /a2, q-2n; q2, a2l/A). (Nassrallah and Rahman [1981]) 3.14 Using (3.4.7) show that the q-Bessel function defined in Ex. 1.24 can be expressed as

for Ixl < 2. (Rahman [1987]) 3.16 Show that 1 a, -qa"2, b, c axq ] 4¢3 [ 1 ; q, -b- -a"2, aq/b, aq/c c

_ (l-xa~)(axq;q)oo J. [a~,-qa~,(aq)~,-(aq)~,aq/bc. ] - () 5'1-'4 / / / ' q, q x; q 00 aq b, aq c, q x, axq (1 - a~)(aq, aq/bc, axq/b, axq/c; q)oo +~~~~~~--~~~~~- (aq/b,aq/c,axq/bc,l/x;q)oo

J. [xa~,-xqd,x(aq)~,-x(aq)~,aXq/bc ] x 5'1-'4 ;q,q . axq/b,axq/c,qx,aqx2 Exercises 105

(cz/ab; q2)00 ( /) ~ n 3.17 If (. 2) 2¢1 a,b;c;q,cz abq = ~anZ , z, q 00 n=O show that

(Singh [1959], Nassrallah [1982])

(cqz/ab; q2)00 ( / ) ~ n 3.18 If ( . 2) 2¢1 a, b; c; q, cz ab = ~ anz ,show that z,q 00 n=O 2¢1(cq/a, cq/b; cq2; q2, z) 2¢1(a, b; c; q2, cz/ab)

(Singh [1959], Nassrallah [1982])

(cqz/ab; q2)00 (/ / /) ~ n 3.19 If ( . 2) 2¢1 a q, b; c q; q, cz ab = ~ anz ,show that z,q 00 n=O 2¢1(cq/a, c/bq; c; q2, z) 2¢1(a, b; c; q2, cz/ab)

(Singh [1959], Nassrallah [1982]) 3.20 Prove that

when max(lpl, Iql, Ibl) < 1, and extend this to the bibasic transformation formulas ~ 1 - apkqk (a;ph(c/b; qh bk 6 1- a (q; qh(abp;ph

= 1 - c ~ (ap;ph(c/b; q)k (bq)k 1 - b 6 (q; q)k(abp;p)k

= 1 - c ~ (ap;p)k(cq/b; qh bk 1- abp 6 (q;q)k(abp2;ph = (1 - c)(ap;p)oo f (b;ph(cqpk; q)oo (ap)k (1 - b) (abp; p)oo k=O (p; p)k (bqpk; q)oo

when max(lpl, Iql, lapl, Ibl) < l. (Gasper [1989a]) 106 Additional summation, transformation, and expansion formulas

3.21 Derive the quadbasic transformation formula

~ (1 - apkqk)(1 - bpkq-k) (a, b;p)de, a/be; q)k t:a (1 - a)(I- b) (q, aq/b; q)k(ap/e, bep;p)k x (CP-n /A, p-n / BC; Ph(Q-n, BQ-n /A; Qh k (Q-n /C, BCQ-n /A; Qh(p-n /A, p-n / B; P)k q (ap, bP;P)n(eq, aq/be; q)n (Q, AQ/ B; Q)n(AP/C, BCP; P)n (q, aq/b; q)n(ap/e, bep;P)n (AP, BP; P)n(CQ, AQ/BC; Q)n x ~ (1 - APkQk)(1 - BpkQ-k) (A, B; P)k(C, A/ BC; Q)k t:a (I-A)(I-B) (Q,AQ/B;Q)k(AP/C,BCP;Ph x (ep-n /a,p-n /be;p)k(q-n, bq-n fa; qh Qk (q-n /e, beq-n fa; q)k(p-n /a,p-n /b;ph

for n = 0,1, .... Use it to derive the mixed bibasic and hypergeometric transformation formula

(C - A - n) k ( - B - C - n) k (/lB - jJ,A - n) k ( -n) k k X (_ jJ,C - n) k (jJ,B + jJ,C - jJ,A - n) k ( - A - n) k ( - B - n) k q (ap, bP;P)n(eq, aq/be; q)n n!(jJ,A + 1 - jJ,B)n(A + 1 - C)n (q, aq/b; q)n(ap/e, bep;P)n (A + l)n(B + l)n(jJ,C + l)n

x -,----..,..-'----'--.".,,---(B+C+l)n (jJ,A + 1 - jJ,B - jJ,C)n

x ~ (A + k + k/jJ,)(B + k - kjJ,) (A)k(B)k (jJ,Ch (jJ,A - jJ,B - jJ,C)k t:a AB k!(jJ,A + 1 - jJ,B)k(A + 1 - Ch(B + C + l)k (e/apn,l/bepn;p)k(q-n,b/aqn;q)k x (l/eqn,be/aqn;q)k(l/apn,l/bpn;p)k'

and the following transformation formula for a "split-poised" 104>9 series

a, qVa, -qVa, b, e, a/be, C/Aqn, 1/ Bcqn, B/Aqn, q-n ] [ 104>9 Va, -Va, aq/b, aq/e, beq, l/cqn, BC/Aqn, 1/ Bqn, I/Aqn; q, q (aq, bq, eq, aq/be, Aq/ B, Aq/C, BCq; q)n (Aq, Bq, Cq, Aq/BC, aq/b, aq/e, beq; q)n A, qVA, -qVA, B, c, A/ BC, e/aqn, l/beqn, b/aqn, q-n ] [ X 104>9 VA,-VA,Aq/B,Aq/C,BCq,l/eqn,be/aqn,l/bqn,l/aqn;q,q .

(Gasper [1989a]) Exercises 107

3.22 Using the observation that, for arbitrary (fixed) positive integers ml, ... , mr,

00

L A(kl , ... , kr)Z~l ... Z~r kl, ... ,kr=O 00 =L M =0 kl ml +···+krmr=M kl, ... ,kr?O show that (3.7.14) implies the multivariable bibasic expansion formula

00

00 ( apn q n , bp n q -no,p ) j n j j X L ~~j+nX q j=O (q, aq2n+1 jb; q)j

(1 - apM qM) (1 - bpM q-M) (q-n, aqn jb; q) M x k1ml +···+krmr=M:S:n kl, ... ,kr?O

X A(kl , ... , kr)(wm1 Zd k1 ... (wmr zr)kr, which is equivalent to (3.7.14) and extends Srivastava [1984, (10)]. (Gasper [1989a]) 3.23 Prove the following q-Lagrange inversion theorem: If 00 Gn(x) = Lbjnxj , j=n where bjn is as defined in (3.6.20), and if

00 00 f(x) = L /jxj = L cnGn(x), j=O n=O then j /j = Lbjncn n=O and, vice versa, n

Cn = Lanj/j, j=O where anj is as defined in (3.6.19). (Gasper [1989a]) 3.24 Derive (3.8.19)-(3.8.22). 108 Additional summation, transformation, and expansion formulas

3.25 Prove Gosper's formula

00 (a2b2c2q-\ q2)n(1- a2b2c2q3n-l ) (abcd, abcd-\ q2)n(a2, b2, c2; q)n n ~ (q; q)n(1- a2b2c2q-l ) (abcd-l, abed; q)n(b2c2q, e2a2q, a2b2q; q2)n q (a2b2c2q,a2q,b2q,c2q, dq/abe, bcdq/a, cdaq/b, dabq/c; q2)00 (q, b2c2q, c2a2q, a2b2q, abcdq, adq/bc, bdq/ae, cdq/ab; q2)00 (a2, b2, c2; q)oo (a2b2c2q; q2)00 (q; q)00(b2e2q, c2a2q, a2b2q, abedq, abed-1q; q2)00 x I:(1- d2q4n+2) (bcdq/a,cdaq/b,dabq/c;q2)n (_~)n q(n+l)2. n=O (adq/bc, dbq/ac, cdq/ab; q2)n+l abc (Rahman [1993]) 3.26 Show that ~ (abcq;q)k(l -abcq3k+l)(d,q/d;q)k(abq,bcq,caq;q2)k k ~ (q2;q2)k(1 -abcq)(abcq3/d,abcdq2;q2)k(cq,aq,bq;q)k q k=O abeq, q2Jabcq, -q2.jabcq, d, q/d, abq, bcq, eaq [ 2 2] 8 ¢7 r:::c:::. r:::c:::. 3 2 2 2 2 ; q ,q = yabcq,-yabcq,abcq /d,abcdq ,cq ,aq ,bq + (abcq3, abq, bcq, caq, d, q/ d; q2)00 q (q2, aq2, bq2, cq2, abcq3 / d, abcdq2; q2)00 (1 - aq) (1 - bq) (1 - cq) q2,abCq2 ,dq,q2/d 2 2] X 4¢3 [ 3 b 3 3 ; q ,q . aq , q ,cq

3.27 Prove I: (bedq-2; q3)n(1- bcdq4n-2)(b, c, d; q)n n2 n=O (q; q)n(1- bcdq-2) (cdq, bdq, bcq; q3)n q

q-n, ql-n, q2-n, bcdq3n [ 3 3] X 4¢3 b 2 2 d 2 ; q ,q q ,eq , q (bcdq, bq, cq,dq;q3)00 (q,cdq, bdq, bcq; q3)00 . (Rahman [1993]) 3.28 Show that ~ (bcdq-\ q3)n(1- bcdq4n-l )(b, c, d; q)n n2+n ~ (q; q)n(l - bedq-l) (edq2, bdq2, beq2; q3)n q

q-n-l, q-n, ql-n, bcdq3n [ 3 3] X 4¢3 b d ; q ,q q,cq, q (bcdq2,bq2, cq2, dq2; q3)00 (q2,cdq2,bdq2,bcq2;q3)00 . (Rahman [1993]) Exercises 109

3.29 Derive the summation formulas: (i) ~ (1 - aq5k)(a, b; q2)k(ab2 j q3; q3)k(q2 jb; q)daq3 jb; q6)k (_ q2) k q(k!l) 6 (1 - a)(q3, aq3 jb; q3h(q5 jb2; q2)k(ab, abq2; q4)k b (aq2,qa3jb;q2)00(ab2,q9jb3;q6)00 (ab, q5 jb2; q2)00(q3, aq6 jb; q6)00 ' (ii) ~ (1 - acq5k)(a,q4ja;q2)k(q5jac;q3)k(acjq2;q)k 6 (1 - ac)(cq3, a2cjq; q3h(a2c2 j q3; q2)k (a 2c2jq;q6)k ( ac)k (k+l) x -- q 2 (q4,q6;q4)k q2 (acq2,acjq;q2)00(q6,a3c2jq3,ac2q,a2c2jq,aq4,q8ja;q6)00 (q4,a2c2jq3;q2)00(cq3,cq6,a2cq2,a2cjq,acq,acq4;q6)00 ' (iii) ~ (1 - acq5k)(a, q2 ja; q2h(qjac; q3h(ac; q)k(a2c2q; q6)k (_ac)kq(k!l) 6 (1 - ac)(cq3,a2cq;q3)k(a2c2q;q2)k(q2,q4;q4)k (acq2,acq;q2)00(q6,a3c2q3,ac2q5,a2c2q,aq2,q4ja;q6)00 (q2,a2c2q;q2)00(cq3,cq6,a2cq4,a2cq,acq2,acq5;q6)00

(Rahman [1989b])

3.30 Derive the quartic summation formula

00 1 _ aq5k (a, b; q)k(qjb, q2 jb, q3 jb; q3)k(a2b2 jq2; q4)k k ~ 1- a (q4, aq4jb;q4h(abq,ab,abjq;q2)k(q3jab2;q)k q + ab3 (aq, bq, 1jb; q)00(a2b2q2; q4)00 q2 (ab, q3 j ab2; q)oo (abj q; q2)00 (q4, ab3 j q2, aq4 jb; q4)00 a2b2jq2 ] x l¢l [ a2b2q2 ;q4,ab3q2 (aq, ab2 j q2; q)oo (ab; q)oo (abj q; q2)00 (aq4 jb, ab3 j q2; q4)00 . (Gasper [1989a])

3.31 Derive the cubic transformation formulas (i) ~ 1- acq4k (a, b; qh(cqjb; qhk(a2bcq3n, q-3n; q3h k 6 1- ac (cq3, acq3 jb; q3h(ab; qhk(ql-3n jab, acq3n+l; q)k q (1 -acq2)(1 -abjq2)(1 -abq3n)(1 _ acq3n) (1 - acq3n+2) (1 - abq3n-2)(1 - ab)(l - ac) 110 Additional summation, transformation, and expansion formulas

X n 1 _ aeq6k+2 (aq2,bq2,eq2/b,eq3/b,a2beq3n,q-3n;q3)k 3k t; 1- aeq2 (eq3,aeq3/b,abq3,abq2,q5-3n/ab,aeq3n+5;q3)k q ,

(ii) ~ (1 - aq4k)(1 - bq-2k) (a, 1/ab; q)k(abq; qhk(e, a/be; q3)k q3k ~ (1 -a)(1 -b) (q3,a2bq3;q3)k(q/b;q)2k(aq/e,beq;q)k (aq,bq;q)n(eq3, aq3/be;q3)n (aq/e,beq;q)n(q3, aq3/b;q3)n x lOWg(a/b; q/ab2, e, a/be, aqn+l, aqn+2, aqn+3, q-3n; q3, q3), where n = 0,1, .... (Gasper and Rahman [1990]) 3.32 Derive the cubic summation formula

00 1_a2q4k (b,q2/b;qh(a2/q;q)2k(e3,a2q2/e3;q3h k t; 1 - a2 (a2q3/b, a2bq; q3h(q2; qhk(a2q/e3, e3/q; q)k q (bq2,q4/b,be3/q,e3q/b,e3/a2,e3q2/a2,a2q,a2q3;q3)00 (q2,q4,e3q,be3/a2,a2q3/b,a2bq,e3q2/a2b;q3)00 (b,bq,bq2,q2/b,q3/b,q4/b,a2/q,a2q,a2q3,e3/a2;q3)00 (q2,q4,e3/q,c3q,a2/e3,a2q/e3,e3q3/a2,c3q3/a2b,a3q3/b;q3)00 (e6q/a2; q3)00 [be3/a 2, c3q2/a2b 3 3] X (a2bq,be3/a2;q3)00 2¢1 e6q/a2 ;q ,q

and show that it has the q ----; 1- limit case Po [ a - 1/2, a, b, 2 - b, c, (2a + 2 - 3c)/3, a/2 + 1 .1] 7 6 3/2, (2a - b + 3)/3, (2a + b + 1)/3, 3c - 1, 2a + 1 - 3e, a/2' r (~) r (~) r (e - ~) r (c + ~) r (2a-3b+3) r(bt2)r(43b)r(b+~-1)r(3c-:+l)r(2atl) x r (~) r (2a-~c+3) r (2a-~c+l) r (2at3) r (2a-3~-b+3) r (2a-3~+b+l)· (Gasper and Rahman [1990]) 3.33 Derive the quartic transformation formula

00 1 - a2b2q5k-2 (a, b; q)k(ab/q, ab, abq; q3)k(a2b2/q2; q4h k t; 1 - a2b2/ q2 (ab2q2, a2bq2; q4 h (abq, ab, ab/ q; q2)k (q; qh q _ (aq, b; q)00(a2b2q2; q4)00 ¢ [ a . 4 b 4] - (q; q)oo (abq; q2)00 (b, ab2q2, a2bq2; q4)00 1 1 aq4' q , q . (Gasper and Rahman [1990]) 3.34 Show that

.t, [ q-2n,e2,a,qa . 2 2] _ (-q,qa/e;q)n 4'f'3 q2 a 2 , eq -n , eq I-n , q , q - ( -aq, q/). c; q n Notes 111

3.35 For k = 1,2, ... , show that f (a; qk)n(b; qhn tn _ (b; q)oo(at; qk)oo f (c/b; q)n(t; qk)n bn n=O (qk; qk)n(c; qhn - (c; q)oo(t; qk)oo n=O (q; q)n(at; qk)n .

(Andrews [1966c]) 3.36 For the q-exponential function defined in (1.3.33) prove Suslov's addition formula

where

( 2 2) 00 n 2 /4 £ (x,y;a) = a .' q 00 L --(ae-''I'tq "'" ( _q-2I-n e'"0+"'" ''I',_q-2 I-n"","o) e''I'-';q q (qa2; q2)00 n=O (q; q)n n

with x = cos e, y = cos cP, 0 ::::; e ::::; 7f and 0 ::::; cP ::::; 7f. (Suslov [1997, 2003]) 3.37 Derive the quadratic transformation formula

(-a; q1/2)00 A-. [q1/4 eiO, q1/4 e-iO. 1/2 ] £(x;a)= ( 2.2) 2'/-'1 1/2 ,q ,-a. qa ,q 00 -q (Ismail and Stanton [2002]) Notes

§3.4 Bressoud [1987] contains some transformation formulas for terminating r+ 1 CPr (a1' a2, ... , ar+1; b1, ... , br; q; z) series that are almost poised in the sense that bkak+1 = a1qbk with 15k = 0,1 or 2 for 1 ::::; k ::::; r. 1fansformations for level basic series, that is r+1CPr series in which a1bk = qak+1 for 1 ::::; k ::::; r, are considered in Gasper [1985]. §3.5 For a comprehensive list of q-analogues of the quadratic transfor­ mation formulas in §2.11 of Erdelyi [1953], see Rahman and Verma [1993]. §3.6 Agarwal and Verma [1967a,b] derived transformation formulas for certain sums of bibasic series by applying the theorem of residues to contour integrals of the form (4.9.2) considered in Chapter 4. Inversion formulas are also considered in Carlitz [1973] and W. Chu [1994b, 1995] and, in connection with the Bailey lattice, in A.K. Agarwal, Andrews and Bressoud [1987]. §3.7 Jackson [1928] applied his q-analogue of the Euler's transformation formula (the p = q case of (3.7.11)) to the derivation of transformation for­ mulas and theta function series. Jackson [1942, 1944] and Jain [1980a] also derived q-analogues of some of the double hypergeometric function expansions in Burchnall and Chaundy [1940, 1941]. §3.8 Gosper [1988a] stated a strange q-series transformation formula con­ taining bases q2, q3, and q6. Krattenthaler [1989b] independently derived the terminating case of (3.8.18) and terminating special cases of some of the other summation formulas in this section. For further results on cubic and quintic summation and transformation formulas, see Rahman [1989d, 1992b, 1997]. 112 Additional summation, transformation, and expansion formulas

§3.9 For multibasic series containing bibasic shifted factorials of the form (a; p, q)r,s = TI;:~ TI~:~(1- api qk) and connections with Schur functions and permutation statistics, see Desarmenien and Foata [1985-1988]. §3.10 Jain and Verma [1986] contains nonterminating versions of some of Nassrallah and Rahman's [1981] transformation formulas. Ex.3.11 q-Differential equations for certain products of basic hypergeo­ metric series are considered in Jackson [1911]. Exercises 3.17-3.19 These exercises are q-analogues of the Cayley [1858] and Orr [1899] theorems (also see Bailey [1935, Chapter X], Burchnall and Chaundy [1949]' Edwards [1923]' Watson [1924]' and Whipple [1927, 1929]). Other q-analogues are given in N. Agarwal [1959], and Jain and Verma [1987]. Ex. 3.20 The formula obtained by writing the last series as a multiple of the series with argument bp is equivalent to the bibasic identity (21.9) in Fine [1988], and it is a special case of the Fundamental Lemma in Andrews [1966a, p. 65]. Applications of the Fundamental Lemma to mock theta functions and partitions are contained in Andrews [1966a,b]. Agarwal [1969a] extended An­ drews' Fundamental Lemma and pointed out some expansion formulas that follow from his extension. Ex. 3.23 For additional material on Lagrange inversion, see Andrews [1975b, 1979a], Bressoud [1983b], Cigler [1980], Fiirlinger and Hofbauer [1985], Garsia [1981]' Garsia and Remmel [1986], Gessel [1980], Gessel and Stanton [1983, 1986], Hofbauer [1982, 1984]' Krattenthaler [1984, 1988, 1989a], Paule [1985b], and Stanton [1988]. 4

BASIC CONTOUR INTEGRALS

4.1 Introduction

Our first objective in this chapter is to give q-analogues of Barnes' [1908] contour integral representation for the hypergeometric function

F ( b. . ) = f(c) _1 jiOO f(a+s)f(b+s)f(-s)(_ )8 d 21 a, ,c,z r(a)r(b)2ni -ioo r(c+s) Z s, (4.1.1) where larg( -z)1 < n, Barnes' [1908] first lemma

~ jiOO r(a + s)r(b + s)r(c - s)f(d - s) ds 2nz -ioo r(a + c)f(a + d)r(b + c)f(b + d) (4.1.2) f(a + b + c + d) and Barnes' [1910] second lemma _1_jiOO f(a + s)r(b + s)f(c + s)r(1 - d - s)f( -s) ds 2ni -ioo f( e + s) f(a)f(b)f(c)f(1 - d + a)f(1 - d + b)f(1 - d + c) (4.1.3) r(e - a)r(e - b)f(e - c) where e = a + b + c - d + 1.

In (4.1.1) the contour of integration is the imaginary axis directed upward with indentations, if necessary, to ensure that the poles of f( -s), i.e. s = 0,1,2, ... , lie to the right of the contour and the poles of f( a + s )f(b + s), i.e. s = -a - n, -b - n, with n = 0,1,2, ... , lie to the left of the contour (as shown in Fig. 4.1 at the end of this section). The assumption that there exists such a contour excludes the possibility that a or b is zero or a negative integer. Similarly, in (4.1.2), (4.1.3) and the other contour integrals in this book it is assumed that the parameters are such that the contour of integration can be drawn separating the increasing and decreasing sequences of poles. Barnes' first and second lemmas are integral analogues of Gauss' 2Fl sum­ mation formula (1.2.11) and Saalschiitz's formula (1.7.1), respectively. In Askey and Roy [1986] it was pointed out that Barnes' first lemma is an exten­ sion of the beta integral (1.11.8). To see this, replace b by b - iw, d by d + iw and then set s = wx in (4.1.2). Then let w ----; CXJ and use Stirling's formula to obtain the beta integral in the form

I: x++ c- 1 (1 - x)~+d-ldx = B(a + c, b + d), (4.1.4)

113 114 Basic contour integrals

where Re(a + c) > 0, Re(b + d) > 0 and

if x 2: 0, x+ -_ {X (4.1.5) o if x < O.

It is for this reason that Askey and Roy call (4.1.2) Barnes' beta integral. Following Watson [1910]' we shall give a q-analogue of (4.1.1) in §4.2, that is, a Barnes-type integral representation for 2 (/JI (a, b; c; q, z). It will be used in §4.3 to derive an analytic continuation formula for the 2¢1 series. We shall give q-analogues of (4.1.2) and (4.1.3) in §4.4. The rest of the chapter will be devoted to generalizations of these integral representations, other types of basic contour integrals, and to the use of these integrals to derive general transformation formulas for basic hypergeometric series.

y

c

• • • • .) -a

______~~~------~~------.------+-x " 0 2 3

• • • • .J -b

Fig. 4.1 4.2 Watson's contour integral 115

4.2 Watson's contour integral representation

for 2¢>1(a,b;c;q,z) series

For the sake of simplicity we shall assume in this and the following five sections that 0 < q < 1 and write

q = e-w , w > o. (4.2.1)

This is not a severe restriction for most applications because the results derived for 0 < q < 1 can usually be extended to complex q in the unit disc by using analytic continuation. The restriction 0 < q < 1 has the advantage of simplifying the proofs by enabling one to use contours parallel to the imaginary axis. Extensions to complex q in the unit disc will be considered in §4.8. For 0 < q < 1 Watson [1910] showed that Barnes' contour integral in (4.1.1) has a q-analogue of the form

(4.2.2) where Iz I < 1, Iarg( - z) I < 1r. The contour of integration (denote it by C) runs from -ioo to ioo (in Watson's paper the contour is taken in the opposite direction) so that the poles of (ql +s; q) 00 / sin 1r s lie to the right of the con­ tour and the poles of 1/(aqS, bqs; q)oo, i.e. s = w-1loga - n + 21rimw-1, s = w-1logb - n + 21rimw-1 with n = 0, 1,2, ... , and m = 0, ±1, ±2, ... , when a and b are not zero, lie to the left of the contour and are at least some E > 0 distance away from the contour. To prove (4.2.2) first observe that by the triangle inequality,

and so

(ql+s,cqs;q)oo I I (aqS, bqS; q)oo

00 (1 + e-(n+H Re(s))w) (1 + Icle-(n+ Re(s))w) (4.2.3) :s; 11 (1-l ale-(n+ Re(s))w) (1 -Ible-(n+ Re(s))w) ' which is bounded on C. Hence the integral in (4.2.2) converges if Re[slog(-z) -log(sin1rs)] < 0 on C for large lsi, i.e. if larg(-z)1 < 1r. Now consider the integral in (4.2.2) with C replaced by a contour CR consisting of a large clockwise-oriented semicircle of radius R with center at the origin that lies to the right of C, is terminated by C and is bounded away 116 Basic contour integrals from the poles (as shown in Fig. 4.2). y r--~

c

• •

• • •

------~~------.------.---_r--~-x o 1 2 R 3

• •

• • •

Fig. 4.2

Setting S = ReiIJ , we have for Izl < 1 that Re [log (~z)S] SIn 7rS = R [cos e log Izl - sine arg( -z) - 7r1 sinel] + 0(1) ~ -R [sine arg( -z) + 7r1 sinel] + 0(1). Hence, when Izl < 1 and larg( -z)1 < 7r - 0, 0 < 0 < 7r, we have

(~z)S = 0 [exp( -oRI sinel)] (4.2.4) SIn 7rS on CR as R ---+ 00, and it follows that the integral in (4.2.2) with C replaced by CR tends to zero as R ---+ 00, under the above restrictions. Therefore, by applying Cauchy's theorem to the closed contour consisting of CR and that part of C terminated above and below by CR and letting R ---+ 00, we obtain that - --21 . Ji~ ... ds equals the sum of the residues of the integrand at 'ITt -'/;00 n = 0,1,2, .... Since . (q1+s, cqS;q) 7r(-z)S hm (s - n) 00 ---'-----'- s-+n (aqs,bqS;q)oo sin7rs 4.3 Analytic continuation of 21>1 (a, b; C; q, z) 117 this completes the proof of Watson's formula (4.2.2). It should be noted that the contour of integration in (4.2.2) can be replaced by other suitably indented contours parallel to the imaginary axis. To see that (4.2.2) is a q-analogue of (4.1.1) it suffices to use (1.10.1) and the reflection formula 7r r(x)r(1 - x) = -.- (4.2.5) SIn 7rX to rewrite (4.2.2) in the form 21>1 (qa,qb;qc;q,z)

= rq(c) 1 fiOO r q(a+s)rq(b+s)r(-s)r(1+s)(_z)Sds. rq(a)rq(b)27ri -ioo rq(c+s)rq(1+s) (4.2.6)

4.3 Analytic continuation of 21>I(a, b; C; q, z)

Since the integral in (4.2.2) defines an analytic function of z which is single­ valued when larg( -z)1 < 7r, the right side of (4.2.2) gives the analytic continu­ ation ofthe function represented by the series 21>I(a, b; c; q, z) when Izl < 1. As in the ordinary hypergeometric case, we shall denote this analytic continuation of 21>I(a,b;c;q,z) to the domain larg(-z)1 < 7r again by 21>I(a,b;c;q,z). Barnes [1908] used (4.1.1) to show that if larg( -z)1 < 7r and a, b, c, a - b are not integers, then the analytic continuation for Izl > 1 of the series which defines 2Fl(a, b; C; z) for Izl < 1 is given by the equation r(c)r(b-a) -a -1 2Fl(a,b;c;z) = r(b)r(c-a)(-z) 2Fl(a,1+a-c; 1+a-b;z ) r(c)r(a - b) -b + r(a)r(c-b)(-z) 2Fl(b,1+b-c; 1+b-a;z-1), (4.3.1) where, as elsewhere in this section, the symbol "=" is used in the sense "is the analytic continuation of". To illustrate the extension of Barnes' method to 21>1 series we shall now give Watson's [1910] derivation of the following q-analogue of (4.3.1): 21>1 (a, b; C; q, z) (b, cl a; q)oo (az, qlaz; q)oo ( I I I) = (bl ) ( I ) 21>1 a, aq C; aq b; q, cq abz c, a; q 00 z, q Z; q 00 (a, clb; q)oo (bz, qlbz; q)oo ( I I I) + ( Ib) ( I ) 21>1 b, bq C; bq a; q, cq abz , (4.3.2) c, a ; q 00 z, q Z; q 00 provided that larg( -z)1 < 7r, c and alb are not integer powers of q, and a, b, z =I=­ o. First consider the integral h = (ql+s, cqS;q)oo7r(_z)S ds _1-/ (4.3.3) 27ri (aqs, bqs; q)oo sin 7rS 118 Basic contour integrals along three line-segments AI, A 2 , B, whose equations are:

Al : Im(s) = ml, A 2 : Im(s) = -m2, B: Re(s) = -M, ( 4.3.4) where ml,m2,M are large positive constants chosen so that A l ,A2 ,B are at least a distance E > 0 away from each pole and zero of (ql+s, cqs; q)oo g(s) = () (4.3.5) aqS, bqS; q 00 and it is assumed that AI, A 2 , B are terminated by each other and by the contour of the integral in (4.2.2), i.e. Re(s) = 0 with suitable indentations. From an asymptotic formula for (a; q)oo with q = e-w , W > 0 , due to Littlewood [1907, §12]' it follows that if Re(s) ---. -00 with Is - sol> E for some fixed E > 0 and any zero So of (qS; q)oo, then W 2 W Re[log(qS; q)ool = "2(Re(s)) +"2 Re(s) + 0(1). (4.3.6) This implies that

g(s) = 0 (I :~ IRe(S)) , (4.3.7) when Re(s) ---. -00 with s bounded away from the zeros and poles of g(s). By using this asymptotic expansion and the method of §4.2 it can be shown that if Izl > Icqjabl, then the value of the integral II in (4.3.3) taken along the contours AI, A2, B tends to zero as ml, m2, M ---. 00. Hence, by Cauchy's theorem, the value of h, taken along the contour C of §4.2, equals the sum of the residues of the integrand at its poles to the left of C when Izl > Icqjabl. Set 0: = -w-lloga, f3 = -w-llogb so that a = q''',b = qf3. Since the residue of the integrand at -0: - n + 27rimw-l is ( -1 l-n -1 -n n+l ) a q ,ca q ,q ; q 00 -l( )-a-n n(n+l)/2 -1 -n 7rW -z q (q, q, ba q ; q)oo x exp {2m7riw-l log( -z)} csc (2m7r2iw-l - 0:7r) , we have 00

m=-oo 7rw- l (cja, qja; q)oo ( )-a ( j j j) x (bj .) -z 2¢1 a, aq c; aq b; q, cq abz a, q, q 00 + idem (a; b). (4.3.8) Thus it remains to evaluate the above sums over m when larg(-z)1 < 7r. Letting c = bin (4.3.8) and using (4.2.2), we find that the analytic continuation of 2¢1(a,b;b;q,z) is

00

m=-oo 7rw- l (a, qja;q)oo( )-a ( j j j) x ( ) -z 2¢1 a, aq b; aq b; q, q az . q,q;q 00 4.4 q-analogues of Barnes' first and second lemmas 119

Since, by the q-binomial theorem, ( ) (az; q)oo 2rP1 a, b; b; q, z = ( ) z;q 00 and the products converge for all values of z, it follows that

00

m=-oo (4.3.9) w(q,q,az, q/az; q)oo 7r(a,q/a,z,q/z;q)oo . Using (4.3.9) in (4.3.8) we finally obtain (4.3.2).

4.4 q-analogues of Barnes' first and second lemmas

Assume, as before, that 0< q < 1, and consider the integral

1 fiOO (q1-c+s q1-d+s. q) S d I _ _ "00 7rq S (4.4.1) 2 - 27ri -ioo (qa+s,qb+s;q)oo sin7r(c-s)sin7r(d-s)' where, as usual, the contour of integration runs from -ioo to ioo so that the increasing sequences of poles of the integrand (i.e. c + n, d + n with n = 0,1,2, ... ) lie to the right and the decreasing sequences of poles (i.e. the zeros of (qa+s, qb+s; q)oo) lie to the left of the contour. By using Cauchy's theorem as in §4.2 to evaluate this integral as the sum of the residues at the poles c + n, d + n with n = 0,1,2, ... , we find that c (q q1+c-d. q) I - 7rq , '00 A.. (a+c b+c. l+c-d. ) 2 -. ( d) (a+c b+c.) 2'/-'1 q ,q ,q ,q, q sm 7r c - q , q ,q 00 + idem (c; d). (4.4.2) Applying the formula (2.10.13) to (4.4.2), we get 1 fiOO (q1-c+s, q1-d+s; q)oo 7rqs ds 27ri -ioo (qa+s,qb+s;q)oo sin7r(c-s)sin7r(d-s)

qC (q, q1+c-d, qd-c, qa+b+c+d; q) 00 ( 4.4.3) sin 7r(c - d) (qa+c, qa+d, qb+c, qb+d; q)oo which is Watson's [1910] q-analogue of Barnes' first lemma (4.1.2), as can be seen by rewriting it in terms of q-gamma functions.

A q-analogue of Barnes' second lemma (4.1.3) can be derived by proceeding as in Agarwal [1953b]. Set c = nand d = c - a - bin (4.4.3) to obtain 1 fiOO (q1-n+s,q1-c+a+b+s;q)00 7rqs ds 27ri -ioo (qa+s,qb+s;q)oo sin7rssin7r(c-a-b-s) (q1+a+b-c, qc-a-b, qC, q; q)oo = csc 7r( C - a - b) -'---,----,------;----:---'-''''- (qa, qb, qc-a, qC-b; q)oo

X (_l)n (q(a,qb;)q)nqn(C-a-bl-G) (4.4.4) qC;q n 120 Basic contour integrals for n = 0,1,2, .... Next, replace c by d in (4.4.4), multiply both sides by (_I)nqn(e-c)+G) (qC; q)n / (q, qe; q)n' sum over n and change the order of inte­ gration and summation (which is easily justified if Iqe-c+sl < 1) to obtain

(ql+a+b-d, qd-a-b, qd, q; q) 00 csc 7f( d - a - b) --'------;----;----;-----;--;--;---'---''-''­ (qa,qb,qd-a,qd-b;q)oo x 3¢2 (qa, l, qC; qd, qe; q, qd+e-a-b-c) 1 jiOO (ql+s,ql-d+a+b+s;q)oo 7fqs = 27fi -ioo (qa+s,qb+s;q)oo sin7fssin7f(d-a-b-s) x 2¢1 (q-S, qC; qe; q, qe-c+s) ds

= (qe-c; q)oo ~ jiOO (ql+s, qe+s, ql-d+a+b+s; q) 00 (qe; q)oo 27f~ -ioo (qa+s, qb+s, qe-c+s; q)oo 7fqs ds x ( )' (4.4.5) sin 7f s sin 7f d - a - b + s by the q-Gauss formula (1.5.1). Now take c = d. Then the series on the left of (4.4.5) can be summed by the q-Gauss formula to give, after an obvious change in parameters,

1 jiOO (ql+S, qd+s, qe+s; q) 00 7fqs ds 27fi -ioo (qa+s, qb+ s, qc+s; q)oo sin 7fS sin 7f( d + s) d (q, qd, ql-d, qe-a, qe-b, qe-c; q) 00 =~7f , ( 4.4.6) (qa, qb, qC, ql+a-d, ql+b-d, ql+c-d; q)oo where d + e = 1 + a + b + c, which is Agarwal's q-analogue of Barnes' second lemma. This integral converges if q is so small that Re [s log q - log(sin 7fS sin 7f(d + s ))] < 0 ( 4.4.7) on the contour for large lsi-

4.5 Analytic continuation of r+l¢r series

By employing Cauchy's theorem as in §4.2, we find that if Izl < 1 and larg(-z)1 < 7f, then

.t.. [al ,a2, ... ,ar+l . ] r+l 'f'r b b ' q, Z 1, ... , r (aI, a2,···, ar+l; q)oo (q, bl , ... , br; q)oo X (2)jiOO (ql+S,b1qS, ... ,brqS;q)00 7f(-z)Sds (4.5.1) 27fi -ioo (alqS, a2qs, ... , ar+lqS; q)oo sm 7fS where, as before, only the poles of the integrand at 0,1,2, ... , lie to the right of the contour. As in the r = 1 case, the right side of (4.5.1) gives the analytic continuation of the r+l¢r series on the left side to the domain larg( -z)1 < 7f. 4.6 Contour integrals representing well-poised series 121

Also, as in §4.3, it can be shown that if Izl > Ib1 ... brq/a1 ... ar+11, then the integral I3 = 2;'i t,:,oo ... ds is equal to the sum of the residues of the integrand at those poles of the integrand which lie on the left of the contour. Set a1 = -w-110g a1. Since the residue ofthe integrand at -a1 -n+27rimw-1 IS n+1 -1 1-n b -1 -n b -1 -n ) ( q ,a1 q ,1 a1 q , ... , ra1 q i q -1 _ _ (+1)/2 --'-----;------,------,------,------'---"oo=7rW (-z) <>1 nqn n (q, q, a2 al 1q-n, ... , ar+1 al 1q-ni q)oo x csc(2m7r2iw-1 - al7r) exp {2m7riw-110g( -z)}, by proceeding as in the proof of (4.3.2) and using (4.3.9) and (4.5.1) we obtain the expansion

(a2, ... , ar+1, bl/a1, ... , br/a1, a1 Z, q/a1z i q)oo (b1, ... ,br,a2/a1, ... ,ar+1/al,z,q/Ziq)00

A, [al,a1q/bl, ... ,alq/br. qb1 ... br ] r (4.5.2) X +1'i/r alq / a2,···,alq /'ar+l q, za1···ar+1 + idem (ali a2,···, ar+d, where the equality holds in the "is the analytic continuation of" sense. The symbol "idem (ali a2, ... , ar+d" after an expression stands for the sum of the r expressions obtained from the preceding expression by interchanging a1 with each ak, k = 2,3, ... , r + 1.

4.6 Contour integrals representing well-poised series

Let us replace a, b, c, d and e in (4.4.6) by a + n, b + n, c + n, d + nand e + 2n, respectively, where e = 1 + a + b + c - d, (4.6.1) and transform the integration variable s to s - n, where n is a non-negative integer. Then we get 1 I n+ioo (q1+s-n,qd+s,qe+s+niq)00 7rqs-n ds 27ri n-ioo (qa+s , qb+s, qc+s i q) 00 sin 7r s sin 7r (d + s) (q,qd+n,q1-d-n,qe-a+n,qe-b+n,qe-c+niq)00 d = CSC7r (b b b) . (4.6.2) qa+n,q +n,qc+n,qe-a- ,qe-a-c,qe- -ciq 00 The limits of integration n ± ioo can be replaced by ±ioo because we always indent the contour of integration to separate the increasing and decreasing sequences of poles. Thus, it follows from (4.6.2) that ( d I-d e-a e-b e-c ) (a b c ) csc(7rd) q,q ,q ,e ,q ,q iq 00 q ,q ,q iq n (l-d)n (qa, qb, qC, qe-b-c, qe-c-a, qe-a-bi q)oo (qe-a, qe-b, qe-ci q)n q 1 fioo (q1+S, qd+s, qe+s i q)oo 7rq(n+l)s(q-Si q)n ds = 27ri -ioo (qa+s, qb+ s, qc+s i q)oo sin 7rssin 7r(d + s)(qe+si q)n' (4.6.3) 122 Basic contour integrals

( 4.6.4) where e = 1 + a + b + c - d, Izl < Iqdl and it is assumed that (4.4.7) holds. Therefore, if we can sum the series on the right of (4.6.4), then we can find a simpler contour integral representing the series on the left. In particular, if we let r = 4,e = 1 +A,al = ql+A/2 = -a2,a3 = qd,a4 = qe,

b1 - qA/2 - -b2, b3 - ql-d+A , b 4 - ql-e+A ,-Z _ q2-d-e+A , then we get a VWP-balanced 6¢5 series which can be summed by (2.7.1). This yields Agarwal's [1953b] contour integral representation for a VWP-balanced 8¢7 series: qA,ql+A/2,_ql+A/2,qa,qb,qc,qd,qe B] [ 8¢7 qA/2, _qA/2, ql+A-a, ql+A-b, ql+A-c, ql+A-d, ql+A-e ; q, q

= sin 7r(a + b + c - A)

7rqS ds x ------~~------~ (4.6.5) sin 7rS sin 7r( a + b + c - A + s) , where B = 2 + 2A - a - b - c - d - e, provided Re(B) > 0 and Re [s log q -log(sin7rssin7r(a + b+ c - A + s))] < o.

If we evaluate the integral in (4.6.5) by considering the residues at the poles of 1 / [sin 7r s sin 7r( a + b + c - A + s ) ] lying to the right of the contour, then we obtain the transformation (2.10.10) of a VWP-balanced 8¢7 series in terms of the sum of two balanced 4¢3 series. In addition, if we replace A, d, e and a by A, A+d-A, A+e-A and A+a-A, respectively, and take A+a+d+e = 1+2A in (4.6.5), then the integral in (4.6.5) remains unchanged. This gives Bailey's transformation formula (2.10.1) between two VWP-balanced 8¢7 series. 4.7 Contour integral analogue of Bailey's formula 123

4.7 A contour integral analogue of Bailey's summation formula

By replacing A, a, b, c, d, e in (4.6.5) by a, d, e, f, b, c, respectively, we obtain the formula

s W7 (qa; qb, qC, qd, qe, qf; q, q) = sin n (d + e + f - a) (ql+a,qd,qe, qf,ql+a-d-e, ql+a-d-f, ql+a-e-f ; q)oo X ~~~~--~----~--~~------~--~~--~--~~-- (q, ql+a-b, ql+a-c, ql+a-d, ql+a-e, ql+a-f ,ql+a-d-e-f ; q)oo

1 (ql+S, ql+a-b+s, ql+a-c+s; q) 00 nqs ds x- jiOO 2ni -ioo (qd+s, qe+s, qf+s; q)oo sin ns sin n(d + e + f - a + s)' (4.7.1) provided the series is VWP-balanced, i.e., 1 + 2a = b + c + d + e + f. (4.7.2) Since 1 + 2(2b - a) = b + (b + c - a) + (b + d - a) + (b + e - a) + (b + f - a) by (4.7.2), it follows that (4.7.2) remains unchanged if we replace a, c, d, e, f by 2b - a, b + c - a, b + d - a, b + e - a, b + f - a, respectively, and keep b unaltered. Then (4.7.1) gives S W7 (q2b-a; qb, qb+c-a, qb+d-a, qb+e-a, qb+ f-a; q, q) = sin nc

( q 1+2b-a ,qb+d-a ,qb+e-a ,qb+f-a ,ql+a-d-e ,ql+a-d-f ,ql+a-e-f. ,qoo ) X ~------~~~~--~~--~~~~~--~~~~--~------~= (q,ql+b-a,ql+b-c, ql+b-d,ql+b-e,ql+b-f, qC; q)oo

1 jiOO (ql+b-a+s, ql+b-c+s, ql+s; q) 00 nqs ds x- 2ni -ioo (qb+ d- a+s, qb+e- a+s , qb+ f -a+s; q) 00 sin n s sin n( c - s) . (ql+2b-a, qb+d-a, qb+e-a, qb+f-a; q)oo = SIn nc """"'------,----:;-:-;-----:;-.,-;------,------'-'''''' (q, q1+b- a, q1+b- c, qC; q)oo

X (ql+a-d-e,ql+a-d- f ,ql+a-e-f ;q)oo 1 jiOO (q1+s;q)oo

(q1+b-d, q1+b-e , q1+b- f ; q) 00 2ni -ioo (qd+s; q) 00

(ql+a-c+s, ql+a-b+s; q) 00 nqs ds x (qe+s, qf+s; q)oo sinn(a-b+s)sinn(c+b-a-s)' (4.7.3) where the second integral in (4.7.3) follows from the first by a change of the in­ tegration variable s ----; a-b+s. Combining (4.7.1) and (4.7.3) and simplifying, we obtain (ql+a-b, ql+a-c, ql+a-d, ql+a-e, ql+a-f ; q)oo

(q1+a, qC, qd, qe, qf ; q) 00

X SW7 (qa;qb,qc,qd,qe,qf;q,q)

(ql+b-a ql+b-c ql+b-d ql+b-e ql+b-f . q) b-a ' , , , '(Xl - q (ql+2b-a, qb+c-a, qb+d-a, qb+e-a, qb+f-a; q)oo 124 Basic contour integrals

X 8 W7 (q2b-a; qb, qb+c-a, qb+d-a, qb+e-a, l+f-a; q, q)

( q I+a-d-e ,q I+a-e-f ,q I+a-d-f.,q(X) ) (q, qC, qb+c-a; q)(X)

l+s ql+a-b+s ql+a-c+s. q) s· (_ b) d 1 ji(x) (q , , '(x) nq SIn n a s x- 2ni -i(X) (qd+s, qe+s, qf+s; q)(X) sin ns sin n(a - b + s) , (4.7.4) when (4.7.2) holds. Evaluating the above integral via (4.4.6), we obtain Bailey's summation formula (2.11.7). Agarwal's [1953b] formula

1 i(X) (ql+S, q~a+s, _q~a+s, ql+a-b+s, ql+a-c+s; q) (X)

2ITi l,~ (q"+", q1+j "+', _q1+ j "+', qb+., qo+,; q) oc (ql+a-d+s,ql+a-e+s,ql+a-f+s;q)(X) nqs ds x ------~~--~--~--~----~------~--~--7 (qd+s , qe+s, qf+s; q)(X) sin ns sin n(a - b + s) b (q, ql+a-b, qb-a, ql+a-d-e, ql+a-e-f; q) (X) =cscn(a-) (qb,qc,qd,qe,qf;q)(X)

( q I+a-d- f ,qI+a-c-d ,qI+a-c-e ,qI+a-c- f. ,q ) (X) (4.7.5) x (qb+c-a, qb+d-a, qb+e-a, qb+ f -a; q) (X) , where 1 + 2a = b + c + d + e + j, follows directly from (2.11.7) by considering the residues of the integrand of the above integral at the poles to the right of the contour, i.e. at s = n, b - a + n with n = 0,1,2, .... Thus (4.7.5) gives an integral analogue of Bailey's summation formula (2.11.7). The integral in (4.7.5) converges if q is so small that Re [s log q - log (sin n s sin n (a - b + s) ) 1 < 0 (4.7.6) on the contour for large lsi-

4.8 Extensions to complex q inside the unit disc

The previous basic contour integrals can be extended to complex q inside the unit disc by using suitable contours. For 0 < Iql < 1, let (4.8.1) where WI = -log Iql > 0 and W2 = -Arg q. Thus q = e-(Wl +iW2). Then a modification of the proof in §4.2 (see Watson [1910]) shows that if 0 < Iql < 1 and Izl < 1, then formula (4.2.2) extends to

2¢1 (a, b; c; q, z) (a, b; q)(X) (4.8.2) (q,c;q)(X) 4.9 Other types of basic contour integral formulas 125 where C is an upward directed contour parallel to the line Re( s(WI + iW2)) = 0 with indentations, to ensure that the increasing sequence of poles 0, 1,2, ... , of the integrand lie to the right, and the decreasing sequences of poles lie to the left of C. Since the above integral converges if Re[s log( -z) - log (sin 7fs)] < 0 on C for large lsi, i.e., if (4.8.3) it is required that z satisfies (4.8.3) in order for (4.8.2) to hold. This restriction means that the z-plane has a cut in the form of the spiral whose equation in polar coordinates is r = eW1 (J/W2. Analogously, when 0 < Iql < 1, the contours in the q-analogues of Barnes' first and second lemmas given in §4.4 and the contours in the other integrals in §§4.4-4.7 must be replaced by upward directed contours parallel to the line Re(s(wl + iW2)) = 0 with indentations to separate the increasing and decreasing sequences of poles.

4.9 Other types of basic contour integrals

Let q = e-w with W > 0 and suppose that (aIz, ... ,aAz, bl/z, ... ,bB/z;q)oo P ()z = ~------~~----~~~~ (4.9.1) (CIZ, ... ,ccz, dl/z, ... ,dD/z;q)oo has only simple poles. During the 1950's Slater [1952c,d, 1955] considered contour integrals of the form

i7r W 1m == 1m(A,B;C,D) = -.W J / P(qS)qmsds (4.9.2) 27f~ -i7r /w with m = 0 or 1. However, here we shall let m be an arbitrary integer. It is assumed that none of the poles of P(qS) lie on the lines Ims = ±7f/w and that the contour of integration runs from -i7f / W to i7f / wand separates the increasing sequences of poles in 11m sl < 7f/w from those that are decreasing. By setting if) = -sw the integral 1m can also be written in the "exponen­ tial" form 1m = ~ J7r P (ei(J) eim(Jdf) (4.9.3) 27f -7r with suitable indentations, if necessary, in the contour of integration. Similarly, setting z = qS we obtain that

( 4.9.4) where the contour K is a deformation of the (positively oriented) unit circle so that the poles of 1/ (CIZ, ... , ccz; q)oo lie outside the contour and the origin and poles of 1/(dl/z, ... ,dD/z;q)oo lie inside the contour. Special cases of (4.9.3) and (4.9.4) have been considered by Askey and Roy [1986]. Although each of the above three types of integrals can be used to derive transformation formulas for basic hypergeometric series, we shall prefer to 126 Basic contour integrals use mainly the contour integrals of the type in (4.9.4) since they are easier to work with, especially when the assumption in Slater [1952c,d, 1966] that 0< q < 1 is replaced by only assuming that Iql < 1, which is the case we wish to consider in the remainder of this chapter.

4.10 General basic contour integral formulas

Our main objective in this section is to see what formulas can be derived by applying Cauchy's theorem to the integrals Im in (4.9.4). Let Iql < 1 and let 15 be a positive number such that 15 -=1= Idjqnl for j = 1,2, ... , D, and 15 -=1= Icjlq-nl for j = 1,2, ... ,0 when n = 0,1,2, .... Also let ON be the circle Izl = blqlN, where N is a positive integer. Then ON does not pass through any of the poles of P(z) and we have that IF (bqN) (bqN) m-l I = I (alb, ... ,aAb, bd 15, ... ,bB/b; q)oo I (clb, ... , ceb, ddb, ... , dD/b; q)oo x I (Clb, ... ,ceb,qb/bl, ... ,qb/bB;.q)N (bl ... bBqm-l)NI (alb, ... , aAb, qb/dl , ... , qb/dD, q)N dl ··· dD

D B D B l X bm-llbNq(Ntl) I - = 0 (I b ~~ .. ~~~:-l IN IbN q(Ntl) I - ) .

(4.10.1) Since ON is of length O(lqIN) it follows from (4.10.1) that if D > B or if D = B and (4.10.2) then lim 1 p(z)zm-ldz = O. (4.10.3) N--+oo eN Hence, by applying Cauchy's residue theorem to the region between K and ON for sufficiently large N and letting N ----+ 00, we find that if D > B or if D = B and (4.10.2) holds, then Im equals the sum of the residues of p(z)zm-l at the poles of 1/ (dd z, ... , dD/ z; q)oo' Therefore, since

. ( 1 ) (_I)ndq2n+G) ReSIdue = , n = 0,1,2, ... , (4.10.4) z = dqn (d/ z; q)oo (q; q)n (q; q)oo it follows that

(4.10.5) 4.10 General basic contour integral formulas 127 if D > B, or if D = Band (4.10.2) holds. In addition, by considering the residues of p(z)zm-l outside of K or by just using the inversion z -+ Z-l and renaming the parameters, we obtain 1 (blCl, ... ,bBcl,aI/cl, ... ,aAlcl;q)oo_m m= ~ (q, dlCl, ... , dDCl, c2/cl, ... , Cc ICl; q)oo x f (dlCl, ... , dDCl, qcI/al, ... , qcI/aA; q)n n=O (q, blCl, ... , bBCl, qCI/C2, ... , qcI/cc; q)n

(n+l)/2)n(C-A) (al ... aAq-m)n x ( -Clq Cl··· Cc + idem (Cl; C2, ... , cc) (4.10.6) if C > A, or if C = A and al ... aAq-ml < 1. (4.10.7) l Cl··· Cc In the special case when C = A we can use the reps notation to write (4.10.5) in the form (A ·A ) - (aldl, ... ,aAdl,bI/dl, ... ,bBldl;q)oo dm Im , B, , D - ( I I) 1 q,cl dl , ... ,CAdl ,d2 dl, ... ,dD dl;q 00

rI, [ cldl , ... , cAdI, qdI/bl , ... , qdI/bB . t( d )D-B] XA+B'I'A+D-l d d did did ,q, q 1 al 1,···,aA l,q 1 2,···,q 1 D + idem (dl ;d2, ... ,dD) (4.10.8) where t = bl b2 ··· bBqm Idl ... dD, if D > B, or if D = Band (4.10.2) holds.

Similarly, from the D = B case of (4.10.6) we have

1 (A B·C B) = (blCl, ... ,bBcl,aI/cl, ... ,aAlcl;q)oo -m m ", (q,dlCl, ... ,dBCl,C2/cl, ... ,cclcl;q)ooCl

rI, [dlCl, ... , dBCl, qCI/al, ... , qcI/aA. ( )C-A] XA+B'I'B+C-l I I' q, U qCl blCl, ... ,bBcl,qcl C2,···,qCl Cc + idem (Cl; C2, ... , cc) (4.10.9) if C > A, or if C = A and (4.10.7) holds, where u = al··· aAq-m ICI ... CC. Evaluations of Im which follow from these formulas will be considered in §4.11. From (4.10.8) and (4.10.9) it follows that if C = A and D = B, then we have the transformation formula (aldl , ... , aAdl , bI/dl , ... , bBldl ; q)oo dm (cldl , ... , cAdI, d2/dl, ... , dBI dl ; q)oo 1

rI, [Cldl, ... ,CAdl,qdI/bl, ... ,qdI/bB. bl ... bBqm ] XA+B'I'A+B-l d d did did ,q, d d al 1,···aA l,q 1 2,···,q 1 B 1··· B

+ idem (dl ; d2, ... , dB) (blCl, ... , bBCl, aI/cl, ... , aAlcl; q)oo -m = ~ (dlCl, ... , dBCl, c2/cl, ... , cAlcl; q)oo 128 Basic contour integrals

'" [dICI, ... ,dBCI,qcI/al, ... ,qcI/aA. al ... aAq- m ] XA+B 'f'A+B-I bICI,···, bBCI,qCI / C2,···,qCI /'CA q, CI···CA + idem (CI; C2, ... , CA) (4.10.10) provided that Ibl ·· ·bBqml < Idl ·· ·dBI, lal·· ·aAq-ml < ICI·· ·cAI and m = 0,±1,±2, .... In some applications it is useful to have a variable Z in the argument of the series which is independent of the parameters in the series. This can be accomplished by replacing A by A + 1, B by B + 1 and setting bB +I = Z and aA+I = q/z in (4.10.10). More generally, doing this to the m = 0 case of (4.10.5) and of (4.10.6) gives the rather general transformation formula (aldl , ... , aAdl , bI/dl , ... , bB/dl , z/dl , qdI/ z; q)(XJ (cldl , ... , CedI, d2/dl , ... , dD/dl ; q)(XJ x f (cldl , ... , CedI, qdI/bl , ... , qdI/bB; q)n n=O (q, aldl , ... , aAdl , qdI/d2, ... , qdI/dD; q)n

x (_dl q(n+IJ/2)n(D-B-IJ (bl ... bBZ) n dl ·· ·dD + idem (dl ; d2, ... , dD) (bl CI, ... , bBCI, aI/CI, ... , aA/ CI, CIZ, q/ CIZ; q)(XJ (dICI, ... , dDCI, C2/CI, ... , Ce /CI; q)(XJ x f (dICI, ... , dDCI, qCI/al, .. . , qcI/aA; q)n n=O (q, bICI, ... , bBCI, qcI/c2, ... , qcI/ce; q)n

X (_Clq(n+IJ/2)n(e-A-IJ (al ... aAq)n CI··· CeZ + idem (CI; C2, ... , ce), (4.10.11) where, for convergence, bl ... bBz I (i) ... D > B + 1, or D = B + 1 and I dl dD < 1 and al .. . aAql (ii) C > A + 1, or C = A + 1 and l < l. CI··· CeZ

This is formula (5.2.20) in Slater [1966]. Observe that by replacing Z in (4.10.11) by zqm and using the identity

(4.10.12) we obtain from (4.10.11) the formula that would have been derived by using (4.10.5) and (4.10.6) with m an arbitrary integer. 4.11 Some additional extensions of the beta integral 129

4.11 Some additional extensions of the beta integral

Askey and Roy [1986] used Ramanujan's summation formula (2.10.17) to show that iO iO iO iO 1 j7r (ce /(3, qe / ca, cae- , q(3e- / c; q) 00 dB 21f -7r (ae'O , be'o, ae-'o, (3e -,0; q) 00 (aba(3, c, q/c, cal(3, q(3/ca; q)oo (4.11.1) (aa, a(3, ba, b(3, q; q)oo where max(lql, lal, Ibl, lal, 1(31) < 1 and ca(3 =I- 0; and they extended it to the contour integral form ~ r (cz/(3,qz/ca,ca/z,q(3/cz;q)oo dz 21f21K (az, bz,a/z,(3/z; q)oo z (aba(3, c, q/c, cal(3, q(3/ ca; q)oo ( 4.11.2) (aa, a(3, ba, b(3, q; q)oo where aa, a(3, ba, b(3 =I- q-n, n = 0,1,2, ... ,ca(3 =I- 0, and K is a deformation of the unit circle as described in §4.9. These formulas can also be derived from the A = B = D = 2, m = 0 case of (4.10.8) by setting al = c/(3, a2 = q/ca, bl = ca, b2 = q(3/c, CI = a, C2 = b, dl = a and d2 = (3 and then using the summation formula (2.10.13) for the sum of the two 2¢1 series resulting on the right side. In Askey and Roy [1986] it is also shown how Barnes' beta integral (4.1.2) can be obtained as a limit case of (4.11.1). Analogously, application of the summation formula (2.10.11) to the A = 3, B = D = 2, m = 0 case of (4.10.8) gives ~ r (8z,qzh,1'z/a(3,1'/z,qa(3hz;q)00 dz 21f2 1 K (az, bz, cz, a/ z, (3/ z; q)oo z b /a, qah, 1'/(3, q(3h, 8/a, 8/b, 8/c; q)oo (4.11.3) (aa, a(3, ba, b(3, ca, c(3, q; q)oo where 8 = abca(3, abca(31' =I- 0, and aa, a(3, ba, b(3, ca, c(3 =I- q-n, n = 0,1,2, ...

Note that (4.11.2) follows from the c -+ 0 case of (4.11.3). In addition, application of Bailey's summation formula (2.11.7) gives the more general formula

1 r (za!, -za!, qaz/b, qaz/c, qaz/d, qaz/ j; q) 00

21fi 1 K (lIbqza2 , -qza2 , z, cz, d z, j z; q ) 00 (qzh, v/a (3, 1'/ z, qa(3hz; q)oo dz x ~~--~~--~~~~--~ (aaz,a(3z,a/z,(3/z;q)oo z (1'/a,qa/1',1'/(3,q(3/1',aq/cd,aq/bd,aq/bc,aq/bj,aq/cj,aq/dj;q)oo (aa(3, ba, ca, da, ja, b(3, c(3, d(3, j(3, q; q)oo (4.11.4) 130 Basic contour integrals where aq = bedfa{3, bedfa{3, -=I- 0, aa{3, ba, ea, da, fa, b{3, e{3, d{3, f {3 -=I- q-n, n = 0,1,2, ... , and K is as described in §4.9; see Gasper [1989c].

4.12 Sears' transformations of well-poised series

Sears [1951d, (7.2)] used series manipulations of well-poised series to derive the transformation formula

(qal/aM+2, ... , qal/a2M, q/aM+2, ... , q/a2M, af, -af, q/af, -q/af; q) 00 (al, ... , aM+l, a2/al, ... , aM+l/al; q)oo ~ [al ,a2, ... ,a2M . ] X2M'f/2M-l / / ,q,-x qal a2,···,qal a2M (qa2/aM+2, ... , qa2/a2M, qal/a2aM+2, ... , qal/a2a2M; q)oo = a2(al/a2,a2,a3/a2, ... ,aM+l/a2,a§/al,a2a3/al, ... ,a2aM+l/al;q)oo

x (af /a2, -af /a2, qa2/a f, -qa2/a f; q) 00

~ [a§/al,a2,a2adal, ... ,a2a2M/al. ] X2M'f/2M-l qa2 / al,qa2 / a3,···,qa2 / a2M ,q,-x + idem (a2; a3,···, aM+l), (4.12.1) where x = (qad M /ala2··· a2M. Slater [1952c] observed that this formula could also be derived from (4.10.10) by taking A = B = M + 1, m = 1, choosing the parameters such that P(z) in (4.9.1) becomes

(qal z/aM+2, ... , qalz/a2M, qzaf, -qzaf; q) 00 (alz, ... , aM+lz; q)oo

(q/zaM+2, ... , q/za2M, l/zaf, -1/zaf; q) x 00 (4.12.2) (l/z,a2/zal, ... ,aM+l/zal;q)00 ' and then using the fact that (a, -a, q/a, -q/a; q)oo - a2(qa, -qa, l/a, -1/a; q)oo = 2(a, -a, q/a, -q/a; q)oo (4.12.3) to combine the terms with the same 2M¢2M-l series. Similarly, taking A = B = M +2 and m = 1 in (4.10.10) and choosing the parameters such that P(z) in (4.9.1) becomes

(qalz/aM+3, ... , qalz/a2M, qzaf, -qzaf, z(qad!, -Z(qal)!' q) 00 (alz, ... , aM+2 Z; q)oo

(q/zaM+3, ... , q/za2M, l/zaf, -1/zaf, q! /zaf, -q! /zaf; q) 00 x ~------~~--~------~--~------~ (1/ z, a2/ zal, ... , aM+2/ zal; q)oo (4.12.4) 4.12 Sears' transformations of well-poised series 131 we obtain (qal/aM+3, ... , qal/a2M, q/aM+3, ... , q/a2M, q/al; q)= (a2' ... ' aM+2, adal, ... , aM+2/al; q)=

A.. [aI, a2, ... , a2M ] 2M\f'2M-l / / ;q,x x qal a2,···,qal a2M (q a2/aM+3, ... , qa2/a2M, qal/a2aM+3, ... , qal/a2a2M; q)= = a2 (al/a2,a3/a2, ... ,aM+2/a2,a2a3/al, ... ,a2aM+2/al;q)=

(al/a§, qa§!al; q)= A.. [a§/al, a2, a2a3/al, ... , a2 a2M/al. ] x ( 2M\f'2M-l / / / ,q,x a2, a2/2 al; q) = qa2 aI, qa2 a3,···, qa2 a2M + idem (a2; a3, ... , aM+2), (4.12.5) where x = (qat)M /al ... a2M, which is formula (7.3) in Sears [1951d]. Finally, if we take A = B = M +2 and m = 1 in (4.10.10) and choose the parameters such that P(z) in (4.9.1) becomes

(qalz/aM+3, ... , qalz/a2M+1, qzaf, -qzaf, ±q!zaf; q) = (alz, ... , aM+2z; q)= (q/zaM+3, ... , q/za2M+l, l/zaf, -l/zaf, ±q! /zaf ;q) = x (/ / /) , (4.12.6) 1 z,a2 zal, ... ,aM+2 zal;q = we obtain (qal/aM+3, ... , qal/a2M+l, q/aM+3, ... , q/a2M+1; q)= (al' ... ' aM+2, a2/al, ... , aM+2/al; q)= x (af, -af, q/af, -q/af, ±(alq)!, ±(q/at)!; q) =

A.. [al,a2, ... ,a2M+l ] x 2M+l\f'2M / /;q, ~y qal a2,···, qal a2M+l (qa2/aM+3, ... , qa2/a2M+l, qal/a2aM+3, ... , qal/a2a2M+l; q)= = a2(al/a2,a3/a2, ... ,aM+2/a2,a2,a§/al,a2a3/al, ... ,a2aM+2/al;q)= x (af /a2, -af /a2, qa2/af, -qa2/af, ±a2(q/at)!, ±(qat)! /a2; q) = a§/al,a2,a2a3/al, ... ,a2a2M+l/al x 2M+l¢2M [ ;q,H] qa2/al,qa2/a3, ... ,qa2/a2M+l (4.12.7) where y = (qal)M+! /al ... a2M+l, which are formulas (7.4) and (7.5) in Sears [1951d]. 132 Basic contour integrals

Exercises

4.1 Let Re e > 0, Re d > 0, and Re(x + y) > 1. Show that Cauchy's [1825] beta integral

_1_liOO ds = r(x + y - 1)(1 + d/e)l-Y(1 + C/d)l-x 27ri -ioo (1 + es)x(1- ds)Y (e + d)r(x)r(y)

has a q-analogue of the form

1 liOO (-esqX,dsqY;q)oo ds 27ri -ioo (-es, ds; q)oo r q(x + y - 1) (_eqX /d, -dqY /e; q)oo r q(x)r q(Y) (e + d)( -eq/d, -dq/e; q)oo'

where °< q < 1. (Wilson [1985])

4.2 Prove that

1 liOO (q1+s, _qa+s, qa-b+1+s, _qa-b+1+s, qc+d+e-a+s; q) 00 27ri -ioo (qc+s, qd+s, qe+s, _qa+1+s, _qa-2b+s; q)oo 7rqs ds x ~--~--~~~------~ sin 7r s sin 7r (e + d + e - a + s) (q,qc+d+e-a, q1+a-c-d-e,q1+a-b; q)oo =csc7r(e+d+e-a) ( 1+ 1+ -2b ) q a, _q a, _qa , qC; q 00

(_q1+a-b, _qa, q1+a-c, q1+a-d, q1+a-e; q) 00 X ~~--~----~~------~--~--~= (qd,qe,q1+a-c-d, q1+a-c-e, q1+a-d-e; q)oo

XlO Wg (qa; iq1+a/2, _iq1+a/2, l, _qb, qC, qd, qe; q, _q1+2a-2b-C-d-e) ,

where 1 + 2a - 2b > e + d + e.

4.3 Show that

ae, be, ad ] 3cP2 [ b dh;q,gh a eg,ae (q, ae, be, ag, bg, eh; q)oo Exercises 133

4.4 Prove that

iO iO iO iO iO iO ~ j1l" (Je , ke Id, qde- Ik, cke- , qe Ick, abcdghe I j; q) 00 dB 27r (ae iO beiO ce-iO de-iO geiO heiO . q) -7r """ 00 (k,qlk,ckld,qdlck,cj,dj,acdg, bcdg, cdgh,abcdhlj; q)oo (q, ac, ad, bc, bd, cg, dg, ch, dh, cdjg; q)oo

x 8 W 7 (cdjglq; cg, dg, j la, jib, j Ih; q, abcdhl1)·

4.5 Prove that

q-n abcdqn-l aeiO ae-iO .,.f... [ , " 4'1-'3 ab, ac, ad 2 (q;q)oo (q!, q!, ab, ac, bc; q)oo

1 1l" (q! eiHi I a, q! ae-i(}-i , aq! eiO - i , q! ei-iO I a, abcei I a; q) 00

X 27r { (aei la bei la cei la aeiO - i ae-iO - i. q) } -7[ "'" 00 x (dae-i,bc;q)n (~eird¢ (abcei la, ad; q) n a and, more generally, (q,az,alz,bz, blz,cz, clz;q)oo (t;;, ql t;;, t;;Z 2, ql t;; Z 2, ab, ac, bc; q)oo 1 {1l" (t;;zei la, qae-i It;;z, t;;aze-i, qei It;;az, abcei la; q)oo x 27r ) -1l" (aei I a, bei I a, cei I a, aze-i, ae-i I Z; q)oo (daci,bc;q) (a .",)n X n -e''I' d¢ (abcei la, ad; q)n a

for n = 0,1,2, ... , where z = ei() and t;; is an arbitrary parameter.

4.6 Prove that

-1 (aqle,aqlj,aqlg,aqlh,qlae,qlaj,qlag,qlah;q)oo a (qa2, ab, ac, ad, bla, cia, dla; q)oo

x lOW9 (a 2; ab, ac, ad, ae, aj, ag, ah; q, q3labcdejgh) + idem (a; b, c, d) = 0,

where Iq31 < labcdejghl. 134 Basic contour integrals

4.7 Prove that

_l(alq/bl,alq/b2, ... ,alq/br,q/albl, ... ,q/albr;q)OO a l (qai, ala2,···, alan a2/al, ... , ar/al; q)oo

X 2r+2 W2r+l (ai; ala2, ... , alan albl , ... , albr ; q, qr-l /al ... arbl ... br) + idem (al; a2,···, ar) = 0,

where r = 1,2, ... , and Iqr-ll < lal ... arbl ... brl.

4.8 Show that

_ 1 lioo (_csqn+l ' bds , as· , q) oods 27ri -ioo (-CS, ds, basqn-l; q)oo (-a/ c, -bd/ c; q)oo (b, a/ d; q)n (C + d)( -dq/c, -ba/cq; q)oo (q, -cq/d; q)n'

where Re(c, d, ba) > °and n = 0,1, .... Show that the q-Cauchy beta integral in Ex. 4.1 follows from this formula by letting n ----+ 00 and then setting b = qY, a = _cqx.

4.9 Extend the integral in Ex. 4.8 to

1 lioo (-acs, ac2s/a, ac2s/ (3, ac2 s/'Y, ac2S/6, ac2 s/ A; q)oo 27ri -ioo (-cs, as, (3s, "Is, 6s, AS; q)oo x (1- a2~s2) ds (a/ q, -ac/ a, -ac/(3, -ach, -ac/6, -ac/ A; q)oo c(q, -a/c, -(3/c, -'Y/c, -6/c, -A/C; q)oo (q2/a, ac2 /a(3, ac2/a'Y, ac2 /(3"1; q)n X ~~7---~~----~~~~~~~- (-cq/a, -cq/(3, -cqh, -a2c3 /a(3'Yq; q)n'

where Re(c,a,(3,'Y,6,A) > 0, a3 c5 = -a(3'Y6Aq2, ac = -Aqn+l, and n = 1,2 ....

4.10 Show that

~ r (q2z/aa'Y,q2z/ba'Y,q2z/ca'Y,'Y/z;q)oo 1- qz2/a'Y dz 27r2 ) K (az, bz, cz, a/ z; q)oo z (a'Y, b'Y, c'Y, aqh; q)oo (aa, ba, ca, q; q)oo '

where q2 = abcaq2, I'Y/al < 1, and the contour K is as defined in §4.9. Notes 135

4.11 Show that ~ r (bqz,qzh,'"'(lz;q)oo 27r~ JK (az, bz, al z; q)oo dz x (qzhqm; q)m(alz; q)ml ... (arz; q)mr - Z b la, aqh, bqla; q)oo (aa, qlaa, ba; q)oo x (qlb'"'(qm; q)m(al/b; q)ml ... (arlb; q)mr (ba)m+ml +.+mr,

provided 1'"'(1 a 1 < 1, where m, ml, ... ,mr are nonnegative integers, q = a'"'(qm+ml +·+mr and K is as defined in §4.9.

4.12 Show that 1 jiOO (-acs, dqs; q)oo ( ) ( ) -. (d .) al s; q ml . .. ars; q mr ds 27r~ -ioo -cs, S, q 00 (-acld;q)oo (I) (I) (c + d)( -cqld; q)oo al d; q ml ... ar d; q mr provided laq-(m1+·+mr)1 < 1, Re (c,d,al,···,ar) > 0, and ml, ... ,mr are nonnegative integers.

4.13 Show that 1 jiOO (-acs,-ac2slf,ac2sla,ac2sl/3;q)00 27ri -ioo (-cs, - f s, as, /3s; q)oo 2 2 x (ac sh ,ac s15;q)00(1 - ac 2 s 21 q) d s ('"'(s, 5s; q)oo (alq, acl f, -acla, -acl/3, -ach, -acI5; q)oo c(q, f Ic, -alc, -/3lc, -'"'( Ic, -5Ic; q)oo (q2 la, ac2la/3, ac2la,",(, ac21/3'"'(; q)n x ~~~~~~~-7~~~~~~~ (-cqla, -cql/3, -cqh, -a2c3 Ia/3'"'(q; q)n (ac2I Pq, acl f, -ac2I fa, -ac2I f/3, -ac2I f'"'(, -ac2I j8; q)oo +~~~~~~~~~~~~~~~~~~~~~ f(q, cl f, -al f, -/31 f, -'"'(If, -51 f; q)oo (Pq2lac 2, ac2/a/3, ac2/a'"'(, ac21/3'"'(; q)n x ~~~--~~~~~~~~~~~~ (-fqla, - fql /3, - fqh, -a2c4 I fa/3,",(q; q)n provided ac = f qn+l, a3 c5 = fa/3'"'(5q2, Re (c, f, a, /3, ,",(, 5) > 0, the inte­ grand has only simple poles, and n = 0,1, .... (For the formulas in Exercises 4.8 - 4.13, and related formulas, see Gasper [1989c].)

Notes

§4.4 Kalnins and Miller [1988, 1989] exploited symmetry (recurrence relation) techniques similar to those used by Nikiforov and Suslov [1986], Nikiforov, 136 Basic contour integrals

Suslov and Uvarov [1991]' and Nikiforov and Uvarov [1988] to give another proof of (4.4.3) and of (4.11.1). §4.6 Contour integrals of the types considered in this section were used by Agarwal [1953c] to give simple proofs of the two-term and three-term trans­ formation formulas for 8¢7 series. §4.12 Sears [1951b] also derived the hypergeometric limit cases of the transformation formulas in this section. Applications of (4.12.1) to some for­ mulas in partition theory are given in M. Jackson [1949]. Exercises 4.1-4.5 For additional q-beta integrals, see Askey [1988a,b, 198ge], Gasper [1989c], and Rahman and Suslov [1994a,b, 1996a, 1998]. 5

BILATERAL BASIC HYPERGEOMETRIC SERIES

5.1 Notations and definitions

The general bilateral basic hypergeometric series in base q with r numerator and s denominator parameters is defined by

r1/Js(z) == r1/Js [ab1,ab2""'bar;q,z] 1, 2,···, s (5.1.1) f (a1, a2,···, ar; q)n (_I)(s-rlnq(s-rlG) zn. n=-oo (b 1, b2,···, bs; q)n In (5.1.1) it is assumed that q, z and the parameters are such that each term of the series is well-defined (i.e., the denominator factors are never zero, q =I=- 0 if s < r, and z =I=- 0 if negative powers of z occur). Note that a bilateral basic hypergeometric series is a series L~=-oo Vn such that Vo = 1 and vn+dvn is a rational function of qn. By applying (1.2.28) to the terms with negative n, we obtain that r1/Js(z) = f (a1,a2, ... ,ar;q)n (_I)(s-rlnq(s-rlG)zn (b1,b2, ... ,bs;q)n n=O (5.1.2) + f (q/b1, q/b2, ... , q/bs;.q)n (b1'" bs ) n n=l (q/a1, q/a2,"" q/ar, q)n a1 ... arz Let R = Ib 1··· bs/a1'" arl. If s < rand Iql < 1, then the first series on the right side of (5.1.2) diverges for z =I=- 0; if s < rand Iql > 1, then the first series converges for Izl < R and the second series converges for all z =I=- O. When r < sand Iql < 1 the first series converges for all z, but the second series converges only when Izl > R. If r < sand Iql > 1, the second series diverges for all z =I=- O. If r = s, which is the most important case, and Iql < 1, the first series converges when Izl < 1 and the second when Izl > R; on the other hand, if Iql > 1 the second series converges when Izl > 1 and the first when Izl < R. We shall assume throughout this chapter that Iql < 1, so that the region of convergence of the bilateral series

(5.1.3)

137 138 Bilateral basic hypergeometric series is the annulus (5.1.4)

When bj = q for some j, the second series on the right sides of (5.1.2) and (5.1.3) vanish and the first series become basic hypergeometric series. If we replace the index of summation n in (5.1.1) by k + n, where k is an integer, then it follows that

= (al, ... ,ar;q)k zk [(_I)kq(~)]s-r (b l , ... , bs ; qh

al qk, ... , arqk k(S-rJ] X r'l/Js [ k k ; q, zq . (5.1.5) blq , ... ,bsq When rand s are small we shall frequently use the single-line notation

r'l/Js(z) == r'l/Js(al,"" ar; bl , ... , bs; q, z).

An r'l/Jr series will be called well-poised if albl = a2b2 = ... = arbn and very­ well-poised if it is well-poised and al = -a2 = qbl = -qb2. Corresponding to the definition of a VWP-balanced r+l¢r series in §2.1, we call a very-well­ poised r'l/Jr series very-well-poised-balanced (VWP-balanced) if (5.1.6) with either the plus or minus sign. The very-well-poised bilateral basic hyper­ geometric series in (5.3.1), (5.5.2), (5.5.3), and in §5.6 are VWP-balanced. Note 1 that if in (5.1.6) we replace r by r+l, al by qar, and then a3 by aI, then (5.1.6) reduces to the condition (2.1.12) for a r+l Wr series to be VWP-balanced. A well-poised r'l/Jr series will be called well-poised-balanced (WP-balanced) if (ala2'" ar)z = -(±(alb1)br (5.1.7) with either the plus or minus sign. The well-poised bilateral basic hypergeo­ metric series in (5.3.3), (5.3.4), (5.5.1), (5.5.4), and in (5.5.5) are WP-balanced.

5.2 Ramanujan's sum for l'l/Jl(a;b;q,z)

The bilateral summation formula

oj, ('b' )_ (q,bja,az,qjaz;q)cxo 1 '1-'1 a, ,q, z - (j j ) , Ibjal < Izl < 1, (5.2.1) b,q a,z,b az;q cxo which is an extension of the q-binomial formula (1.3.2), was first given by Ramanujan (see Hardy [1940]). In Chapter 2 we saw that this formula follows as a special case of Sears' 3¢2 summation formula (2.10.12). Andrews [1969, 1970a], Hahn [1949b]' M. Jackson [1950b], Ismail [1977] and Andrews and Askey [1978] published different proofs of (5.2.1). The proof given here is due to Andrews and Askey [1978]. 5.2 Ramanujan's sum for l'l/Jl(a;b;q,z) 139

The first step is to regard 1 'l/Jl (a; b; q, z) as a function of b, say, f (b). Then

f(b) = l'l/Jl(a;b;q,z) (5.2.2) = f (a; q)n zn + f (q/b; q)n (b/azt n=O (b; q)n n=l (q/a; q)n so that, by (5.1.4), the two series are convergent when Ib/al < Izl < 1. As a function of b, f(b) is clearly analytic for Ibl < min(l, lazl) when Izl < 1. Since

= f (a; q)n+l zn n=-oo (b; q)n = z-l(l- b/q) f (a;q)n+l zn+l n=-oo (b/q; q)n+l = z-l(l- b/q) l'l/Jl(a; b/q; q, z), we get f(bq) - z-l(l- b)f(b) = a l'l/Jl(a;bq;q,qz). (5.2.3) However, a l'l/Jl(a;bq;q,qz) ~ (a;q)n ( )n =a ~ qz n=-oo (bq; q)n (5.2.4) = -ab-1 f (a; q)n(1- bqn - 1) zn n=-oo (bq; q)n = -ab-1(1- b)f(b) + ab-1f(bq).

Combining (5.2.3) and (5.2.4) gives the functional equation (1- ab-1)f(bq) = (1- b)(Z-l - ab-1)f(b), that is, 1- b/a f(b) = (1 _ b)(l _ b/az) f(bq). (5.2.5)

Iterating (5.2.5) n - 1 times we get

f(b) = (b/a; q)n f(bqn). (5.2.6) (b, b/az; q)n

Since f(b) is analytic for Ibl < min(l, lazl), by letting n ----+ 00 we obtain

f(b) = (b/a; q)oo f(O). (5.2.7) (b, b/az; q)oo 140 Bilateral basic hypergeometric series

However, since f(q) = f= (a; q)n zn = (az; q)oo n=O (q; q)n (z; q)oo by (1.3.2), on setting b = q in (5.2.7) we find that

f(O) = (q, q/az; q)oo f(q) = (q, q/az, az; q)oo. (q/ a; q)oo (q/ a, z; q)oo

Substituting this in (5.2.7) we obtain formula (5.2.1). Jacobi's triple-product identity (1.6.1) is a limit case of Ramanujan's sum. First replace a and z in (5.2.1) by a-I and az, respectively, to obtain

f= (a-\q)n (az)n = (q,ab,z,q/z;q)oo , (5.2.8) n=-oo (b; q)n (b, aq, az, b/ z; q)oo when Ibl < Izl < la-II. Now set b = 0, replace q by q2, Z by zq, and then take a ---+ 0 to get (1.6.1).

5.3 Bailey's sum of a very-well-poised 6'ljJ6 series

Bailey [1936] proved that

'ljJ [ qa~,-qa~,b,c,d,e . qa2 ] 6 6 a~, -a~, aq/b, aq/c, aq/d, aq/e ,q, bcde (5.3.1) (aq, aq/bc, aq/bd, aq/be, aq/cd, aq/ce,aq/de, q,q/a; q)oo (aq/b, aq/c, aq/d, aq/e, q/b, q/c, q/d, q/e,qa2/bcde; q)oo ' provided Iqa 2/bcdel < 1. Since this VWP-balanced 6'ljJ6 reduces to a VWP­ balanced 6¢5 series when one of the parameters b, c, d, e equals a, (5.3.1) can be regarded as an extension of the 6¢5 summation formula (2.7.1). There are several known proofs of (5.3.1). Bailey's proof depends crucially on the identity

(aq/d,aq/e,aq/f,q/ad,q/ae,q/af;q)oo a(qa2, ab, ac, bfa, cia; q)oo (5.3.2)

x 8 W 7 (a 2; ab, ac, ad, ae, af; q, q2/abcdef) + idem (a; b, c) = 0, when Iq2/abcdefl < 1, which is easily proved by using the q-integral represen­ tation (2.10.19) of an 8¢7 series (see Exercise 2.15). If we set c = q/a, the first and third series in (5.3.2) combine to give, via (5.1.3),

(aq/d,aq/e,aq/f, q/ad,q/ae,q/af; q)oo a(qa2,q/a2, q,ab, b/a; q)oo qa, -qa, ab, ad, ae, af q ] [ x 6'ljJ6 a, -a, aq / b, aq / d, aq / e, aq /;f q, bd e f ' 5.4 A general transformation formula 141 while, by (2.7.1), the second series reduces to

b2, bq, -bq, bd, be, bj q] [ 6¢5 b, -b, bqjd, bqje, bqj j; q, bdej (qb2, qjde,qjdj,qjej; q)oo (bqjd, bqje, bqj j, qjbdej; q)oo .

This gives (5.3.1) after we replace a2, ab, ad, ae, aj by a, b, e, d, e, respectively, and use the same square root of a everywhere. Slater and Lakin [1956] gave a proof of (5.3.1) via a Barnes type integral and a second proof via a q-difference operator. Andrews [1974a] gave a simpler proof and Askey [1984c] showed that it can be obtained from a simple difference equation. The simplest proof was given by Askey and Ismail [1979] who only used the 6¢5 sum (2.7.1) and an argument based on the properties of analytic functions. Setting e = a! in (5.3.1), we obtain

oj, -qa21, b, e, d . qa 23] 4 '/-'4 [ 1 ,q,- -a2 , aqjb, aqje, aqjd bed

(aq,aqjbe,aqjbd,aqjed,qa~jb,qa~je,qa~jd,q,qja;q)oo (5.3.3) (aqjb, aqje, aqjd, qjb, qje, qjd, qa!, qa-~, qd jbed; q)oo provided Iqd jbedl < 1. This is an extension of the q-Dixon formula (2.7.2). 1 1 If we set d = a 2 , e = -a2 in (5.3.1) and simplify, we get the sum of a WP-balanced 2'l/J2 series

2'l/J2(b, e; aqjb, aqje; q, -aqjbe) (aqjbe; q)oo(aq2 jb2, aq2 je2, q2, aq, qja; q2)oo (5.3.4) (aqjb, aqje, qjb, qje, -aqjbe; q)oo I~~I < 1.

5.4 A general transformation formula for an r'l/Jr series

In this section we shall derive a transformation formula for an r'l/Jr series from those for r+l¢r series in Chapter 4. First observe that (5.1.3) gives

(5.4.1) 142 Bilateral basic hypergeometric series

In (4.10.11) let us now make the following specialization of the parameters

C = A+ 1, D=B+1, A=B=r, cldl = C2 d2 = ... = CA+ldA+l = q, qdI/b 0:1, qdI/b 0:2,"" qdI/bA O:A, l = 2 = = (5.4.2) aldl = (31, a2 dl = (32, ... , aAdl = (3A,

bl ... bAZ ----=x. dl ... dA+l

Then, combining the pairs ofthe resulting r+l¢r series in (4.10.11) via (5.4.1), simplifying the coefficients and relabelling the parameters, we obtain Slater's [1952b, (4)] transformation formula

(b l , b2, ... , br, q/al, q/a2,"" q/ar, dz, q/dz; q)oo r'¢r [aI, a2,···, ar ; q, z] (Cl,C2, ... ,Cr,q/Cl,q/C2,'" , q/cr;q)oo bl , b2, ... ,br

q (Cl/al,Cl/a2"",Cl/ar,qbl/Cl,qb2/Cl, ... ,qbr/Cl,dclZ/q, q2/dclZ;q)oo Cl (Cl,q/Cl,Cl/C2,'" ,Cl/Cr,qC2/Cl," .,qCr/Cl;q)oo

qaI/cl, qa2/Cl,"" qar/Cl ] X r'¢r [ ; q, Z qbl /Cl,qb2/Cl,'" ,qbr/Cl + idem (Cl; C2, ... , Cr ), (5.4.3)

where d = ala2'" ar/clc2'" CTl I bl ... br I < Izl < l. al'" ar Note that the c's are absent in the r'¢r series on the left side of (5.4.3). This gives us the freedom to choose the c's in any convenient way. For example, if we set Cj = qaj, where j is an integer between 1 and r, then the lh series on the right becomes an r¢r-l series. So if we set Cj = qaj, j = 1,2, ... , r, in (5.4.3), then we get an expansion of an r'¢r series in terms of r r¢r-l series:

(b l , b2, ... , bTl q/al, q/a2,"" q/aTl z, q/z; q)oo '¢ [aI, a2,···, ar . ] (qal, qa2, ... , qar, l/al, 1/a2,"" l/ar; q)oo r rbI, b2, ... , br ' q, Z a~-1(q,qal/a2, ... ,qal/ar,bl/al,b2/al, ... ,br/al,alz,q/alz;q)oo (qal,1/al,al/a2, ... ,al/ar,qa2/al, ... ,qar/al;q)oo

qaI/bl, qaI/b2, ... , qaI/br bl ··· br ] x r¢r-l [ ; q, ---- qal/a2, ... ,qal/ar al···arz + idem (al; a2,"" ar), (5.4.4) provided I bl ... br I < Izl < l. al ... ar On the other hand, if we set Cj = bj , j = 1,2, ... , r in (5.4.3), then we 5.5 A transformation formula for a 2r'IjJ2r series 143 obtain the expansion formula (q/al,q/a2, ... ,q/an dz,q/dz;q)oo 'IjJ [aI, a2, ... ,ar . ] (q/bl ,q/b2, ... ,q/br;q)oo r rbI, b2, ... ,br ,q,z q (q,bl/al,bl/a2, ... ,bl/ar,dbIZ/q,q2/dblz;q)oo (5.4.5) bl (b l ,q/bl ,bl /b2, ... ,bl/br;q)oo qadbl, ... ,qar/bl ]. x r¢r-l [ / /; q, z + Idem (b l ; b2,···, br ), qb2 bl, ... ,qbr bl with d = ala2'" ar/bl b2 ··· br.

5.5 A general transformation formula for a very-well-poised 2r'IjJ2r series

Using (4.12.1) and (5.4.1) as in §5.4, we obtain Slater's [1952bj expansion of a WP-balanced 2r'IjJ2r series in terms of r other WP-balanced 2r'IjJ2r series: 1 1 1 1 (q/bl , ... , q/b2n aq/h, ... , aq/b2n a"2, -a"2, qa-"2, -qa-"2; q)oo

x (ar/a,qa/ar,ala2/a, ... ,alar/a,qa/ala2, ... ,qa/alar;q)oo albl/a, alb2/a, ... ,alb2r/a arqr ] X 2r 'ljJ2r [ / / /; q, - b b alq bl , alq b2,···, alq b2r I'" 2r +idem(al;a2, ... ,ar). (5.5.1)

For the very-well-poised case when al = bl = qa!, a2 = b2 = -qa!, the first two terms on the right side vanish and we get (q/b3, ... , q/b2r , aq/b3, ... , aq/b2r ; q)oo (aq,q/a,a3, ... ,ar,q/a3, ... ,q/ar;q)oo x (a3/a, ... , aria, aq/a3, ... , aq/ar; q)~l

! ! of, 2 2 , b b r-l r-2] [ qa ,-qa 3,···, 2r a q X 2r'/'2r 1 1 ; q, ---- a"2, -a"2, aq/b3, ... , aq/b2r b3 ··· b2r (a3q/b3, ... ,a3q/b2r,aq/a3b3, ... ,aq/a3b2r;q)oo (a3,q/a3,a3/a,aq/a3,qa§/a,aq/a§;q)oo x (a4/a3, ... ,ar/a3,qa3/a4, ... ,qa3/ar;q)~1 x (a3a4/a, ... , a3ar/a, aq/a3a4,"" aq/a3ar; q)~l 144 Bilateral basic hypergeometric series

(5.5.2) In particular, for r = 3 we have the following transformation formula for VWP­ balanced 6 '¢6 series (q/b3, ... ,q/b6,aq/b3, ... ,aq/b6;q)oo (aq,q/a,a3,q/a3,a3/a,aq/a3;q)oo

.J, [ qa!, -qa!, b3, b4, b5, b6 a2q ] x 6 '/-"6 1 1 ; q, b b b b a"2, -a"2, aq/b3, aq/b4, aq/b5, aq/b6 3 4 5 6 (a3q/b3, ... ,a3q/b6,aq/a3b3, ... ,aq/a3b6;q)oo (a3,q/a3,a3/a,aq/a3,qa~/a,aq/a~;q)oo qa3a-!, -qa3a-!, a3 b3/a, a3 b4/a, a3 b5/a, a3 b6/a a2q 1 x 6'¢6 [ 1 1 ; q, b b b b . a3a-"2, -a3a-"2, qadb3, qa3/b4, qa3/b5, qa3/b6 3 4 5 6 (5.5.3)

If we now set a3 = b6, then the 6'¢6 series on the right side becomes a 6¢5 with sum (qb~/a,aq/b3b4,aq/b3b5,aq/b4b5;q)oo (qb6/b3, qb6/b4, qb6/b5, qa2/b 3b4b5b6; q)oo .

This provides another derivation of the 6'¢6 sum (5.3.1); see M. Jackson [1950a]. As in Slater [1952b], Sears' formulas (4.12.5) and (4.12.7) can be used to obtain the transformation formulas (q/bl, ... ,q/b2r,aq/bl, ... ,aq/b2r;q)oo

(a2/al, ... ,ar+l/al,qal/a2, ... ,qal/ar+l,aq/aIa2, ... ,aq/aIar+l;q)oo x (al,q/al,aq/al,al/a,aIa2/a, ... ,aIar+l/a;q)~1 aIbda, aIb2/a, ... , aIb2r /a qrar ] [ X 2r '¢2r / / /; q, b b b aIq bl , aIq b2,···, aIq b2r I 2'" 2r

(5.5.4) and (q/bl , ... , q/b2r - l , aq/bl , ... , aq/b2r- l ; q)oo (a,al, ... ,ar,q/a,q/al, ... ,q/ar;q)oo (a!, -a!, q/a!, -q/a!, ±(aq)!, ±(q/a)!; q)oo x ~~--~~--~~--~~~--~~~~- (al/a, ... ,ar/a,aq/al, ... ,aq/ar;q)oo 5.6 Transformation formulas for 8'¢8 and 1O'¢1O series 145

al (al q/bl , ... ,al q/b2r- l , aq/ albl , ... ,aq/al b2r- l ; q)oo (aq/ai,aq/ala2, ... ,aq/alar ,a2/al, ... ,ar/al;q)oo (a! /al, -a! /al, alq/a!, -alq/a!; q)oo x (a/al,q/al,ai/a,ala2/a, ... ,alar/a;q)oo (±(aq)! /al, ±al(q/a)!; q)oo x-,-----'--:-..:...... ::"----'---=.:,---=...:~--'----..:,.=...:....::....,- (q al/a,al,qal/a2, ... ,qal/ar;q)oo al bI/a, ... ,al b2r _I/a =fqr-!ar-!] X 2r-I'¢2r-1 [ ;q, -----=--- qaI/bl , ... , qaI/b2r - 1 bl b2 ··· b2r- 1 + idem (al; a2, ... , ar). (5.5.5)

5.6 Transformation formulas for very-well-poised s'¢s and 1O'¢1O series

In this section we consider two special cases of (5.5.2) that may be regarded as extensions of the transformation formulas for very-well-poised S¢7 and 1O¢9 series derived in Chapter 2. First, set r = 4 in (5.5.2) and replace b3, b4, b5 , b6 , b7 , bs by b, e, d, e, j, g, respectively, choose a3 = j, a4 = 9 and simplify to get (aq/b,aq/e, aq/d, aq/e, q/b, q/e,q/d, q/e; q)oo (j,g,j/a,g/a,aq,q/a;q)oo

of, [ qa!,-qa!,b,e,d,e,j,g . a3q2 ] x s~s 1 1 'q'------bd j a"2, -a"2, aq/b, aq/e, aq/d, aq/e, aq/ j, aq/g e e 9 (q,aq/bj,aq/ej,aq/dj,aq/ej,jq/b,jq/e, jq/d,jq/e; q)oo (j,q/j,aq/j,j/a,g/j,jg/a,qj2/a;q)oo p la, qja-!, -qja-!, jb/a, je/a, jd/a, je/a, jg/a. a3q2 ] x S¢7 [ 1 1 ,q, ------bd j ja-"2, - ja-"2, jq/b, jq/e, jq/d, jq/e,Jq/g e e 9 + idem (f;g), (5.6.1) where I b:;:;9 I < 1. Replacing a, b, e, d, e, j, 9 by a2, ba, ea, da, ea, j a, ga, respectively, we may rewrite (5.6.1) as (aq/b, aq/e, aq/d, aq/e, q/ab, q/ae, q/ad, q/ae; q)oo (ja,ga, j/a,g/a, qa2,q/a2; q)oo qa, -qa, ba, ea, da, ea, ja, ga q2 ] [ X s'¢s a, -a, aq/b, aq/e, aq/d, aq/e, aq/ j, aq/g ; q, bedejg (q,q/bj,q/ej,q/dj,q/ej,qj/b,qj/e, qj/d,qj/e; q)oo (ja, q/ja, aq/j,j/a,g/j,jg,qj2;q)oo ¢ [ p,qj,-qj,jb,je,jd,je,jg. q2] X 8 7 j, _ j,Jq/b, jq/e, jq/d, jq/e, jq/g' q, bedejg 146 Bilateral basic hypergeometric series

+ idem (f; g), (5.6.2) provided I q2/bcdefgl < 1. Note that no a's appear in the 8¢7 series on the right side of (5.6.2). This is essentially the same as eq. (2.2) in M. Jackson [1950a]. For the next special case of (5.5.2) we take r = 5 and replace b3 , ... ,blO by b, c, d, e, f, g, h, k, respectively, choose a3 = g, a4 = h, a5 = k, and finally, replace a, b, ... , k by a2, ba, ... , ka and simplify. This gives (aq/b, aq/c, aq/d, aq/e, aq/f,q/ab, q/ac, q/ad, q/ae,q/af; q)oo (ag,ah,ak,g/a,h/a, k/a, qa2,q/a2; q)oo qa, -qa, ba, ca, da, ea, fa, ga, ha, ka q3 ] [ X 1O'¢1O a, -a, aq/b, aq/c, aq/d, aq/e, aq/ f, aq/g, aq/h, aq/k ; q, bcdefghk (q, q/bg, q/cg, q/dg, q/eg, q/fg, qg/b, qg/c, qg/d, qg/e,qg/f;q )00 (gh,gk,k/g,h/g, ag, q/ag, g/a, aq/g, qg2; q)oo g2,qg,-qg,gb,gc,gd,ge,gf,gh,gk q3] [ x 1O¢9 g, _g, qg/b, qg/c, qg/d, qg/e, qg/ f, qg/h, qg/k ; q, bcdefghk + idem (g; h, k), (5.6.3) where I q3/bcdefghkl < 1. Notice that all of the very-well-poised series in this section are VWP-balanced.

Exercises

5.1 Show that

5.2 Letting c ----+ 00 in (5.3.4) and setting a = 1, b = -1, show that 1 +4 00 (_1)nqn(n+1)/2 = [ (q;q)oo ]2 ~ 1 + qn (-q; q)oo

5.3 In (5.3.1) set b = a, c = d = e = -1 and then let a ----+ 1 to show that

1 + 8 ~ (_q)n = [ (q; q)oo ] 4 ~ (1 + qn)2 (-q; q)oo See section 8.11 for applications of Exercises 5.1-5.3 to Number Theory.

5.4 Set b = c = d = e = -1 and then let a ----+ 1 in (5.3.1) to obtain 00 q2n(4 _ qn _ q-n) 1+16~ (1+qn)4 = [(q;:)00]8 (-q, q)oo Exercises 147

5.5 Show that

00 L q4n 2 z2n(1 + zq4n+1) = (q2, -zq, -q/z; q2)00, z yf O. n=-oo

5.6 Prove the quintuple product identity

00

n=-CXJ = (q, -Z, -q/ Z; q)00(qZ2, q/ Z2; q2)00, z yf O. See the Notes for this exercise. 5.7 Show that

q,3 q 3 5 5 ] 6 6; q ,q . q, q Deduce that

00 {q5n+1 q5n+2 q5n+3 q5n+4} ~ (1 - q5n+1)2 - (1 - q5n+2)2 - (1 - q5n+3)2 + (1 - q5n+4 )2

(q5; q5)~ = q ,Iql < l. (q; q)oo See Andrews [1974a] for the above formulas. 5.8 Deduce (5.4.4) directly from (4.5.2).

5.9 Deduce (5.3.1) from (5.4.5) by using (2.7.1).

5.10 Show that

q (e/a, e/b, e/ab, qc/e, q2/e, q2 I/e; q)oo ni, [e/c, e/q . ] 'd (. I) - (/ / /) 2 0/2 , q, q + 1 em e, e e,q e,e I,ql e;q 00 e/a,e/b (q,q/a,q/b, c/a,c/b, c/el,qel/c; q)oo (e, I, q/e, q/ I, c/ab; q)oo

5.11 Show that 1/J [ qa~, -qa~, c, d, e, I, aq-n, q-n a2q2n+2] 8 8 a~, -a ~ ,aq/c, aq/ d, aq/ e, aq/ I, qn+l, aqn+1 ; q, cdel (aq, q/a,aq/cd,aq/el; q)n (q/c, q/d, aq/e, aq/I; q)n e, I, aqn+1 lcd, q-n ] [ n = 0,1, ... , X 41/J4 aq / c,aq / d,qn +1 ,el/ aqn ; q, q , 148 Bilateral basic hypergeometric series

and deduce the limit cases e,j aq ] 2'lP2 [ ; q, - aq/ e, aq/ d ej (q/e,q/d, aq/e,aq/j; q)= (aq,q/a,aq/ed,aq/ej;q)= f (1- aq2n)(e,d,e,j;q)n (qa3 )n n2 X n=-= (1 - a)(aq/e, aq/d, aq/e, aq/ j; q)n edej q

and

which reduces to the first and second Rogers-Ramanujan identities when a = 1 and a = q, respectively. (Bailey [1950a] and Garrett, Ismail and Stanton [1999])

5.12 Using (5.6.2) and (2.11.7), show that qa,-qa,ab,ae,ad,ae,aj,ag ] [ s'l/Js a, -a, aq/b, aq/ e, aq/ d, aq/ e, aq/ j, aq/ 9 ; q, q (q, qa2, q/a2, ag, g/a, q/be, q/bd,q/be,q/bj,q/ed,q/ee, q/ej; q)= (bg,eg,dg,eg,jg,aq/b,aq/e,aq/d,aq/e,aq/j,q/ab,q/ae;q)= (q/de, q/dj, q/ej; q)= x (q/ad, q/ae, q/aj; q)=' provided bedejg = q and (bj, q/bj, ej, q/ej, dj, q/dj, ej, q/ej, ag, q/ag, g/a, aq/g; q)= = 1. (bg,q/bg,eg, q/eg,dg, q/dg, eg, q/eg,aj,q/aj,j/a,aq/j; q)= Following Gosper [1988b], we may call this the bilateral Jackson formula.

5.13 Deduce from Ex. 5.12 the bilateral q-Saalschiitz formula

f (a, b, e; q)n qn n=-= (d, e, j; q)n (q,d/a,d/b,d/e,e/a,e/b,e/e,q/j;q)= (d, e, aq/ j, bq/ j, eq/ j, q/a, q/b, q/ e; q)= ' provided dej = abeq2 and (e/a, aq/e,e/b, bq/e, e/e, eq/e, j,q/j; q)= = (f la, aq/ j, j /b, bq/ j, j /e, eq/ j, e, q/e; q)=. Exercises 149

5.14 Show that l d (-qt/c, qt/d: q)oo dqt -c (-at/c, bt/d, q)oo d(1 - q) = b l'l/h(q/a;bq;q,-ad/c) 1- cd(1 - q)(q, ab, -c/d, -d/c; q)oo (c + d)(a, b, -bc/d, -ad/c; q)oo when labl < lad/cl < l. (Andrews and Askey [1981]) 5.15 Show that 100 (ct, -dt: q)oo dqt -00 (at, -bt, q)oo 2(1 - q)(c/a, d/b, -c/b, -d/a, ab, q/ab; q)00(q2; q2)~ (cd/abq,q;q)00(a2,q2/a2,b2,q2/b2;q2)00 (Askey [1981]) 5.16 Show that roo (aat, aft, abt, bit, act, cit, adt, d/t; q)oo dqt 10 (aqt2, q/at2; q)oo t (1 - q)(aa, a, ab, b, ac, c, ad, d; q)oo (aq,q/a;q)oo

.1. [q..ja,-q..ja,q/a,q/b,q/C,q/d. a 2abcd] x 6'/-'6 t;;, t;;, , q, --3- Y a, -y a, aa, ab, ac, ad q (1 - q)(q, aab/q, aac/q, aad/q, abc/q, abd/q, acd/q; q)oo (a2abcd/q3; q)oo 2 when la abcd/q3 1 < l. 5.17 Show that roo (alt, ... ,art,bdt, ... ,bs/t;q)OOt'Y-ld t 10 (CIt, ... , crt, ddt, ... , ds/t; q)oo q (1- q)(al, ... , an bl , ... , bs; q)oo (CI"'" cr, dl , ... , ds; q)oo CI, ... ,Cnq/bl, ... ,q/bs bl ... bs ] X r+s'l/Jr+s [ / /; q, d d q'Y aI, ... , ar, q dl ,···, q ds I . .. s when

5.18 Derive Bailey's [1950b] summation formulas:

(i) 'l/J [ b, c, d . -.!L] _ (q, q/bc, q/bd, q/cd; q)oo 3 3 q/b, q/c, q/d' q, bcd - (q/b, q/c, q/d, q/bcd; q)oo' 150 Bilateral basic hypergeometric series

'¢ [ b,e,d . (q,q2/be,q2/bd,q2/ed;q)00 (ii) L] _ 3 3 q2/b, q2/e, q2 /d' q, bed - (q2/b, q2/e, q2/d, q2/bed; q)oo' b, e, d, e, q-n ] (iii) [ 5'¢5 q /b ,q/ e,q /d ,q/ e,q n+l ; q, q (q, q/be, q/bd, q/ed; q)n h b d n+l = W en e e = q , (q/b,q/e,q/d,q/bed;q)n b, e, d, e, q-n ] (iv) [ 5'¢5 q 2/b ,q2/ e,q 2/d ,q2/ e,qn+2 ; q, q (1 _q)(q2,q2/be,q2/bd,q2/ed;q)n (q2/b,q2/e,q2/d,q2/bed;q)n when bede = qn+3, where n = 0,1, ....

5.19 Show that

(i) t (_1)k [ 2n ]3 qk(3k+l)/2 = (q;qhn k=-n n+k q (q,q,q;q)n

and

(ii) t (_1)k [ 2n+ 1 ]3 l (3k+l)/2 = (q;qhn+l n+k+1 (q,q,q;q)n k=-n-l q for n = 0,1, .... (Bailey [1950b]) 5.20 Derive the 2'¢2 transformation formulas

(i) '¢ [a, b. ] _ (az, d/a, e/b, dq/abz; q)oo 2 2 e, d' q, z - (z, d, q/b, ed/abz; q)oo a, abz/d. d] X 2'¢2 [ ,q, - , aZ,e a

n/, [a, b. ] _ (az, bz, eq/abz, dq/abz; q)oo (ii) 2 '1-'2 , q, Z - (/ /b d ) e, d q a, q ,e, ; q 00 abz/e, abz/d. Cd] x 2'¢2 [ b' q, -b . az, z a z

(Bailey [1950a])

5.21 Verify that Ex. 2.16 is equivalent to

(b/a, aq/b, dj/a,aq/dj,ej/a, aq/ej,bde/a, aq/bde; q)oo = (f la, aq/ j, bd/a, aq/bd, be/a, aq/be, dej la, aq/dej; q)oo

b 2 2 - -(d,q/d,e,q/e,j/b,qb/j,bdej/a ,a q/bdej;q)oo' a (See Bailey [1936]) Exercises 151

5.22 Extend the above identity to

b (be aq bd aq be aq b f aq g aq !!.. aq g hq. ) a aca'b' 'bd' aea'b' 'bf" aga"h'h' goo,q _ b (Ch aq dh aq eh aq f h aq ~ aq g aq g bq. ) a ac'h ' a , dh' ae'h ' a , fh' a 'b' ag, 'b ' goo , q _ (cg aq dg aq eg aq f 9 aq ~ aq !!.. aq ~ hq. ) - ag , , 'd' , , 'f" b' 'h' h' b ,q a cg a 9 a eg a 9 a a 00 q q q qbhq9hq9bq) ( - bh c, ~' d, d' e, ~' f, 7' h' b' h' g' b' g; q 00'

where a3 q2 = bcdefgh. (See Slater [1954a]) 5.23 More generally, show that it follows from the general formula for sigma functions in Whittaker and Watson [1965, p. 451, Example 3] and Tannery and Molk [1898, §400], and also from (5.4.3) that

~ (ak/bl,ak/b2, ... ,ak/bn;q)00 k=l (ak/al, ak/a2,"" ak/ak-l, ak/ak+l,"" ak/an; q)oo (qb1/ak,qb2/ak, ... ,qbn /ak;q)00 x = 0, (qal/ak,qa2/ak, ... ,qak-l/ak,qak+l/ak, ... ,qan/ak;q)00 where ala2 ... an = b1b2 ··· bn · (See Slater 1954a]) 5.24 Extend the summation formula (1.9.6) to

oj, [a,b,blq'ml, ... ,brq'mr -ll-n] r+2'f/r+2 db b b ;q,a q , q, 1,"" r = (q, q, bq/a, d/b; q)oo (bdb; q)'ml ... (br/b; q)'mr bn , (bq, q/a, q/b, d; q)oo (b1; q)'ml ... (br ; q)'mr where ml, ... , mr are nonnegative integers, n is an integer, and Iq/al < Iqnl < Iq'm1+'+'mr /dl. (W. Chu [1994a]) 5.25 More generally, extend the transformation formula in Ex. 1.34(ii) to

oj, [a, b, b1q'ml, ... , brq'mr -1 1-n] r+2'f/r+2 d bcq b b ;q,a q , ,1, .. ·, r = (q, cq, bq/a, d/b; q)oo (bdb; q)'ml ... (br/b; q)'mr bn (bcq, q/a, q/b, d; q)oo (b1; q)'ml ... (br; q)'mr

A.. [ c-1, bq/d, bq/b1, ... , bq/br . d n-('ml +'+'mr)] x r+2'f/r+l bq/a, bql-'m1 /b1, ... , bql-'mr /br ' q, C q ,

where ml, ... , mr are nonnegative integers, n is an integer, and Iq/al < Iqnl < Iq'ml +'+'mr /cdl. (W. Chu [1994a]) 152 Bilateral basic hypergeometric series

5.26 Extend the summation formula in Ex. 2.33(i) to

qa 1/2 ,-qa1/2 ,b,a/ b,c,d,el, ... ,ek,--,""--aqn1 +1 aqn k+1 aq I-N] '¢ [ e1 ek' q 6+2k 6+2k 1/2 1/2 aq b aq ad aq aq -n -nk' '--d- a ,-a 'b' q'c'd'e-;""'ek,elq 1, ••• ,ekq c (q,q,aq, q/a,aq/bc, aq/bd, bq/c, bq/d; q)oo (aq/b,aq/c, aq/d,bq, bq/a,q/b,q/c, q/d; q)oo

k (aq/bej,bq/ej;q)n' x rr J j=1 (aq/ej, q/ej; q)nj ,

where nl,"" nk are nonnegative integers, N = nl +-. +nk, and laql-N /cdl < 1 when the series does not terminate. (W. Chu [1998a]) 5.27 Prove that S(a-2,bc, bd, cd) . ( ) S(b/a, cia, d/a, abcd) + Idem a; b, c, d = 2, where S is defined in Ex. 2.16. See Askey and Wilson [1985, pp. 10, 11], where it is used to evaluate the integral in (6.1.1).

Notes

§5.2 Andrews [1979c] used Ramanujan's sum (5.2.1) to prove a continued fraction identity that appeared in Ramanujan's [1988] "lost" notebook. For­ mal Laurent series and Ramanujan's sum are considered in Askey [1987]. A probabilistic proof of (5.2.1) can be found in Kadell [1987b]. Milne [1986, 1988a, 1989] derived multidimensional U(n) generalizations of (5.2.1). §5.3 Gustafson [1987b, 1989, 1990] derived a multilateral generalization of (5.2.1), (5.3.1) and related formulas by employing contour integration and Milne's [1985d, 1987, 1993, 1994a,b, 1997] work on U(n) generalizations of the q-Gauss, q-Saalschiitz, and very-well-poised 6¢5 summation formulas. §5.4 M. Jackson [1954] employed (5.4.3) to derive transformation formu­ las for 3'¢3 series. §5.6 A transformation formula between certain 4¢3 and s'¢s series was found by Jain [1980b], along with transformation formulas for particular 7'¢7 series, and then used to deduce identities of Rogers-Ramanujan type with moduli 5, 6, 8, 12, 16, 20 and 24. Some recent results on bilateral basic hypergeometric series are given in Schlosser [2003a,b,c]. Ex.5.6 Watson [1929b] derived this identity in an equivalent form. For various proofs of the quintuple product identity (and of its equivalent forms) and applications to number theory, Lie algebras, etc., see Adiga, Berndt, Bhar­ gava and Watson [1985], Andrews [1974a], Atkin and Swinnerton-Dyer [1954], Bailey [1951]' Carlitz and Subbarao [1972]' Gordon [1961]' Hirschhorn [1988], Kac [1978, 1985], Sears [1952]' and Subbarao and Vidyasagar [1970]. Exercises 5.12 and 5.13 See Rahman and Suslov [1994a, 1998] for more general formulas. Notes 153

Exercises 5.15 and 5.16 For integrals of Ramanujan-type that corre­ spond to the summation formulas of basic bilateral series, see Rahman and Suslov [1994b, 1998] and Ismail and Rahman [1995]. Ex.5.18 Using (i) and (ii) one can show that the formula in Ex. 2.31 holds even when aI, a2, a3 are not nonnegative integers, provided that Iql < 1 and Iqa 1 +a 2 +a3 +11 < l. Exercises 5.21-5.23 Additional identities connecting sums of infinite prod­ ucts are given in Slater [1951, 1954b, 1966] and Watson [1929b]. 6

THE ASKEY-WILSON q-BETA INTEGRAL AND SOME ASSOCIATED FORMULAS

6.1 The Askey-Wilson q-extension of the beta integral

It should be clear by now that the beta integral and extensions of it that can be evaluated compactly are important. A significant extension of the beta integral was found by Askey and Wilson [1985]. Since it has five degrees of freedom, four free parameters and the parameter q from basic hypergeometric functions, it has enough flexibility to be useful in many situations. This integral is j 1 h(x;I,-I,q"2,-q"2)1 1 dx -1 h(x; a, b, e, d) viI - x 2 2n( abed; q)oo (6.1.1) (q, ab, ae, ad, be, bd, ed; q)oo' where

h(x; aI, a2,···, am) = h(x; aI, a2,···, am; q) = h(x; adh(x; a2) ... h(x; am),

00 h(x; a) = h(x; a; q) = II (1 - 2axqn + a2q2n) n=O (6.1.2) and max (Ial, Ibl, lei, Idl, Iql) < 1. (6.1.3)

As in (6.1.1), we shall use the h notation without the base q displayed when the base is q. Askey and Wilson deduced (6.1.1) from the contour integral

1 { (Z2,z-2;q)00 dz 2ni ) K (az, az-l, bz, bz-l, ez, ez-l , dz, dz- l ; q)oo z 2(abed; q)oo (6.1.4) (q, ab, ae, ad, be, bd, ed; q)oo' where the contour K is as defined in §4.9 and the parameters a, b, e, d are no longer restricted by (6.1.3), but by the milder restriction that their pairwise products are not of the form q-j, j = 0, 1,2, .... Askey and Wilson's original proof of (6.1.4) required a number of interim assumptions that had to be removed by continuity and analytic continuation arguments. In their paper

154 6.1 The Askey-Wilson q-extension of the beta integral 155 they also provided a direct evaluation of the reduced integral 1 h(x; 1, -1) dx 1-1 h(x; a, b) VI - x2 27r( -abq; q)oo 1 1 1 1 (6.1.5) (q, -q, aq2, -aq2, bq2, -bq2, ab; q)oo by using summation formulas for 1 'l/J1 and 4'l/J4 series. Simpler proofs of (6.1.1) were subsequently found by Rahman [1984] and Ismail and Stanton [1988]. In the following section we shall give Rahman's proof since it only uses formu­ las that we have already proved, whereas the Ismail and Stanton proof uses some results for certain orthogonal polynomials which will not be covered until Chapter 7. We shall conclude this section by showing that the beta integral

1 (1 _ x)<>(1 + x)f3dx = 2<>+f3+1 r(a + l)r(fi + 1) (6.1.6) 1-1 r(a+fi+2) is a limit case of (6.1.5). Let 0 < q < 1, a = q<>+~, b = _qf3+~ and use the notation (z; q)oo () (6.1. 7) Z; q <> = (zq<>; q)oo and the definition (1.10.1) of the q-gamma function to express the right side of (6.1.5) in the form

22<>+2f3+ 2 r q(a + l)rq(fi + 1) _7r_ (-q; q)<>+f3( -q~; q)<>+~ (-q~; q)f3+~ rq(a+fi+2) q(~) 22<>+2f3+1

By (1.10.3) this tends to 22<>+2f3+1 r(a + 1) r(fi + 1)/r(a + fi + 2) as q ---+ 1-, since r (~) = ..fif. For the integrand in (6.1.5) we have h(x;I,-I) (i()) (-i()) (i()) (-i()) . b) = e ;q <>+1 e ;q <>+1 -e ;q f3+1 -e ;q f3+1 h(x, a, 2 2 2 2 and hence h(x;I,-I) lim q--+1- h(x; q<>+~, _q!3+~) = [(1- ei()) (1- e-i())r+~ [(1 + ei()) (1 + e-i())lf3+~ = 2<>+f3+ 1(1 - cos e)<>+~ (1 + cos e)!3+~, which shows that (6.1.6) is a limit of (6.1.5). Formula (6.1.1) is substantially more general than (6.1.5) since it contains two more parameters. It is the freedom provided by these extra parameters which will enable us to prove a number of important results in this and the subsequent chapters. 156 The Askey-Wilson q-beta integral

6.2 Proof of formula (6.1.1)

Denote the integral in (6.1.1) by J (a, b, c, d). Since x = cos B is an even function of B, one can write

J(a, b, c, d) = ~ j7r h(x; 1, -1, y'ij, -y'ij) dB. (6.2.1) 2 -7r h(x; a, b, c, d) Let us assume, for the moment, that a, b, c, d and their pairwise products and quotients are not of the form q-j, j = 0,1,2, .... It is easy to check that, by (2.10.18), h(x; l)/h(x; a, b) (6.2.2) -:-:-:-_--:('-;-a_---'l,-;cb:---:-'\....::q;.-)00----:----:_ lb (qu / a, qu / b, u; q) 00 ~ , b(l - q) (q, alb, bq/ a, ab; q)oo a (u/ ab; q)oo h(x; u) h(x; -l)/h(x; c, d) (-c-l,-d-\q)oo Id (qv/c,qv/d:-v;q)oo dqv (6.2.3) d(l-q)(q,c/d,dq/c,cd;q)oo c (-v/cd,q)oo h(x;v) , and

1 1 1 1 1 h 1 h q2(-q2U- ,-q2v- ;q)oo (X;-q2)/ (X;U,V) = v(l-q)(q,U/V,vq/U,uv;q)oo

(6.2.4)

Also, ~ j7r h(x; q!) dB 2 -7r h(x; tq!) = ~ j7r (q! eiO , q! e-iO ; q)oo dB 2 -7r (tq!eiO,tq!e-iO;q)oo = ~ j7r f f (r\ qh(r\ q)R (tq!) kH ei(k-R)OdB 2 -7rk=O R=O (q;q)k(q;q)R = ~ f f (r\ qh(r\ q)R (tq!) kH j7r ei(k-R)OdB 2 k=O R=O (q; qh(q; q)R -7r 1 = 7r f (r , rl; q)k (qt2)k = 7r (qt, qt; q)oo (6.2.5) k=O (q, q; q)k (q, qt2; q)oo for Itq! I < 1, by (1.5.1). Since (qt2; q)oo = (qt2, q2t2; q2)00 = (tq! , -tq! , qt, -qt; q)oo, (6.2.6) we have h(x; q!) dB = 7r( qt; q)oo ~ j7r (6.2.7) 2 -7r h(x; tq!) (q, tq! , -tq! , -tq; q)oo . 6.3 Integral representations for very-well-poised 81>7 series 157

Thus

I(a,b,e,d)

bd(1 - q)3(q; q)'!xo(a/b, bq/a, ab, e/d, dq/e, cd; q)oo I b d (qu/a, qu/b, u; q)oo Id d (qv/e, qv/d, -v, -q! /u, -q! Iv; q)oo X a qU (u/ab;q)oo c qV v(-v/ed,vq/u,u/v,uv;q)oo 1 X lv:~'2 dqt (t'!~ /u, ~q~ lv, qt; q)oo uq 2 (tq'2, -tq'2, -qt/uv; q)oo n(a-1 , b-1 , -e-1 , -d-1 ; q)oo bd(1 - q)2(q; q)'!xo(a/b, bq/a, ab, e/d, dq/e, cd; q)oo b d (qu/ a, qu/b; q)oo Id d (qv / e, qv / d, -uv; q)oo X I c a qU (-u,u/ab;q)oo qV (v,uv,-v/ed;q)oo n( a-l, b-l, q! , -q! , -1; q)oo Ib d (qu/ a, qu/b, edu; q)oo = b(1 - q)(q; q)~(a/b, bq/a, ab, e, d, cd; q)oo a qU (eu, du, u/ab; q)oo n( -1, q! , -q!, abed; q)oo (6.2.8) (q, ab, ae, ad, be, bd, cd; q)oo' by repeated applications of (2.10.18). Since (-I,q!,-q!;q)oo = 2(q!,-q!,-q;q)oo = 2, which follows from (6.2.6) by setting t = 1, we get (6.1.1). By analytic continuation, the restric­ tions on a, b, e, d mentioned above may be removed.

6.3 Integral representations for very-well-poised 81>7 series

Formulas (2.10.18) and (2.10.19) enable us to use the Askey-Wilson q-beta integral (6.1.1) to derive Riemann integral representations for very-well-poised 81>7 series. Let us first set

1 1 2 _!h(x;I,-I,q'2,-q'2) w(x; a, b, e, d) = (1 - x) 2 h(. b d) (6.3.1) x,a, ,e, and 1 h(x;g) J(a, b, e, d, f, g) = 1-1 w(x; a, b, e, d) h(x; f) dx, (6.3.2) where max(lal, Ibl, lei, Idl, If I, Iql) < 1 and 9 is arbitrary. Since, by (2.10.18), h(x;g) (g/d,g/f;q)oo h(x;d,f) f(1 - q)(q, d/ f, qf /d, fd; q)oo f d (qu/ d, qui f, gu; q)oo (6.3.3) X id qU (gu/df; q)ooh(x; u) , 158 The Askey-Wilson q-beta integral we have J( b d f ) (g/d, g/ f; q)oo a, ,e, , ,g = f(l-q)(q,d/f,qf/d,df;q)oo {f d (qu/d, qui f, gu; q)oo 11 ( ) (6.3.4) X Jd qU (gu/df; q)oo -1 w x; a, b, e, u dx. By (6.1.1),

1 (. b ) d _ 27f(abeu; q)oo W x, a, ,e, u x - . (6.3.5) 1-1 (q, ab, ae, be; q)oo(au, bu, eu; q)oo Substituting this into (6.3.4), we obtain J( b d f ) = 27f(g/d, g/ f; q)oo ff a, ,e, , , 9 f( 1 - q) ( q, . q)2 00 (d/f ,q f/d ,J,d a b,ae, b·)e, q 00 f d (qu/d, qui f, gu, abeu; q)oo x qU ( /). (6.3.6) i 00 d au,bu,eu,gu df;q The parameters in this q-integral are such that (2.10.19) can be applied to obtain J( b d f ) = 27f(g/f,fg,abef,bedf,edaf,dabf;q)00 a, ,e, "g (q,ad,bd,ed,af, bf,ef,df,ab,ae, be, abedf2; q)oo x 8W7(abedf2q-\af,bf,ef,df,abedfg-\q,g/f), (6.3.7) provided Ig/ fl < 1, if the series does not terminate. By virtue of the transfor­ mation formula (2.10.1) many different forms of (6.3.7) can be written down. Two particularly useful ones are b d f = 27f(ag,bg,eg,abed,abef;q)00 J( a, ,e, , ,g) ( ) q, ab, ae, ad, af, be, bd, bf, cd, ef, abeg; q 00 x 8 W7(abegq-\ ab, ae, be, g/d, g/ f; q, df), (6.3.8) which was derived in Nassrallah and Rahman [1985], and J( b d f ) = 27f(ag, bg,eg, dg, fg,abedf/g; q)oo a, ,e, "g (q,ab,ae,ad,af,be,bd,bf,ed,ef,df,g2;q)00

x 8 W7(lq-\ g/a, g/b, g/e, g/d, g/ f; q, abedfg-1). (6.3.9) If the series in (6.3.9) does not terminate, then we must impose the condition labedfg-11 < 1 so that it converges. Note that, if in (6.3.8) we let 0 < q < 1, a = -b = q!, e = qa+!, d = z, f = _qf3+!, 9 = zq' with Re(a, (3) > -~ and then take the limit q --+ 1 -, we obtain, after some simplification,

. . _ r(a + (3 + 2) r1 a f3 -, 2F1(--y,a+1,a+(3+2,z)-r(a+1)r((3+1)Jo X (I-x) (l-xz) dx. (6.3.10) This shows that (6.3.8) is a q-analogue of Euler's integral representation (1.11.10). 6.4 Integral representations for very-well-poised 1O¢9 series 159

Another limiting case of (6.3.8) was pointed out in Rahman [1986b]. To derive it, replace a,b,e,d,f,g in (6.3.8) by qa,qb,qc,qd,qf,qg, respectively. Also, replace x in the integral (6.3.2) by cos(t log q), which corresponds to replacing ei () by qit. Now let q ---+ 1- and use (1.10.1) and (1.10.3) to get the formula ~ 100 r(a + it)r(a - it)r(b + it)r(b - it)r(e + it)r(e - it) 47f -00 r(2it)r( -2it) x r(d + it)r(d - it)r(j + it)r(j - it) dt r(g + it)r(g - it) r(a + b)r(a + e)r(a + d)r(a + f)r(b + e)r(b + d) r(a + g)r(b + g)r(e + g) r(b + f)r(e + d)r(e + f)r(a + b + e + g) x ~--~~--~~~~~----~~ r(a + b + e + d)r(a + b + e + f) F. a+b+e+g-l,~(a+b+e+g+l),a+b,a+e, b+e x 7 6 [ 1 2(a + b + e + 9 - 1), e + g, b + g, a + g,

9 - d, 9 - j, ., 1] , (6.3.11) a + b + e + d, a + b + e + j where Re (a, b, e, d, f) > O.

6.4 Integral representations for very-well-poised 1O¢9 series

If we set 9 = abedj in (6.3.7), then the 8 W7 series collapses to one term with value 1 and so we have the formula 11 h(x;I,-I,q!,-q!,abedf) dx -1 h(x; a, b, e, d, f) VI - x 2 27f(abed,abej,bedj,abdj,aedj;q)00 (q, ab, ae, ad, aj, be, bd, bj, cd, ej, dj; q)oo = go(a, b, e, d, f), say, (6.4.1) where (max lal, Ibl, lei, Idl, Ijl, Iql) < 1. This is a q-analogue of the formula 1 10 x a- 1(1 - x)b-1(1 - tx)-a-b dx = ~~~~(:? (1 - t)-a, Re(a, b) > o.

Replace j by jqn in (6.4.1), where n is a nonnegative integer, to get

1 (jei () , je-i(); q)n 1 v(x;a,b,e,d,f)( b dj -i() b dj i(). ) dx -1 ace ,a e e ,q n ( ) (aj, bj, ej, dj; q)n = go a,b,e,d,j (bedj,aedj,abdj,abej;q)n' (6.4.2) where

. b d j) = (1- h(x; 1, -1, q!, -q!, abedf) ( 2)-! (6.4.3) v x, a, ,e, , x h(.x,a, b,e, d , j) . 160 The Askey-Wilson q-beta integral

Let u = abedj. If Izl < 1, then (6.4.2) gives the formula

11 v(x;a,b,e,d,f) r+sWr+4(Ujq-\al, ... ,ar,jei!J,je-i!J;q,z) dx -1 = go(a, b, e, d, f) r+7Wr+6(u jq-\ al, ... , ar, aj, bj, ej, dj; q, z). (6.4.4)

In particular, for r = 3 and z = u2/ala2a3, we have the formula

1 v(x;a,b,e,d,j) 8W7(ujq-\al,a2,a3,jei!J,je-i!J;q,u2/ala2a3) dx 1-1 = go(a, b, e, d, f) lOW9(U jq-\ aI, a2, a3, aj, bj, ej, dj; q, u2/ala2a3), (6.4.5) where lu2/ala2a31 < 1, if the series do not terminate. Let us assume that (6.4.6) which ensures that the very-well-poised series on either side of (6.4.5) are bal­ anced. Then, by (2.11.7)

8W 7(u jq-l; aI, a2, a3, jei!J, je-i!J; q, q) (u j, qaI/u j, a2, qaI/a2, a3, qaI/a3; q)ooh(x; j, qaI/ f) +~~--~~~~~~~~~~~~~~~~~~~----~ (u j /qal, u j /a2, u j /a3, qala2/u j, qala3/u j, q2 aI/u j; q)ooh(x; u, qaI/u) x 8 W7(qai!u j; aI, qala2/u j, qala3/uj, qalei!J /u, qale-i!J /u; q, q) (u j, u j /a2a3, qaI/u j, u / j; q)ooh(x; u/a2, u/a3) (6.4.7) and hence 11 h(x;I,-I,q!,-q!,u/a2,u/a3) dx -1 h(x;a,b,e,d,j,u/a2a3) viI -x2 = go(a, b, e, d, f) { (~j /a2, ~ j /a3, uj ja2; u / j a3; ~)oo U j, Uj a2a3, u j, u ja2a3; q 00 x lOW9(U jq-l; aI, a2, a3, aj, bj, ej, dj; q, q) + (aj, bj, ej, dj, a2, a3, qaI/a2, qaI/a3; q)oo (u/a, u/b, u/e, u/d, qaal/u, qbal/U, qeal/u,qdal/U; q)oo (qal/aj,qal/bj,qal/ej,qal/dj,qal/uj;q)oo x (u / j, u f!qal, u j /a2a3, u/ ja2a3, q2 aI/u j; q)oo

x lOWg(qai!uj; aI, u/ ja2, u/ ja3, qaaI/u, qbaI/u, qeaI/u, qdaI/u; q, q) },

(6.4.8) where u = abedj and ala2a3q = u 2. Since the integrand on the left side of (6.4.8) is symmetric in a, b, e, d and j, the expression on the right side must have the same property. This provides an alternate proof of Bailey's four-term transformation formula (2.12.9) for VWP-balanced 1O

If we set a3 = q-n, n = 0,1,2, ... , then the coefficient of the second 10 Wg on the right side of (6.4.8) vanishes and we obtain

1 h(x;I,-I,q!,-q!,a) (aeiIJ/g,ae-iIJ/g;q)n dx 1 iIJ 2 -1 h(x; a, b, e, d, f) (aeiIJ, ae- ; q)n VI - x = go(a, b, e, d, f) (at 7' a!fg;)q)n a ,a ,q n ¢ [v' qy'v, -qy'v, g, aj, bj, ej, dj, a 2 qn-l/g, q-n ] X 10 9 y'v, _y'v, af/g, a/a, a/b,a/e,a/d, jgql-n/a,ajqn;q,q , (6.4.9) where v = a j /q. By applying the iteration of the transformation formula (2.9.1) given in Exercise 2.19, this can be written in the more symmetric form 11 h(x;I,-I,q2,-q2,aqn,T)1 1 dx -1 h(x; a, b, e, d, j, Tqn) vff=X2 = ( b d j) (Ta,Tb,Te,Td,Tj,a/T;q)n go a, ,e" (/a a, a /b ,a/ e, a /d , a /j ,T;2 q ) n

T 2 q -1 , Tq 12 , -Tq 12 ,T / a, T /b ,T / e, T /d , T /j , aTq n-l , q -n 1 [ x 1O¢9 _1 _1 I-n 2 n ;q,q, Tq 2, -Tq 2, Ta, Tb, Te, Td, T j, Tq / a, T q (6.4.10) where a = abedj and T is arbitrary. Similarly, by applying (2.12.9) twice one can rewrite (6.4.8) in the form 11 h(x;I,-I,q!,-q!,A,JL) dx -1 h(x; a, b, e, d, j, g) VI - x 2 2~(AJL/aj,AJL/bj,AJL/ej,AJL/dj,Ag,JLg,A/g,JL/g;q)oo (q, ab, ae, ad, ag, be, bd, bg, cd, eg, dg, jg, j /g, AJLg/ j; q)oo ¢ [VI, qy'Vl, -qy'Vl, ag, bg, eg, dg, AI j, JL/ j, AJL/q ] XlO 9 y'Vl, _y'Vl, AJL/aj, AJL/bj, AJL/ej, AJL/dj, JLg, Ag, gq/j;q,q 2~(AJL/ag,AJL/bg,AJL/eg,AJL/dg,Aj,JLj,A/j,JL/j;q)oo +~~~~~~~~~~~~~~~~~~-=~ (q, ab, ae, ad, aj, be, bd, bj, cd, ej, dj, gj, g/ j, AJLj /g; q)oo ¢ [V2' qyVi, -qyVi, aj, bj, ej, dj, A/g, JL/g, AJL/q ] XlO 9 yVi, _yVi, AJL/ag, AJL/bg, AJL/eg, AJL/dg, JLj, Aj, jq/g;q,q (6.4.11) where VI = AJLg/ jq, V2 = AJLj /gq, AJL = abedjg and max(lal, Ibl, lei, Idl, Ijl, Igl, Iql) < 1. For these and other results see Rahman [1986b]. If A or JL equals a, b, e or d, then the right side of (6.4.11) is summable by (2.11.7), while the integral on the left side gives the sum by (6.4.1). For this reason the integral in (6.4.1) may be considered as an integral analogue of Bailey's formula (2.11.7) for the sum of two balanced very-well-poised S¢7 series. Likewise, the integral in (6.4.2) is an integral analogue of Jackson's sum (2.6.2). Indeed, a basic integral may be called well-poised if it can be written in the form 162 The Askey-Wilson q-beta integral

n a _eiO a _e- iO . q) 11" I1 ( J' J ,00 1 j:,l d(). (6.4.12) -7r (b-eiO b-e-iO'q) I1 J 'J ,00 j=l The integrals in (6.1.1), (6.3.2), (6.4.1) and (6.4.2) are all well-poised.

6.5 A quadratic transformation formula for very-well-poised balanced 1O¢9 series

In (6.4.11) let us set J1 = -A, 9 = - f and b = -a so that A2 = -a2edj2. Then the expression on the right side of (6.4.11) becomes 2n(A2jef,A2jdf;q)00(A4ja2f2,A2f2,A2jf2;q2)00 (q, -1, -a2, - j2, A2; q)00(a2e2, a2d2, a2j2; q2)00(ed, ef, df; q)oo x 10 W g (A2q-1; _A2q-1, af, -af, Aj f, -Aj f, ef, df; q, q) + 2n( _A2 jef, _A2jdf; q)00(A4 ja2j2, A2 j2, A2 j j2; q2)00 (q, -1, -a2, - j2, A2; q)00(a2e2, a2d2, a2j2; q2)00(ed, -ef, -df; q)oo

x 10 W g (A2q-1; _A2q-1, af, -af, Aj f, -Aj f, -ef, -df; q, q). (6.5.1) We now turn to the integral on the left side of (6.4.11). Observing that

(6.5.2) where x = cos() and ~ = cos2() = 2x2 -1, it follows from (2.10.18) that

h(X;A- A) h(~;A2;q2) h(x; a, -a, f, - f) h(~; a2, j2; q2) (A2ja2, A2jf2; q2)00 j2(1- q2)(q2, a2 j j2, q2 j2 ja2, a2j2; q2)00

X (q 2uj a2, q2uj j2, A2U; q2)00 d If2 (6.5.3) a 2 (A2uja2 j2; q2)ooh(~; u; q2) q2U. Hence the integral on the left side of (6.4.11) can be expressed as

(6.5.4) by (6.1.1). Since A2 = -a2edj2, the q-integral on the right side of (6.5.4) 6.6 The Askey-Wilson integral when max (Ial, Ibl, lei, Idl) ~ 1 163 reduces to i 2 (q 2u/a 2, q2u/ p, -edqu, -a2edpu; q2)00 l ( 2 2 . 2) dq2U a2 e u, d u, -u, -uq, q 00 p(1- q2)(q2, a2/ p, q2 P /a2; q2)00 (a 2e2, a2d2, e2 j2, d2 j2; q2)00 X (_a2e2 p, _a2d2 p, a2e2d2 p, -edqp, -a2edj4; q2)00 (-a2, - j2, -qj2, -a2e2d2 j4; q2)00 X 8 W7( _a2e2d2 j4q-2; e2 j2, d2 j2, _ j2, cd, eda2 j2q-\ q2, _qa2), (6.5.5) by (2.10.19). Using this in (6.5.4) and equating with (6.5.1) we obtain the desired quadratic transformation formula

(-a2ej, -a2dj; q)oo 2 -1 2-1 (dj j. ) lOW9 ('\ q ;-,\ q ,aj,-aj,A/j,-'\/j,ej,dj;q,q) ,e ,q 00 (a2ej, a2dj; q)oo 2 -1 2-1 + (-dj,-ej;q)oo lOW9 ('\ q ;-,\ q ,aj,-aj,A/j,-'\/j,-ej,-dj;q,q) (_qa2 _a2e2p _a2d2 p. q2) 2 = (-1, -edj2, -eda j2; q)oo (2 jd d2j2 _'2 2d2 j4. '2) 00 e , ,ae ,q 00 X 8 W7( _a2e2d2 j4q-2; e2 j2, d2 j2, - j2, cd, eda2 j2q-\ q2, _qa2), (6.5.6) with ,\2 = -eda2 p. This gives a nonterminating extension of the transfor­ mation formula (3.10.3) when the bibasic q, series is a balanced 1O¢9 . For an extension of (3.10.3) when the q, series is not balanced, see Nassrallah and Rahman [1986].

6.6 The Askey-Wilson integral when max (Ial, Ibl, lei, Idl) ~ 1

Our aim in this section is to extend the Askey-Wilson formula (6.1.1) to cases in which Iql < 1 and the absolute value of at least one of the parameters is greater than or equal to 1. Since the integral in (6.1.1), which we have already denoted by I(a, b, e, d), is symmetric in a, b, e, d, without loss of generality we may assume that lal ~ 1. Let us first consider

lal ~ 1 > max(lbl, lei, Idl)· (6.6.1) If a = ±1, then the functions h(x; ±1) and h(x; a) in the integrand in (6.1.1) cancel and by continuity it follows that = 27f(±bed; q)oo I( ±1, b,e, d) ( ) (6.6.2) q, ±b, ±e, ±d, be, bd, cd; q 00 However, if lal = 1 and a -I=- ±1, then h(x; a) = 0 for some x in the interval (-1,1) and so the integral in (6.1.1) does not converge. Similarly, this integral does not converge if laqnl = 1 and aqn -I=- ±1 for some positive integer n. 164 The Askey-Wilson q-beta integral

If there is a nonnegative integer m such that

(6.6.3) and if ab, ac and ad are not of the form q-n for any nonnegative integer n, then the integral in (6.1.1) converges and we can evaluate it by the following technique. Observe that, since

h(x;a) = (aei !:l,ae- i!:l;q)m+1 h(x; aqm+1) = a2m+2qm+m2 h(x; aqm+1, q-m la)lh(x; qla), (6.6.4) we have 2 J( a, b,c, d) = a -2m-2 q -m-m 1 1 1 xj h(x;1,-1,q2-q2,qla) dx (6.6.5) -1 h(x; b, c, d, aqm+1, q-m fa) V1- x 2' where the parameters b, c, d, aqm+1, q-m I a in the denominator of the integrand are now all less than 1 in absolute value. By (6.3.8),

J( b d) = 2n(qbla,qcla,qdla,abcdqm+l,bcdq-ml a;q)00 a, ,c, (q, bc, bd, cd, abqm+1, acqm+l, adqm+l, bcdqla; q)oo a-2m- 2q-m-m2 x ~--~----~~~--~~-- (bq-m la, cq-m la, dq-m la; q)oo x 8 W 7(bcda-\ bc, bd, cd, q-ma-2, qm+\ q, q). (6.6.6)

The series in (6.6.6) is balanced and so we can apply Bailey's summation formula (2.11.7). After some simplification we find that

J( b d) 2n(abcd;q)00 L (·b d) a, ,c, = ( ) + m a, ,c, , (6.6.7) q,ab,ac,ad,bc,bd,cd;q 00 where

L . b d _ 2n( aql d, bql d, cql d, qm+1, a2bcqm+1, q-m I a2 , bcq-m; q)oo m (a, ,c, ) - (q, ab, ac, bc, abqm+1, acqm+1, adqm+l, aqm+l I d, abcql d; q)oo a-2m-1d-1q-m-m2 x ~--~------~~~~----~~~ (bq-m I a, cq-m I a, dq-m I a, q-m I ad; q)oo x 8 W 7(abcd-\ bc, q-m lad, aqm+1 Id, ab, ac; q, q). (6.6.8)

By (2.10.1),

8 W 7(abcd-\ bc, q-m lad, aqm+1Id, ab, ac; q, q) (abcqld,qlad,abq, acq; q)oo (bqld,cqld,q, a2bcq; q)oo x 8 W7(a2 bc; abcd, ab, ac, a2 qm+1, q-m; q, qlad). (6.6.9) 6.6 The Askey-Wilson integral when max (Ial, Ibl, Icl, Idl) ~ 1 165

Since m is a nonnegative integer, the series on the right side of (6.6.9) termi­ nates and hence, by Watson's formula (2.5.1), 2 TXT ( 2b . b d b 2 m+1 -m. I d) _ (a bcq, q; q)m 8 yy 7 a c, a c ,a ,ac, a q ,q , q, q a - ( b ) a q,acq;q m ¢ [q-m, ab, ac, q-m I ad . ] _ (a2bcq, q, aqlb, aql c; q)m X43 bcq-m, aqld, q-m,q,q - (qa2,abq,acq,qlbc;q)m a2, qa, -qa, ab, ac, ad, a2qm+1, q-m q] [ X 8¢7 a, -a, aq Ib ,aqI c, aq Id , q -m ,a2 q m+1; q, -b a c d (a2bcq,q,aqlb,aqlc;q)m (qa2,abq,acq,qlbc;q)m (a2;q)k(l-a2q2k)(ab,ac,ad;q)k ( q )k f (6.6.10) X (q; qh(1 - a2)(aqlb, aqlc, aqld; q)k abcd . k=O Using (6.6.10) and (6.6.9) in (6.6.8), we obtain 2 L ( . b d) = _ 2n(a- ; q)= m a, ,c, ( q,ab,ac,ad,bI a,c I a,d I a;q ) = (a2; q)k(1 - a2q2k)(ab, ac, ad; q)k ( q )k f (6.6.11) X (q; qh(1 - a2)(aqlb, aqlc, aqld; q)k abcd . k=O Hence 1 h(x;I,-I,q!,-q!) dx 2n(a-2;q)= 1-1 h(x; a, b, c, d) VI - x 2 + (q, ab, ac, ad, bla, cia, dla; q)= f (a2;q)k(l-a2q2k)(ab,ac,ad;q)k ( q )k X (q; q)k(1 - a2)(aqlb, aqlc, aqld; qh abcd k=O 2n(abcd; q)= (6.6.12) (q, ab, ac, ad, bc, bd, cd; q)= ' where max(lbl, Icl, Idl, Iql) < 1, laqm+ll < 1 < laqml for some nonnegative integer m, and the products ab, ac, ad are not of the form q-n, n = 1,2, .... Askey and Wilson [1985] proved this formula by using contour integration. By continuity, formula (6.6.12) also holds if the restriction (6.6.3) is replaced by aqm = ±l. Note that, if one of the products ab, ac or ad is of the form q-n for some nonnegative integer n, the integral in (6.6.12) converges even though the de­ nominator on the right side of (6.6.12) equals zero as does the denominator in the coefficient of the sum in (6.6.12). If we let ab tend to q-n then, since Ibl < 1 and laqm+11 < 1 < laqml, we must have n:S; m. We may then multiply (6.6.12) by 1 - abqn and take the limit ab ~ q-n. The result is a terminat­ ing 6¢5 series on the left side and its sum on the right, giving the summation formula (2.4.2). If max(lcl, Idl, Iql) < 1 and there are nonnegative integers m and r such that (6.6.13) 166 The Askey-Wilson q-beta integral then the above technique can be extended to evaluate I(a, b, e, d) provided the products ab, ae, ad, be, bd are not of the form q-n and alb i=- q±n for any nonnegative integer n. Splitting h(x; b) in the same way as in (6.6.4) and using (6.3.4) we get I(a,b,e,d) = b-2r- 2q-r-r2 J(a, bqr+1, q-r /b, e, d, q/b) - 2r- 2 -r-r2 (q/be, q/bd; q)oo = b q d(l - q)(q, e/d, dq/e, cd; q)oo

X d d (qu/e, quid, qu/b; q)oo 11 (. b r+1 -rib ) d j qU (/b d. ) W x, a, q , q ,u x. c qu e, q 00 -1 (6.6.14) However, by (6.6.12), 1 w(x; q, bqr+l, q-r /b, u) dx 1-1r 27r( aqu; q)oo (q, q, abqr+1, aq-r /b; q)oo(au, buqr+l, uq-r /b; q)oo 27r(a-2; q)oo (q, abqr+1, q-r lab, bqr+1/a; q)oo(au, u/a; q)oo ~ (a2; q)k(l - a2q2k)(au; q)k ( )-k (6.6.15) X ~ (q; qh(l _ a2)(aq/u; q)k au .

Since by (2.10.18), j d dqu (qu/e, quid, qu/b; q)oo (au; qh (qu)-k c (qu/bed, au, u/a; q)oo (aq/u; q)k = (_l)k -2k -k(k+1)/2 jd d (qu/e, quid, qu/b; q)oo qU ( k k/ ) a q c qu /b e d ,auq ,uq- a;q 00 d(l - q)(q, e/d, dq/e, cd, q/ab, aq/b; q)oo (q/be,q/bd, ae, ad, e/a, d/a; q)oo (ab,ae, ad; qh (q)k (6.6.16) X (aq/b,aq/e,aq/d;q)k abed ' we find that I(a, b, e, d) = Lm(a; b, e, d) + 27r(adq, edq, dq/b, qr+2, q1-r /b2; q)oob-2r- 2q-r-r2 (q,q,ad,ed,dq2/b,abqr+1,beqr+1,bdqr+1,aq-r/b,eq-r/b,dq-r/b;q)oo x 8 W7(dq/b; bdqr+1, q, dq-r /b, q/be, q/ab; q, ae), (6.6.17) where Lm (a; b, e, d) is as defined in (6.6.11). The reduction of this 8 W 7 will be done in two stages. First we use (2.10.1) twice to reduce it to a balanced 8 W 7 and then apply (2.11.7) to obtain

8W7(dq/b;bdqr+1,q,dq-r/b,q/be,q/ab;q,ae) 6.6 The Askey-Wilson integral when max (Ial, Ibl, Icl, Idl) ~ 1 167

(dq2/b,q/bd,abcdqr+1,acdq-r/b;q)00 (q1-r/b2,qr+2,ac,qacd2;q)00

X 8 W7( acd2; abcd, bdqr+1, dq-r /b, ad, cd; q, q/bd) (dq2/b,aq/b,cq/b,q, abcdqr+1,acdq-r/b; q)oo (ac,adq, cdq, acdq/b,qr+2, q1-r/b2; q)oo

x 8 W 7 (acdb-\ cd, ad, ac, qr+1, q-r /b2; q, q) (dq2/b, aq/b, bfa, q, q/ab, abcd, bcqr+1, bdqr+1, cq-r /b, dq-r /b; q)oo (ac,adq, cdq,dq/b, bd, bc,qr+2, bqr+1/a, q-r/ab, q1-r/b2; q)oo + ~ (dq2/b, q, bq/a, cq/a, dq/a, ad, qr+1, b2cdqr+1, cdq-r, q-r /b2; q)oo a (bcdq/a, dq/b, bc, bd, adq, cdq, qr+2, bqr+1 la, q-r lab, q1-r /b2; q)oo

X 8 W 7 (bcda-\ bc, bd, cd, bqr+1 la, q-r lab; q, q). (6.6.18) Substituting (6.6.18) into (6.6.17) and simplifying the coefficients we get 27r(abcd; q)oo I(a, b, c, d) - Lm(a; b, c, d) - (b d b bd d ) q,a ,ac,a ,c, ,c;q 00 27r(bq/a, cq/a, dq/a, qr+1, b2cdqr+1, cdq-r, q-r/b2; q)oo (q,bc,bd,cd,abqr+1,bcqr+1,bqr+1/a,aq-r/b,cq-r/b,dq-r;q)00 b-2r-1 -1 -r-r2 X ( -r / b : d q / .) 8 W 7 (bcda-\ bc, bd, cd, bqr+1 la, q1-r lab; q, q). q a, c q a, q 00 (6.6.19) The expression on the right side of (6.6.19) is the same as that in (6.6.8) with a, b, c, d, m replaced by b, d, c, a and r, respectively, and so has the value 27r(b- 2; q)oo (q,ba, bc,bd,a/b,c/b,d/b; q)oo t (b2; q)k(l- b2q2k)(ba, bc, bd; q)k ( q )k X (q; q)k(l - b2 )(bq/a, bq/c, bq/d; q)k abcd k=O = Lr(b; c, d, a), (6.6.20) by (6.6.11). So we find that I(a, b, c, d) - Lm(a; b, c, d) - Lr(b; c, d, a) 27r(abcd; q)oo (6.6.21) (q, ab, ac, ad, bc, bd, cd; q)oo' where the parameters satisfy the conditions stated earlier. It is now clear that we can handle the cases of three or all four of the parameters a, b, c, d exceeding 1 in absolute value in exactly the same way. For example, in the extreme case when min(lal, Ibl, lei, Idl) > 1 > Iql with laqm+11 < 1 < laqml, Ibqr+11 < 1 < Ibqrl, Icqs+11 < 1 < IcqSI, Idqt+11 < 1 < Idqtl, (6.6.22) for some nonnegative integers m, r, s, t such that the pairwise products of a, b, c, d are not of the form q-n and the pairwise ratios of a, b, c, d are not 168 The Askey-Wilson q-beta integral of the form q±n for n = 0,1,2, ... , we have the formula I(a, b, c, d) - Lm(a; b, c, d) - Lr(b; c, d, a) - Ls(c; d, a, b) - Lt(d; a, b, c) 27r( abcd; q)oo (6.6.23) (q, ab, ac, ad, bc, bd, cd; q)oo' where Ls(c; d, a, b) and Lt(d; a, b, c) are the same type of finite series as those in (6.6.11) and (6.6.20), and can be written down by obvious replacement of the parameters.

Exercises

6.1 Prove that 17r sin2 () d() 7r(1 - a1 a2a3a4) 2 I1 (1 - ajak) o IT (1- 2aj cos() + a;) 1<;j

where 0 < q < 1 and Izq!1 < 1. 6.4 If 0 < q < 1 and max(lal, Ibl, Icl, Idl) < 1, show that 1 h(x;I,-I,q!,j) dx 1-1 h(x; a, b, c, d) VI - x 2 27r(aj, bj, cj, -abcq!, abcd; q)oo (q, ab, ac, ad, bc, bd, cd, -aq2,1 -bq2, 1 -Cq2, 1 abcj; q)oo

x 8 W 7 (abcjq-\ ab, ac, bc, - jq-!, Jld; q, -dq!). 6.5 If max(lal, Ibl, Icl, Idl, Iql) < 1, show that 1 h(X;I,~I,q!,-q!,-q/c) dx 1-1 h(x, a, -a, b, -b, c) VI - x 2 27r( -q; q)oo( _a2b2c2, _q2 /c2; q2)00 (q; q)oo( -a2, -b2, a2b2, a2c2, b2c2; q2)00· Exercises 169

6.6 If b = a-I, Icl < 1, Idl < 1 and laqm+11 < 1 < laqml for some nonnegative integer m such that ac, ad are not of the form q-n, n = 0,1,2, ... , show that

I( -1 d) 27f(qa2)-1 a, a , c, + (q,q,acq,adq,c / aq,d / aq;q )=

X m-1 (1 - a2q2k+2) (acq, adq; q)k(q/cd)k {; (1 - qk+l )(1 - a2qk+l )(aq2 /c, aq2/d; q)k 27f (q,q,ac,ad,c/a,d/a;q)=

x ~ {a2 ~ qr + (Cd)-~ _ qr - ac_ l1_ qr - ad_l1 _ qr } qr,

where I(a, b, c, d) is as defined in (6.2.1). If m = 0, the series on the left side is to be interpreted as zero.

6.7 Applying (2.12.9) twice deduce from(6.4.8) that

r h(cos();I,-I,q!,-q!,A,J-L)d() Jo h(cos();al,a2,a3,a4,as,a6) = A (J-L 2;?)= IT (Aa j ;?)= (J-L/ A, q)= j=l (Ajaj, q)=

x 10 W g (A2q-1; AJ-L/q, Ajal, A/a2, Aja3, Aja4, A/as, Aja6; q, q) + A (A2; .q)= IT (J-Laj;.q)= (A/ J-L, q)= j=l (J-L/aj, q)=

x 10 W g (J-L2 q-l; AJ-L/ q, J-L/ aI, J-L/ a2, J-L/ a3, J-L/ a4, J-L/ as, J-L/ a6; q, q),

6 where AJ-L = I1 aj, max(lql, lall,···, la61) < 1, and j=l

(Rahman [1986b, 1988b])

6.8 Prove that

8 W 7 (abczq-\ az, bz, cz, abcdflq, 1/ J; q, dqJ (q,az, a/z,bz, b/z, cz, abcz, abc/Az, AJ-L2 q/ abdz,J-L2, ab/J-L2; q)= 27f(ab, ac, bc, q/dz, zq/ Jd, abczJ, qJ-L2/dz, aq/ JdJ-L2; q)= 170 The Askey-Wilson q-beta integral

Aq a2be .) 11 h(x;l,-l,q!,-q!,I!j,/aJ-L) x ( -, -2 ,bef,q A bd AJ-L 00 -1 h(x· Q 12. HZ I!:. abe ~) , J-L' J-L' "" , z' AJ-L ' abd a aei!) ae-iO q ab Aq ) ( x SW7 dfJ-L2; ---;;-' -J-L-' bdf' fAJ-L2' abedi q, bef J-L 2 J-LeiO J-Le- iO q ab AJ-L2q ) dx XSW7 ( -d ;-'--'-d"'-bd;q,ez ~' z z z Z /\ a e V 1 - x2

where Izl = 1 andmax(lal, Ibl, lei, Idl, If I, IJ-LI, la/J-LI, Ib/J-LI, labe/AJ-LI, IAJ-Lq/abdl, Iq/dzl, Iql) < 1.

6.9 Choosing A = abedq-1 and f = qn deduce from Ex. 6.8 that

where z = cos'lj; and max(lal, Ibl, IJ-LI, Iql) < 1.

6.10 Show that

roo (iax, -ia/x, ibx, -ib/x, iex, -ie/x, idx, -id/x; q)oo dx Jo (-qx2,_q/x2;q)00 x (log q-1 )(ab/q, ae/q, ad/q, be/q, bd/q, ed/q, q; q)oo (abed/ q3; q)oo

when labed/q31 < 1 and Iql < 1. (Askey [1988b])

6.11 Show that

when Re x > 0, Re y > 0, Iql < 1, and k = 0, 1, .... (Askey [1980b]) Exercises 171

6.12 Show that 00 100 (-at qx+y+2k -at qx+y+2k. q) 1 t x - 1t x - 1 1 ,2 ,00 o 0 1 2 (-at1,-at2;q)00

x tik(t2q1-k It1; qhk dqt 1dqt2 rq(x)rq(X + k)rq(y)rq(y + k)rq(k + l)rq(2k + 1) rq(x + y + k)rq(x + y + 2k)rq(k + l)rq(k + 1) x (-aqX, _ aqx+2k, _q1-x I a, _q1-x-2k I a; q)oo (-a, -a, -qla, -qla; q)oo when Re x > 0, Re y > 0, Iql < 1, and k = 0, 1, .... (Askey [1980b]) 6.13 With w(x; a, b, c, d) defined as in (6.3.1), prove that 1111 w(x; a, aq2,1 b, bq2)1 w(y; a, aq2,1 b, bq2)1 -1 -1 x l(q!ei(lIH),q!ei(lI-1»;qhI2dxdy

2 2n = J1 [(1 - ab)(q, abq(j-1)k+1/2, abq(j-1)k+1; q)ooJ2

2 (a 2b2qjk+1,qk+l,q;q)00 x II (a2q(j-1)k+1/2 b2q(j-1)k+1/2 qjk+1. q) , j=l ' " 00 where x = cosO, y = coscp, Iql < 1, and k = 0,1, .... (Rahman [1986a]) 6.14 Let C.T. f(x) denote the constant term in the Laurent expansion of a function f(x). Prove that if j and k are nonnegative integers, then C.T. (qx, x-\ q)j(qx2, qx-2; q2)k

= 2~ 17r(qe2i!:1,e-2ill;q)j(qe4ill,qe-4ill;q2)k dO

(q; qhj(q2; q2hk(q2j+\ q2h (q; q)j(q2; q2h(q; q)j+2k (Askey [1982b]) 6.15 Verify Ramanujan's identities (i) I: e-x2+2mX(_aqe2kX,_bqe-2kX;q)00 dx vfif( abq; q)ooem2 (ae2mk..fil, be-2mk ..fil; q)oo ' 00 e-x2+2mxdx (ii) 1 -00 (ae2ikx ..fil, be-2ikx ..fil; q)oo vfif em2 (_aqe2imk , _bqe-2imk ; q)oo (abq; q)oo 172 The Askey-Wilson q-beta integral

where q = e-2k2 and Iql < l. (See Ramanujan [1988] and Askey [1982a])

6.16 Derive the q-beta integral formulas

{OO (-tqb,-qa+~/t;q)oo dqt = rq(a)rq(b) (i) Jo (-t,-qlt,q)oo t rq(a+b) and {OO (_tqb, _qa+l It; q)oo dt = -log q r q(a)r q(b) , (ii) Jo (-t,-qlt;q)oo t 1-q rq(a+b)

where 0 < q < 1, Re a > 0 and Re b > o. (Askey and Roy [1986], Gasper [1987])

6.17 Extend the above q- beta integral formulas to

{OO t - 1 (-tqb, _qa+l It; q)oo d t (i) c Jo (-t, -qlt; q)oo q (_qC, _ql-c; q)oo r q(a + c)rq(b - c) (-l,-q;q)oo rq(a+b) and

{OO t c - 1 (_tqb, _qa+~ It; q)oo dt (ii) Jo (-t, -qlt, q)oo r(c)r(l- c)rq(a + c)rq(b - c) rq(c)rq(l- c)rq(a + b)

where 0 < q < 1, Re (a + c) > 0 and Re (b - c) > o. (See Ramanujan [1915]' Askey and Roy [1986], Gasper [1987], Askey [1988b], and Koornwinder [2003])

6.18 Use Ex. 2.16(ii) to show that, for y = cos ¢;,

if x> y, if x < y, if x = y.

(Ismail and Rahman [2002b])

6.19 Denoting S(al' ... ' am) = S(al' ... ' am; q) = n:=l (ak, qlak; q)oo, use Ex. 2.16(i) to prove that Notes 173

Deduce that n 7r (2W -2iIJ.) rr h(cosf); Ak, q/Ak) r e , e ,q 00 k=1 df) io h(cos f); a, b) n +l (f) /) rr h cos ;am,q am m=1 21f n+l n = (1 _ ab)( .) L II S(Ak/aj, Akaj) q, q, q 00 j=1 k=1 n+l X II [S(araj, ar/aj)(aaj, baj, aq/aj, bq/aj; q)oo]-I. r=l r=l-j (Ismail and Rahman [2002b]) 6.20 Show that 1 h(x; 1, -1, ql/2, _ql/2, aql/2, ql/2/a ) dx 1-1 h(x;al,a2, ... ,a6) viI -x2 6 (al a2a3a4a5a6/ q; q)oo rr (aajql/2, ajql/2 / a; q)oo j=1 (q, q, qa2, q/a 2; q)oo rr (ajak; q)oo l::;j

provided a i=- q±n, n = 0,1,2, ... , lala2a3a4a5a6/ql < 1, and lakl < 1, k = 1, ... ,6. (See Gustafson [1987b] and Rahman [1996a]) 6.21 Prove that

sin V2 cosh VI (q, _qe2V1 , _qe-2V1 , tl t2t3t4/ q3; q)oo I(qe 2iv2 ; q)oo 12 ' where V = VI + iV2, 0 < V2 < 1f/2 or V2 = 1f/2 and VI :::; 0, Itjl < 1, Itlt2t3t4/q31 < l. (Ismail and Masson [1994])

Notes

Ex.6.7 Setting al = -a2 = a, a3 = -a4 aq~, a5 = b, a6 = C, A = a 2 bq, JL = a 2 c and using Ex. 2.25, Rahman [1988b] evaluated this integral in closed form and found the corresponding systems of biorthogonal rational functions. 174 The Askey-Wilson q-beta integral

Ex. 6.8 This may be regarded as a q-extension of the terminating case of Erdelyi's [1953, 2.4(3)] formula

1 F( a, b; c; x) = r(c) r t lL - 1 (1 _ t)c-IL -1 (1 _ tx ).. -a-b r(JL)r(c - JL) 10 x F(A - a, A - b; JL; tx) F(a + b - A, A - JL; c - JL; (1 - t)x/(1 - tx)) dt, where Ixl < 1, 0 < Re JL < Re c. Also see Gasper [1975c, 2000]. Ex.6.10 Askey [1988b] found the 4¢3 polynomials which are orthogonal with respect to the weight function associated with this integral. Exercises 6.11- 6.13 The double integrals in these exercises are q-analogues of the n = 2 case of Selberg's [1944] important multivariable extension of the beta integral: 2z

= IT r(x + (j - l)z)r(y + (j - l)z)r(jz + 1) j=l r(x+y+(n+j-2)z)r(z+l) , where Re x > 0, Re y > 0, Re z > - min(l/n, Re x/(n - 1), Re y/(n - 1)). Aomoto [1987] considered a generalization of Selberg's integral and uti­ lized the extra freedom that he had in his integral to give a short elegant proof of it. Habsieger [1988] and Kadell [1988b] proved a q-analogue of Selberg's integral that was conjectured in Askey [1980b]. For conjectured multivariable extensions of the integrals in Exercises 6.11-6.13, other conjectured q-analogues of Selberg's integral, and related constant term identities that come from root systems associated with Lie algebras, see Andrews [1986, 1988], Askey [1980b, 1982b, 1985a, 1989f, 1990], Evans, Ismail and Stanton [1982]' Garvan [1990], Garvan and Gonnet [1992]' Habsieger [1988], Kadell [1988a-1994]' Macdon­ ald [1972-1995]' Milne [1985a, 1989], Morris [1982]' Rahman [1986a], Stanton [1986b, 1989], and Zeilberger [1987, 1988, 1990a]. Also see the extension of Ex. 6.11, the q-Selberg integrals, and the identities in Aomoto [1998], Evans [1994]' Ito [2002]' Kaneko [1996-1998], Lassalle and Schlosser [2003], Mac­ donald [1998a,b], Rains [2003a], Stokman [2002] and Tarasov and Varchenko [1997]. 7

APPLICATIONS TO ORTHOGONAL POLYNOMIALS

7.1 Orthogonality

Let a(x) be a non-constant, non-decreasing, real-valued bounded function de­ fined on (-00,00) such that its moments

/Ln = I: xnda(x), n = 0,1,2, ... , (7.1.1) are finite. A finite or infinite sequence Po (x), Pl (x), ... of polynomials, where Pn(x) is of degree n in x, is said to be orthogonal with respect to the measure da(x) and called an orthogonal system of polynomials if

(7.1.2)

In view of the definition of a(x) the integrals in (7.1.1) and (7.1.2) exist in the Lebesgue-Stieltjes sense. If a(x) is absolutely continuous and da(x) = w(x)dx, then the orthogonality relation reduces to I: Pm(X)Pn(X)W(X) dx = 0, m -=I- n, (7.1.3) and the sequence {Pn (x)} is said to be orthogonal with respect to the weight function w(x). If a( x) is a step function (usually taken to be right-continuous) with jumps Wj at x = Xj,j = 0,1,2, ... , then (7.1.2) reduces to

LPm(Xj)Pn(Xj)Wj = 0, m -=I- n. (7.1.4) j In this case the polynomials are said to be orthogonal with respect to a jump function and are usually referred to as orthogonal polynomials of a discrete variable. Every orthogonal system of real valued polynomials {Pn(x)} satisfies a three-term recurrence relation of the form (7.1.5) with p-l(X) == O,Po(x) == 1, where An, B n, Cn are real and AnCn+l > 0. Conversely, if (7.1.5) holds for a sequence of polynomials {Pn(x)} such that P-l(X) == O,Po(x) == 1 and An, B n, Cn are real with AnCn+l > 0, then there exists a positive measure da(x) such that

(7.1.6)

175 176 Applications to orthogonal polynomials where _ rrn A k - l Vn - C' Vo = 1. (7.1.7) k=l k If {Pn(X)} = {Pn(x)}~=o and AnCn+1 > °for n = 0,1,2, ... , then the measure has infinitely many points of support, (7.1.5) holds for n = 0,1,2, ... , and (7.1.6) holds for m, n = 0,1,2, .... If {Pn(x)} = {Pn(x)};;=o and AnCn+1 > ° for n = 0,1,2, ... , N -1, where N is a fixed positive integer, then the measure can be taken to have support on N + 1 points xo, Xl,"" XN, (7.1.5) holds for n = 0,1, ... , N - 1, and (7.1.6) holds for m, n = 0,1,2, ... , N. This characterization theorem of orthogonal polynomials is usually at­ tributed to Favard [1935], but it appeared earlier in published works of Perron [1929]' Wintner [1929] and Stone [1932]. For a detailed discussion of this theo­ rem see, for example, Atkinson [1964], Chihara [1978], Freud [1971] and Szego [1975]. In the finite discrete case the recurrence relation (7.1.5) is a discrete ana­ logue of a Sturm-Liouville two-point boundary-value problem with boundary conditions p-I(X) = O,PN+I(X) = 0. If Xo, Xl, ... , XN are the zeros of PN+l(X), which can be easily proved to be real and distinct (see e.g., Atkinson [1964] for a complete proof), then the orthogonality relation (7.1.6) can be written in the form N N LPm(Xj)Pn(Xj)Wj = V;;:-l L Wj bm,n, (7.1.8) j=O j=O m, n = 0,1, ... , N, where Wj is the positive jump at Xj and Vn is as defined in (7.1.7). The dual orthogonality relation N N LPn(Xj)Pn(Xk)Vn = wjl L Wn bj,k, (7.1.9) n=O n=O j, k = 0,1, ... , N, follows from the fact that a matrix that is orthogonal by rows is also orthogonal by columns. It can be shown that N Wj = [ANvNPN(Xj )P~+l (Xj) r l L Wn, j = 0, 1, ... ,N, (7.1.10) n=O where the prime indicates the first derivative. In general, the measure in (7.1.6) is not unique and, given a recurrence re­ lation, it may not be possible to find an explicit formula for o:(x). Even though the classical orthogonal polynomials, which include the Jacobi polynomials

P~Q,(3)(x) = (0: :!I)n 2FI ( -n, n + 0: + (3 + 1; 0: + 1; 1; X) , (7.1.11) and the ultraspherical polynomials

C~(x) = ((2>')t) p2-~')..-~)(x) >'+"2n _ Ln (>'h(>')n-k i(n-2k)e e -'-----'---:----'----'------:-:- e X = cos , (7.1.12) - k!(n-k)! ' k=O 7.2 The finite discrete case 177 are orthogonal with respect to unique measures (see Szego [1975]), it is not easy to discover these measures from the corresponding recurrence relations (see e.g., Askey and Ismail [1984]). However, for a wide class of discrete or­ thogonal polynomials it is possible to use the recurrence relation (7.1.5) and the formulas (7.1.8)-(7.1.10) to compute the jumps Wj and hence the mea­ sure. We shall illustrate this in the next section by considering the q-Racah polynomials (Askey and Wilson [1979]).

7.2 The finite discrete case: the q-Racah polynomials and some special cases

Suppose {Pn(x)} is a finite discrete orthogonal polynomial sequence which sat­ isfies a three-term recurrence relation of the form (7.1.5) and the orthogonality relations (7.1.8) and (7.1.9) with the weights Wj and the normalization con­ stants Vn given by (7.1.10) and (7.1.7), respectively. We shall now assume, without any loss of generality, that Pn(XO) = 1 for n = 0,1, ... , N. This enables us to rewrite (7.1.5) in the form

(x - XO)Pn(x) = An [Pn+l(X) - Pn(x)]- en [Pn(x) - Pn-l(X)] , (7.2.1) where n = 0,1, ... ,N. Setting j = k = 0 in (7.1.9) we find that N N L Vn = W;;l L W n· (7.2.2) n=O n=O It is clear that in order to obtain solutions of (7.2.1) which are representable N in terms of basic series it would be helpful if Vn and L Vn were equal to n=O quotients of products of q-shifted factorials. Therefore, with the 6¢5 sum (2.4.2) in mind, let us take (abq; q)n (1 - abq2n+l) (aq, eq, bdq; q)n ( )-n Vn = (q; q)n(1- abq) (bq, abq/e, aq/d; q)n edq n (1 _ abqk) (1 - abq2k+l) (1 - aqk)(l - eqk)(l - bdqk) = (1- qk) (1- abq2k-l) (1- bqk)(l- abqk/e)(l- aqk/d)edq' k=lII (7.2.3) where bdq = q-N, 0 < q < 1, so that ~ _ [abq, q(abq)~, -q(abq)~, aq, eq, q-N . bqN ] ~ Vn - 6¢5 1 1 N 2' q, n=O (abq)'i, -(abq)'i, bq, abq/e, abq + e (abq2, b/e; q)N (7.2.4) (bq, abq/e; q)N' where it is assumed that a, b, e, d are such that Vn > 0 for n = 0,1, ... , N. In view of (7.1. 7) we can take A _ (1 - abqk) (1 - aqk) (1 - eqk) (1 - bdqk) k-l - (1 _ abq2k-l) rk, (7.2.5) 178 Applications to orthogonal polynomials

(7.2.6) where {rdf=l is an arbitrary sequence with rk =I- 0,1 ::::; k ::::; N. Since Co = 0 and Ao = (1 - aq)(l - eq)(l - bdq)rl' we have from the n = 0 case of (7.2.1) that -1 (1- q-l)(Xj - xo)qr11 Pl ( X· ) -- (7.2.7) J (1 - q)(l - aq)(l - eq)(l - bdq)

This suggests that we should look for a basic series representation of Pn (x j ) whose (k + l)-th term has (q, aq, eq, bdq; qh as its denominator, which in turn suggests considering a terminating 4¢3 series. In view of the product ( -n) k (n+ a+,8 + l)k in the numerator ofthe (k+ l)-th term in the hypergeometric series representation of p~

Replacing q

Observing that (q-j, edqJ+\ q) k is a polynomial of degree k in the variable q-j + edqj+1, we find that if we take

(7.2.9) then Xj - Xo = -(1- q-j)(l- edqJ+l) , and so (7.2.7) is satisfied with

(7.2.10)

Then AkCk+1 > 0 for 0 ::::; k ::::; N - 1 if, for example, a, b, e are real, d = b-1q-N-1, be < 0, max(laql, Ibql, leql, lablel) < 1 and 0 < q < 1. We shall now verify that _ [q-n,abqn+l,q-j,edqj+l. ] Pn(Xj) - 4¢3 bd' q, q (7.2.11) aq, eq, q satisfies (7.2.1) with x = Xj. A straightforward calculation gives

(7.2.12) 7.2 The finite discrete case 179 o < n ::::: N. So we need to verify that q-n, abqn+l, q-j, edqj+l ] [ 4¢3 bd; q, q aq, eq, q _ Anq-n (1 - abq2n+2) [q-n, abqn+2, ql-j , edqj+2 . ] - (1 - aq) (1 - eq) (1 - bdq) 4¢3 aq2, eq2, bdq2 ' q, q Cnql-n (1- abq2n) [ql-n, abqn+l, ql-j , edqj+2 . ] - (l-aq)(l-eq)(l-bdq) 4¢3 aq2,eq2,bdq2 ,q,q, (7.2.13) where An and Cn are given by (7.2.5) and (7.2.6) with Tk as defined in (7.2.10). Use of (2.10.4) on both sides of (7.2.13) reduces the problem to verifying that q-n, abqn+l, bq-j Ie, bdqi+l ] [ 4¢3 bq, a b q I e, bd q ; q, q (1- abqn+l) (1- bdqn+l) (1 - abq2n+l) (1 - bdq) q-n, abqn+2, bq-j Ie, bdqi+l ] [ x 4¢3 bq, a b q I e, bd q 2 ; q, q (1 - qn) (1 - aqn Id) (1 - abq2n+l) (1 _ bdq) bdq ql-n, abqn+l, bq-j Ie, bdqi+l ] [ (7.2.14) X4¢3 bq, a blbd2 q e, q ;q,q, which follows immediately from the fact that (q-n; qh (abqn+l; q) k+l (1 - bdqn+l) (1 - abq2n+l) (1 - bdqk+l) (q-n;q)k+l (abqn+\q)k (1- af) bdqn+l + (1 - abq2n+l) (1 _ bdqk+l) = (q-n, abqn+1; q) k .

The verification that (7.2.11) satisfies the boundary condition PN+l(Xj) = o for 0 ::::: j ::::: N is left as an exercise (Ex. 7.3). Note that the 4¢3 series in (7.2.11) remains unchanged if we switch n, a, b, respectively, with j, e, d. This implies that the polynomials Pn(Xj) are self-dual in the sense that they are of degree n in Xj = q-j + edqi+l and of degree j in Yn = q-n + abqn+l and that the weights Wj are obtained from the Vn in (7.2.3) by replacing n, a, b, e, d, by j, e, d, a, b, respectively, i.e., (edq; q)j (1 - edq2i+l) (eq, aq, bdq; q)j ( )_j W· = abq , (7.2.15) J (q; q)j(l - edq) (dq, edqla, eqlb; q)j o ::::: j ::::: N. Thus {Pn(Xj)};;=o is orthogonal with respect to the weights Wj, while {Pj (Yn)}f=o is orthogonal with respect to the weights Vn (see Ex. 7.4). The calculations have so far been done with the assumption that bdq = q-N, but the same results will hold if we assume that one of aq, eq and bdq is q-N. 180 Applications to orthogonal polynomials

The polynomials Pn(Xj) were discovered by Askey and Wilson [1979] as q-analogues of the Racah polynomials

(,\.) = F [-n,n+a+ f3 +1,-j,j+1'+6"+1.1] (7.2.16) Pn J 4 3 a + 1, l' + 1, f3 + 6" + 1 " where ,\j = j(j + l' + 6" + 1), which were named after the physicist Racah, who worked out their orthogonality without apparently being aware of the polynomial orthogonality. In the physicist's language the Pn ('\j) 's are known as Racah coefficients or 6j-symbols. The connection between the 6j-symbols and the 4F3 polynomials (7.2.16) was first made in Wilson's [1978] thesis. The q-analogues in (7.2.11) are called the q-Racah polynomials. We shall now point out some important special cases of the q-Racah poly­ nomials. First, let us set d = b-1q-N-l in (7.2.11) and replace e by be to rewrite the q-Racah polynomials in the more standard notation q-n,abqn+1, q-X, eqx-N ] Wn(x;a,b,e,N;q) = 4¢3 [ -N ;q,q . (7.2.17) aq,q ,beq Then, by (7.1.8), (7.2.3) and (7.2.15),

N LP(x;q)Wm(X;q)Wn(x;q) = ~m(,n), (7.2.18) x=o n q where Wn(X; q) == Wn(x; a, b, e, N; q), p(x; q) == p(x; a, b, e, N; q) (eq-N. q) (1 eq2X-N) (aq beq q-N. q) = 'x ' , 'x (abq)-X (7.2.19) (q; q)x(l - eq-N)(ea-1q-N, b-1q-N, cq; q)x and hn(q) == hn(a, b, e, N; q) = (bq, aqle; q)N (abq; q)n (1 - abq2n+l) (aq, beq, q-N; q)n (qN ler . (abq2, lie; q)N (q; q)n(1- abq) (bq, aqle, abqN+2; q)n (7.2.20)

Setting e = 0 in (7.2.17) gives the q-Hahn polynomials _ . . _ [q-n, abqn+l, q-X . ] Qn(x) = Qn(x, a, b, N, q) - 3¢2 aq, q-N ,q, q , (7.2.21 ) which were introduced by Hahn [1949a] and, by (7.2.18), satisfy the orthogo­ nality relation

N L Qm(x)Qn(x) (aq; q)x(bq; q)N-x (aq)-X x=o (q; q)x(q; q)N-x (abq2; q)N(aq)-N -----'--(q_;q--'-,)_n('-:-1_-_a_b q--'::),.:-(b.,...."q,-:-a,-bq_N_+----:;c:-'q)---,n_(_2 )n (n)-Nn.r -:- N aq q 2 U m n (q; q)N (abq; q)n(1- abq2n+l )(aq, q- ; q)n ' (7.2.22) 7.3 The infinite discrete case 181 for m, n = 0,1, ... , N. Setting a = 0 in (7.2.17) we obtain the dual q-Hahn polynomials (Hahn [1949b]) Rn(P,(x)) == Rn(P,(x); b, e, N; q) _ [q-n, q-X, eqx-N . ] - 3¢2 q-N, beq , q, q , (7.2.23) where p,(x) = q-X + eqx-N, which satisfy the orthogonality relation N (eq-N. q) (1 eq2X-N) (beq q-N. q) x '" Rm(P,(x))Rn(P,(x)) 'x - " x qNX-(2)(_beq)-X f='o (q; q)x(1- eq-N)(b-1q-N, eq; q)x

= e; ~)N (b (q, b~;;)n) (eq-Nt bm,n, (7.2.24) (!tq;q N eq,q ;q n m, n = 0,1, ... , N. For some applications of q-Hahn, dual q-Hahn polynomials, and their limit cases, see Delsarte [1976a,b, 1978], Delsarte and Goethals [1975], Dunkl [1977-1980] and Stanton [1977-1986c].

7.3 The infinite discrete case: the little and big q-Jacobi polynomials

As a q-analogue of the Jacobi polynomials (7.1.11), Hahn [1949a] (also see Andrews and Askey [1977]) introduced the polynomials Pn(x;a,b;q) = 2¢1(q-n,abqn+\aq;q,xq). (7.3.1) It can be easily verified that

. (1- x. a (3. ) _ p~a,(3)(x) hm Pn --, q ,q ,q - ((3) . (7.3.2) q--->l 2 Pn a, (1) He proved that

~ (x. b.) (x. b.) (bq; q)x ( )X _ bm,n ~ Pm q ,a, ,q Pn q ,a, ,q ( .) aq - h ( b. )' (7.3.3) x=o q, q x n a, ,q where 0 < q,aq < 1 and . ) = (abq; q)n(1- abq2n+l )(aq; q)n(aq; q)= ( )-n h ( b (7.3.4) n a, ,q (q, . q ) n ( 1 - abq ) ( bq, . q ) n (abq 2.),q = aq. Observe that (7.3.3) and (7.3.4) also follow from (7.2.22) when we replace x by N - x and then let N ---+ 00. To prove (7.3.3), assume, as we may, that o :::; m :::; n, and observe that f (bq; q)x (aq)xqxkPm(qx; a, b; q) x=o (q; q)x m (q-m abqm+1. q) . =2: ' 'J qj l¢o(bq;-;q,aq1+k+j ) j=O (q,aq;q)j 182 Applications to orthogonal polynomials

(7.3.5) by the q-binomial and q-Saalschiitz formulas. Then the orthogonality relation (7.3.3) follows immediately by using (1.5.2) and (7.3.5). It is easy to verify that Pn(x; a, b; q) satisfies the three-term recurrence relation (7.3.6) for n 2: 0, where A _ (1- abqn+l) (1- aqn+l) _ n n - (1 _ abq2n+l) (1 _ abq2n+2) ( q ), (7.3.7) (l-qn)(l-bqn) n Cn = (1 _ abq2n) (1 _ abq2n+l) (-aq ), (7.3.8) so the condition that AnCn+l > 0 for n = 0,1, ... , is satisfied if 0 < q, aq < 1 and bq < 1. When b < 0 the polynomials Pn(x; a, b; q) give a q-analogue of the Laguerre polynomials L~(1) (x) since

liml Pn ((1 - q)x; qOl, _q(3; q) = L~OI) (x) / L~OI) (0). (7.3.9) q-+ Andrews and Askey [1985] introduced a second q-analogue of the Jacobi polynomials,

(7.3.10) which has the property that

• • 01 (3 ,. _ p~OI,(3)(X) hm Pn(x, q ,q ,-q ,q) - ((3) , (7.3.11) q-+l Pn 01, (1) where'Y is real. In view of the third free parameter in (7.3.10) they called the Pn(x; a, b, c; q) the big q-Jacobi and the Pn(x; a, b; q) the little q-Jacobi polyno­ mials. We shall now prove that the big q-Jacobi polynomials satisfy the orthog- onality relation aq ( ) ( ) (x/a, x/c; q)oo Pm x; a, b, c; q Pn x; a, b, c; q ( b / .) dqx lcq x,xc,qoo

(7.3.12) hn(a, b, c; q)' where

h ( b . ) = M-l(abq;q)n(1_abq2n+l)(aq,cq;q)n(_ 2)-n -G) n a, ,c, q ()q; q n ( 1 - abq ) ( bq, abq /)c; q n acq q (7.3.13) 7.3 The infinite discrete case 183 and M = jaq (x/a, x/c; q)oo dqx cq (x, bx/c; q)oo aq(1 - q)(q, cia, aq/c, abq2; q)oo (7.3.14) (aq,bq,cq, abq/c; q)oo by (2.10.20). Since

aq (x/a, X/C; q)oo j (bx/c;q)j(X;q)k ( b / .) dqx cq x, x c, q 00 _ja q (x/a, x/c; q)oo d - (k b qX cq xq, xql0/ C; q )00 _ M(bq,abq/c;q)j(aq,cq;q)k - (abq2; q)j+k ' the left side of (7.3.12) becomes

M (bq, abq/c; q)m (c/b)m (aq, cq; q)m

m n (q-m abqm+1. q) 0 (q-n abqn+1. q) '"' '"' ' '1 ' 'k j+k x ~ ~ ( . ) ( . ) ( 2.) q, (7.3.15) j=O k=O q, q j q, q k abq ,q j+k where we used (7.3.10) and the observation that, by (3.2.2) and (3.2.5),

Pm(x; a, b, c; q)

= (bq,abq/c;q)m( /b)m r/, [q-m,abqm+1,bX/C. ] C 3'1'2 b b / ' q, q . ( aq,cq;q ) m q,a q c (7.3.16)

Assume that 0 ::; m ::; n. Since, by (1.5.3)

the double sum in (7.3.15) equals

(q -n abqn+1. q) n-m (qm-n abqm+n+1. q) , 'm ( b m+2) m '"' ' , k k (abq2; qhm a q ~ (q, abq2m+2; qh q (q -n abqn+1. q) _ ' 'n ( b n+2)n J: - (b 2. ) a q Um,n' (7.3.17) a q ,q 2n

Substituting this into (7.3.15), we obtain (7.3.12). 184 Applications to orthogonal polynomials

7.4 An absolutely continuous measure: the continuous q-ultraspherical polynomials

In this and the following section we shall give two important examples of orthogonal polynomials which are orthogonal with respect to an absolutely continuous measure da(x) = w(x)dx. In his work of the 1890's, in which he discovered the now-famous Rogers­ Ramanujan identities, Rogers [1893b, 1894, 1895] introduced a set of orthogo­ nal polynomials that are representable in terms of basic hypergeometric series and have the ultraspherical polynomials (7.1.12) as limits when q ---+ 1. Fol­ lowing Askey and Ismail [1983], we shall call these polynomials the continuous q-ultraspherical polynomials and define them by the generating function

((3te iO , (3te-iO ; q) 00 00 n (7.4.1) (·0te' te-'·"0) q = "Cn(x;~ (3lq)t , , '00 n=O where x = cos 0, 0 ::; 0 ::; 7f and max(lql, Itl) < 1. Using the q-binomial theorem, it follows from (7.4.1) that Cn(x; (3lq) = t ((3; qh((3; q)n-k ei(n-2k)O k=O (q; qh(q; q)n-k

- ((3; q)n inO rI-. ( -n (3. (3-1 1-n. (3-1 -2iO) (742) - ( ) e 2'1-'1 q " q ,q, q e . . . q;q n Note that

1· C ( . AI ) - ~ (A)k(A)n-k i(n-2k)O n q--->l1m x, q q - ~ k'(• n -k)' . e k=O = C~(x). (7.4.3) Before considering the orthogonality relation for Cn(x; (3lq), we shall first derive some important formulas for these polynomials. For 0 < 0 < 7f, 1(31 < 1, set a = q(3-1 e2iO, b = q(3-1, C = qe2iO and z = (32 qn in (3.3.5) to obtain 2cP1 (q-n, (3; (3-1 q1-n; q, q(3-1 e-2iO) ((3qn, (3e- 2iO ; q) " " = 00 rI-. (q(3-1 q(3-1 e2,O. qe2,o. q (32qn) ( qn+1 ,e -2iO.),qoo 2'1-'1 , " ,

((3, q(3-1, (3qne2iO , (3-1 q1-ne-2iO; q) 00 + (qn+1, e2iO , q(3-1 e-2iO, (3-1 q1-n; q)oo X 2cP1 (q(3-l,q(3-1e-2iO;qe-2iO;q,(32qn). (7.4.4) Then, application of the transformation formula (1.4.3) to the two 2cP1 series on the right side of (7.4.4) gives 7.4 An absolutely continuous measure 185

(7.4.5) where x = cos B. (7.4.6)

Rewriting the right side of (7.4.5) as a q-integral we obtain the formula

Cn (x· ;3lq) = 2i sin B (;3,;3; q)oo (;32; q) n , (l-q)W{3(xlq) (q,;32;q)00 (q;q)n

e - iO (queiO,que-iO;q)oo n l (7.4.7) X e iO (;3ue iO , ;3ue- iO ; q)oo u dqu, which was found by Rahman and Verma [1986a]. Now use (1.4.1) to obtain from (7.4.5) that

Cn(x;;3lq) = (;3,;3;;q) ooWi 1(xlq) (;32;q)n (q,;3 ; q)oo (;3q; q)n X {(I - e2iO ) einO 2¢1 (q;3-1, qn+\ ;3qn+\ q, ;3e2iO )

+ (1 - e-2iO ) e-inO 2¢1 (q;3-1, qn+\ ;3qn+l; q, ;3e- 2iO )}

00 = 4sinB Wil(xlq) L b(k, n;;3) sin(n + 2k + l)B, (7.4.8) k=O where 0 < B < 7[, 1;31 < 1 and

b(k,n;;3) = (;3,;3;;q)oo (;32;q)n(q;3-\q)k(q;q)n+k ;3k. (7.4.9) (q,;3 ; q)oo (q; q)n(q; q)d;3q; q)n+k The series on the right side of (7.4.8) is absolutely convergent if 1;31 < 1. For Ixl < 1, Iql < 1 and large n it is clear from (7.4.5) that the leading term in the asymptotic expansion of Cn(cosB; ;3lq) is given by 2iO 2i C ( B·;3I) rv (;3;q)oo {(;3e ;q)00 -inO + (;3e- O;q)00 ino} n cos, q (q;q)oo (e 2iO ;q)00 e (e- 2iO ;q)00 e

=2~;3;q?00 IA(eiO)lcos(nB-a), (7.4.10) q;q 00 where (7.4.11) and, as elsewhere, f(n) rv g(n) means that limn--+oo f(n)jg(n) = 1. For fur­ ther results on the asymptotics of Cn(x; ;3lq), see Askey and Ismail [1980] and Rahman and Verma [1986a]. 186 Applications to orthogonal polynomials

If we use (3.5.4) to express 2¢1 (q-n, (3; (3-1 q1-n; q, q(3-1 e-2iO) as a termi­ nating VWP-balanced S¢7 series in base q~ and then apply (2.5.1), we obtain

( ~-n -2iO. ) (_ (1-n)/2(3-1. ~) ,/-. ( -n (3. (3-1 1-n. (3-1 e-2iO) = q e ,q n q ,q n 2'/"1 q " q ,q,q ((3 - 1 q21-n ; q21) n -n/2 (31 -iO (31 -iO -n/2 ] [ q ,2 e ,- 2 e , -q 1 1 X 4¢3 (3 1_ /2 -iO 1_ /2 -iO ; q2, q2 - ,q4 n e ,-q4 n e ((32. q) [ -n/2 (3 n/2 (31 iO (31 -iO ] _ 'n (3-n/2 -inO,/-. q ,q , 2 e , 2 e . ~ ~ - ((3 ) e 4 '/"3 1 1 1 1 ,q, q (7.4.12) ; q n -(3, (32 q4, -(32 q4 by (2.10.4). However, by (3.10.13),

(7.4.13) and hence, from (7.4.2), (7.4.12) and (7.4.13), we have

Cn(cosB; (3lq) _ ((32;q)n -n/2 [q-n,(32qn,(3~eiO,(3~e-iO. ] (7.4.14) - ( q,q . ) n (3 4¢3 (3 q2,-1 (3 ,-(3 q21 , q, q .

Since W,a(cosBlq) = IA(eiO )I-2 for real (3, it follows from Theorem 40 in Nevai [1979] and the asymptotic formula (7.4.10) that the polynomials Cn( cos B; (3lq) are orthogonal on [O,n] with respect to the measure W,a( cos Blq)dB, -1 < (3 < 1. One can also guess the weight function by setting (3 = q>' and comparing the generating function (7.4.1) and the expansion (7.4.8) with the q ---+ 1 limit cases and the weight function (1 - e2iO )>. (1 - e-2iO )>' for the ultra­ spherical polynomials C~ (cos B). We shall now give a direct proof of the orthogonality relation

1" Cm( cos B; (3lq)Cn( cos B; (3lq)W,a( cos Blq) dB = h:(~lq)' (7.4.15) where Iql < 1,1(31 < 1 and

hn((3lq) = (q, (32; ~)oo (q; ;!n(1- (3qn). (7.4.16) 2n((3, (3q, q)oo ((3 ,q)n(1- (3) As we shall see in the next section, (7.4.15) can be proved by using (7.4.14) and the Askey-Wilson q-beta integral (6.1.1); but here we shall give a direct proof by using (7.4.2), (1.9.10) and (1.9.11), as in Gasper [1981b]' to evaluate the integral. Since the integrand in (7.4.15) is even in B, it suffices to prove that

(7.4.17) 7.4 An absolutely continuous measure 187 when 0 ::; m ::; n. We first show that, for any integer k,

7r ike ( I) { 0, if k is odd, j e W/3 cos () q d() = ((3I)·f k . (7.4.18) -7r Ck/2 q, 1 IS even, where (7.4.19)

By the q-binomial theorem, I: eikeW/3( cos ()Iq) d() = f f ((3-\ q)r((3-\ q)s (3r+s r ei(k+2r-2s)e d(), r=Os=O (q;q)r(q;q)s J-7r which equals zero when k is odd and equals 2n f ((3-1: q)s((3.-\ q)s+j (31+2s s=o (q, q)s(q, q)s+j

= 2n(~-.\~)j (3j 2¢1 ((3-1, (3-1 qj ; q1+\ q,(32) q, q J when k = 2j, j = 0,1, .... By (1.4.5), the above 2¢1 series equals

((3, (3q1+\ q)oo rI-. ( -1 (3-1. (3. (3 j+1) 2 j+1.) 2\f'1 q, "q, q ((3 ,q ,q 00

= ((3,(3,;!q)oo (q;.q)j (1+qj). (q,(3 ,q)oo ((3q,q)j

From this and the fact that W /3 ( cos () 1 q) is symmetric in (), so that we can handle negative k's, we get (7.4.18). Hence, from (7.4.2),

(7.4.20) equals zero when n - k is odd and equals

(7.4.21 ) when n - k = 2j is even. From (1.9.11) it follows that this 4¢3 series and hence the integral (7.4.20) are equal to zero when n > Ikl. Hence (7.4.17) holds when m -I- n. If k = ±n -I- 0, then (1.9.10) gives I: eikOCn(cos();(3lq)W/3(cos()lq) d() 2n((3, (3q; q)00((32; q)n (7.4.22) (q, (32; q)oo((3q; q)n 188 Applications to orthogonal polynomials from which it follows that (7.4.17) also holds when m = n.

7.5 The Askey-Wilson polynomials

In view of the 4¢3 series representation (7.4.14) for the continuous q-ultraspherical polynomials it is natural to consider the more general polyno­ mials (7.5.1) which are of degree n in x = cosO, and to try to determine the values of cv, /3, ,,(, 8, E for which these polynomials are orthogonal. Because terminating balanced 4¢3 series can be transformed to other balanced 4¢3 series and to VWP-balanced 8¢7 series which satisfy three-term transformation formulas (see, e.g., (7.2.13), (2.11.1), Exercise 2.15 and the three-term recurrence re­ lation for the q-Racah polynomials), one is led to consider balanced 4¢3 se­ ries. From Sears' transformation formula (2.10.4) it follows that if we set cv = abcdq-l, /3 = a, "( = ab, 8 = ac and E = ad, then the polynomials

q-n abcdqn-l aeiIJ ae-iIJ ] = (ab, ac, ad; q)na-n 4¢3 [ ' b 'd' ; q, q a ,ac,a (7.5.2) are symmetric in a, b, c, d. In addition, for real 0 these polynomials are analytic functions of a, b, c, d and are, in view of the coefficient (ab, ac, ad; q)na-n, real­ valued when a, b, c, d are real or, if complex, occur in conjugate pairs. Askey and Wilson [1985] introduced these polynomials as q-analogues of the 4F3 polynomials of Wilson [1978, 1980]. Since they derived the orthogo­ nality relation, three-term recurrence relation, difference equation and other properties of Pn(x; a, b, c, dlq), these polynomials are now called the Askey­ Wilson polynomials. Since the three-term recurrence relation (7.2.1) for the q-Racah polynomi­ als continues to hold without the restriction bdq = q-N, by translating it into the notation for Pn(x; a, b, c, dlq) we find, as in Askey and Wilson [1985], that the recurrence relation for these polynomials can be written in the form

2xPn(x) = AnPn+l(X) + BnPn(x) + CnPn-l(X), n 2 0, (7.5.3) with P-l(X) = O,Po(x) = 1, where 1 - abcdqn-l An = ~------~~~------~~ (7.5.4) (1 - abcdq2n-l) (1 - abcdq2n) ' (1 - qn)(1 - abqn-l )(1 - acqn-l )(1 - adqn-l) (1 - abcdq2n-2) (1 - abcdq2n-l) X (1- bcqn-l)(I- bdqn-l)(I- cdqn-l), (7.5.5) 7.4 An absolutely continuous measure 189 and

Bn = a + a-I - A na-l (l- abqn)(l - aeqn)(l - adqn) - Gna/ (1 - abqn-I) (1 - aeqn-I) (1 - adqn-I). (7.5.6)

It is clear that An, B n , Gn are real if a, b, e, d are real or, if complex, occur in conjugate pairs. Also AnGn+1 > 0, n = 0,1, ... , if the pairwise products of a, b, e, d are less than 1 in absolute value. So by Favard's theorem, there exists a measure da(x) with respect to which Pn(x; a, b, e, dlq) are orthogonal. In order to determine this measure let us assume that max(lal, Ibl, lei, Idl, Iql) < 1. Then, by (2.5.1),

Pn( cos B; a, b, e, dlq) (ab, ae, be, de-iO , ; q) "0 = netn (abeeiO ; q)n

X S W7 (abeeiO q-\ aeiO , beiO , eeiO , abedqn-I, q-n; q, qd-Ie-iO ) , (7.5.7) and, by (2.11.1), W (abeeiOq-l. aeiO beiO eeiO abedqn-I q-n. qd-Ie-iO ) S 7 "" " iO iO iO iO (abee , bedeiOqn , beiOqn+1 , eeiOqn+1 , ae- , be- , ee- , de-iOqn.'00 q) (ab ae be qn+1 bdqn edqn beqn+Ie2iO e-2iO . q) , " , " " CXJ X S W7 (beqne2iO ; beqn, beiO , eeiO , qa-IeiO , qd-IeiO ; q, adqn)

(abeeiO, abee-iO qn, bede-iO qn, be-iO qn+l, ee-iO qn+l; q) 00 +~--~~~~--~--~~~~--~~~~~--~ (qn+l, bdqn, edqn, abeeiOqn, beqn+Ie-2iO ; q)oo aeiO beiO eeiO deiO . q) " ( , , , '00 -2tnO X ( 2"0) e ab, ae, ad, e t ; q 00

X S W 7 (beqne-2iO; beqn, be-iO , ee-iO , qa-Ie-iO , qd-Ie-iO ; q, adqn) , (7.5.8) where °< B < 7L Hence Pn(cosB; a, b, e, dlq) = (be, bd, cd; q)n { Qn (eiO ; a, b, e, dlq) + Qn (e-iO ; a, b, e, dlq) }, (7.5.9) where

Qn(z; a, b, e, dlq) (abezqn,bedzqn,bzqn+l,ezqn+l,a/z,b/z,e/z,d/z;q)oo n = z (be bd cd abqn aeqn qn+1 bez2qn+1 z-2. q) , " , " " 00 X SW7 (beqnz 2; beqn, bz,ez, qa-Iz,qd-Iz; q,adqn) . (7.5.10)

It is clear from (7.5.10) that

Qn(z; a, b, e, dlq) '" zn B(Z-I )/(be, bd, cd; q)oo, (7.5.11) where (7.5.12) 190 Applications to orthogonal polynomials as n ----; 00, uniformly for z, a, b, c, d in compact sets avoiding the poles z2 = q-k, k = 0,1, .... Using (7.5.9) we find that Pn (cos e; a, b, e, dlq) '" ein () B(e-i()) + e-in () B(ei()) = 2IB(ei ())1 cos(ne - (3), (7.5.13) where (3 = arg B(ei ()) and 0 < e < 7f (see Rahman [1986c]). Then

IB(ei ())1 2 = [sine w(cose;a,b,e,dlq)r1 , (7.5.14) where, in order to be consistent with the Pn(x; a, b, e, dlq) notation, we have used w(x; a, b, e, dlq) to denote the weight function w(x; a, b, e, d) defined in (6.3.1). It follows from Theorem 40 in Nevai [1979] that the polynomials Pn(x;a,b,e,dlq) are orthogonal on [-1,1] with respect to the measure w(x; a, b, e, dlq)dx when max(lal, Ibl, lei, Idl, Iql) < 1. We shall now give a direct proof of the orthogonality relation

(7.5.15)

where w(x) == w(x; a, b, e, dlq) and hn == hn(a, b, e, dlq) -1 (abedq-1; q) n (1 - abedq2n-1) =/i (a,b,e,dlq)(.)q, q n (-1)(1 - abedq ab, ae, ad, be, bd, cd,. q )n , (7.5.16) with

/i(a, b, e, dlq) = [11 w(x; a, b, e, dlq) dx

27f( abed; q)oo (7.5.17) (q, ab, ae, ad, be, bd, cd; q)oo' by (6.1.1). First observe that, by (7.5.17), 11 (aei(),ae-i();q)j (bei(),be-i();q)k w(x;a,b,e,dlq) dx -1 = 11 w(x; aqj, bqk, e, dlq) dx -1 . k = /i(aqJ, bq ,e, dlq). (7.5.18)

By using (7.5.2) and the fact that Pn(x; a, b, e, dlq) = Pn(x; b, a, e, dlq) we find that the left side of (7.5.15) equals /i(a, b, e, dlq)(ab, ae, ad; q)m(ba, be, bd; q)na-mb-n

~ (q-m,abedqm-\q)j j [q-n,abedqn-1,abqj. ] x ~ (b d.) q 3¢2 abedqj ab ,q, q . (7.5.19) j=O q, a e ,q j , 7.4 An absolutely continuous measure 191

Assuming that 0 ::::: n ::::: m and using the q-Saalschiitz formula to sum the 3¢2 series, the sum over j in (7.5.19) gives (cd abcdqm-1 q-m. q) , " n n A. (n-m b d n+m-1 b d 2n ) 1-nj b. ) ( b d.) q 2'/-'1 q ,a c q ; a c q ; q, q ( q a ,q n a c ,q 2n (cd, abcdqm-1, q-m; q)n (q1+n-m; q)m-n (q1-n jab; q)n (abed; q)2n (abcdq2n; q)m-n (q,cd;q)n (1- abcdq-1) ()n 1 2 1 m (7.5.20) (abcdq- ,ab; q)n (1 - abcdq n- ) ab 15 ,n· Combining (7.5.19) and (7.5.20) completes the proof of (7.5.15). Askey and Wilson proved a more general orthogonality relation by using contour integration. They showed that if Iql < 1 and the pairwise products and quotients of a, b, c, d are not of the form q-k, k = 0,1, ... , then

[11 Pm (X)Pn(X)W(X) dx + 27r ~Pm(Xk)Pn(Xk)Wk

(7.5.21 ) hn(a, b, c, dlq)' where Xk are the points ~ (Jqk + j-1q-k) with j equal to any of the parame­ ters a, b, c, d whose absolute value is greater than 1, the sum is over the k with Ijqkl > 1, and Wk == wk(a, b, c, dlq) (a-2; q)oo (q, ab, ac, ad, bja, cja, dja; q)oo (a2;qh(l-a2q2k)(ab,ac,ad;q)k ( q )k (7.5.22) X (q; q)k(1 - a2)(aqjb, aqjc, aqjd; qh abed when Xk = ~ (aqk + a-1q-k). For a proof and complete discussion, see Askey and Wilson [1985]. Also, see Ex. 7.3l. In order to get a q-analogue of Jacobi polynomials, Askey and Wilson set a = q(2a+1)/4, b = q(2a+3)/4, C = _q(2,6+1)/4, d = _q(2f3+3 )/4 (7.5.23) and defined the continuous q-Jacobi polynomials by

p(a,,6)(xlq) = (qa+1; q)n n (q;q)n q-n, qn+a+,6+1, q(2a+1)/4eiO, q(2a+1)/4e-iO . ] [ X 4¢3 qa+1, _q(a+,6+1)/2, _q(a+,6+2)/2 ,q, q. (7.5.24)

On the other hand, Rahman [1981] found it convenient to work with an ap­ parently different q-analogue, namely, ( a+1 ,6+1.) p~a,,6)(x; q) = q ,-q ,q n (q, -q; q)n

(7.5.25) 192 Applications to orthogonal polynomials

However, as Askey and Wilson pointed out, these two q-analogues are not really different since, by the quadratic transformation (3.10.13),

p(a,(3)(Xlq2) = (-q; q)n qan p(a,(3) (X' q) n (_qa+(3+I; q)n n ,. (7.5.26)

Note that limI p~a,(3)(xlq) = limI p~a,(3)(x; q) = p~a,(3)(x). (7.5.27) q--+ q--+

The orthogonality relations for these q-analogues are

(7.5.28) and

(7.5.29) where 0 < q < 1, a 2 -~, jJ 2 -~,

2 (7.5.30)

(q, qa+l, q(3+I, _q(a+(3+I)/2, _q(a+(3+2)/2; q) 00 an (a, jJlq) = 27rq ((a+(3+2)/2 ,q(a+(3+3)/2. ,qoo ) (1 - q2n+a+(3+I) (q qa+(3+I _q(a+(3+I)/2. q) x '" n -n(2a+I)/2 (1 - qa+(3+I) (qa+l, q(3+I, _q(a+(3+3)/2; q) n q (7.5.31 ) and

From (7.4.14), (7.5.24) and (7.5.25) it is obvious that

Cn( cos 0; qAlq) (q2\q)n -nA/2p,(A-!,A-!)( 01) 1) q n cos q (7.5.33) (qA+2; q n 7.4 An absolutely continuous measure 193

It can also be shown that

C ( . AI ) - (q\ -q; q)n -n/2 ~(A-~'-~)(2 2 - l' ) 2n x, q q - (1 1.) q n x, q , (7.5.35) q2,-q2,q n

C (x' qAlq) = x (q\ -1; q)n+l q-n/2 ~(A-~'~)(2x2 - l' q) 2n+ 1 , n , , (1q2,-q2;q 1) n+l (7.5.36) which are q-analogues of the quadratic transformations

C A (x) = (A)n p,(A-~'-~)(2x2 - 1) (7.5.37) 2n (~)n n and CA () = (A)n+l n(A-~'~)(2 2 _ 1) 2n+ 1 X X ( 1 ) r n X , (7.5.38) '2 n+l respectively. To prove (7.5.35), observe that from (7.4.2) C2n ( cos e; qAlq)

_ (q\ q)2n 2inO A, (-2n A. l-A-2n. I-A -2iO) - ( ) e 2'/-'1 q ,q, q , q, q e , q; q 2n and hence, by the Sears-Carlitz formula (Ex. 2.26), C2n(cose; qAlq)

= (q\ q)2n ( q21 e -2iO ,q21-n e -2iO ; q) e 2inO (q;q)2n n

x A, [ q-n, _q-n, _q~-n, q~-A-n. ] (7.5.39) 4 '/-'3 ql-A-2n, q~ -ne2iO, q~ -ne-2iO , q, q .

Reversing the 4¢>3 series, we obtain C2n (cos e; qAlq)

(qA qA+~. q) 1 . 1 . _ ' 'n -n/2 [q-n,qn+A,q2e2.0,q2e-2.0. ] - ( 1 ) q 4¢>3 A+~ _ ~ _ ' q, q . q, q2; q q ,q, q n (7.5.40) This, together with (7.5.25), yields (7.5.35). The proof of (7.5.36) is left as an exercise. Following Askey and Wilson [1985] we shall obtain what are now called the continuous q-Hahn polynomials. First note that the orthogonality relation (7.5.15) can be written in the form 1: Pm(cose; a, b, c, dlq)Pn(cose; a, b, c, dlq)w(cose; a, b, c, dlq) sine de (7.5.41) hn(a, b, c, dlq)' 194 Applications to orthogonal polynomials

Replace B by B + ¢, a by aei¢, b by ae-i¢ and then set c = bei¢, d = be-i¢ to find by periodicity that if -1 < a, b < 1 or if b = a and lal < 1, then

[: Pm(cos(B + ¢); a, blq)Pn(cos(B + ¢); a, blq)W(B) dB

(7.5.42) where

(7.5.43)

(7.5.44) and

2i 2i x (1 _ a2b2q-l) (q , a2 , b2 , ab " ab abe ¢ , abe- ¢.'n q) (7.5.45) The recurrence relation for these polynomials is 2 cos(B + ¢ )Pn(cos(B + ¢); a, blq) = AnPn+l (cos(B + ¢); a, blq) + BnPn(cos(B + ¢); a, blq) + CnPn-l(cos(B + ¢); a, blq) (7.5.46) for n = 0,1, ... , where P-l (x; a, blq) = 0, 1 - a2 b2 qn-l A - -,---..".-:-;:,...... "..----:-..,...-7-----,,-:-:::-;:--:- (7.5.4 7) n - (1 _ a2b2q2n-l) (1 _ a2b2q2n) '

and

Bn = aei¢ + a-1e-i¢ - Ana-1e-i¢(1 - a2qn)(1 - abqn)(l - abe2i¢qn) - Cnaei¢ /(1- a2qn-l)(1_ abe2i¢qn-l)(1_ abqn-l). (7.5.49)

If we set a = qO<, b = qf3, B = x log q in (7.5.42) and then let q ----> 1 we obtain

[: Pm(x; 0:, (3)Pn(x; 0:, (3)lr(o: + ix)r((3 + ix)1 2dx = 0, m -=I- n, (7.5.50) 7.6 Connection coefficients 195 where

. (3) _·n D + + 2(3 - 1, ix Pn (x, a, - 2 3£2 [-n, n 2a (3 2 a- ,..1] (7.5.51) a+ , a For further details and an extension to one more parameter, see Askey [1985b], Askey and Wilson [1982, 1985], Atakishiyev and Suslov [1985], and Suslov [1982, 1987].

7.6 Connection coefficients

Suppose fn(x) and gn(x), n = 0,1, ... , are polynomials of exact degree n in x. Sometimes it is of interest to express one of these sequences as a linear combination of the polynomials in the other sequence, say, n gn(X) = L ck,nfk(X). (7.6.1) k=O The numbers Ck,n are called the connection coefficients. If the polynomials fn(x) happen to be orthogonal on an interval I with respect to a measure da(x), then Ck,n is the k-th Fourier coefficient of gn(x) with respect to the orthogonal polynomials fk(X) and hence can be expressed as a multiple of the integral II fk(X)gn(x)da(x). A particularly interesting problem is to determine the conditions under which the connection coefficients are nonnegative for particular systems of orthogonal polynomials. Formula (7.6.1) is sometimes called a projection for­ mula when all of the coefficients are nonnegative. See the applications to posi­ tive definite functions, isometric embeddings of metric spaces, and inequalities in Askey [1970, 1975], Askey and Gasper [1971]' Gangolli [1967] and Gasper [1975a]. As an illustration we shall consider the coefficients Ck,n in the relation

n Pn(x; a, (3, 1', dlq) = L Ck,nPk(X; a, b, c, dlq)· (7.6.2) k=O Askey and Wilson [1985] showed that (ad,(3d,1'd,q;q)n (a(31'dqn-\q)k k2-nkdk-n Ck,n = k- q (ad, (3d, 'I'd, q, abcdq 1; qh (q; q)n-k x [qk-n, a(31'dqn+k-1, adqk, bdqk, cdqk . ] S¢4 abcdq2k, adqk, (3dqk, 1'dqk ,q, q . (7.6.3)

To prove (7.6.3), temporarily assume that max (Ial, Ibl, Icl, Idl, Iql) < 1, and observe that, by orthogonality,

bk,j = 11 w(x;a,b,c,dlq)Pk(x;a,b,c,dlq)(deiO,de-iO;q)j dx (7.6.4) -1 vanishes if j < k, and that bk,j = /i(a, b, c, dlq)(ab, ac, ad; q)ka-k 196 Applications to orthogonal polynomials

x (ad,bd,cd;q)j A-, [q-k,abedqk-1,adqj. ] (abcd;q)j 3'1'2 abedqj,ad ,q,q = A:(a, b, c, dlq)(ab, ae, bc, q-j; q)k(ad, bd, cd; q)j(dqj)k /(abed; q)j+k (7.6.5) if j 2 k. Since Pn(x; a, (3, 'Y, dlq) _ . -n n (q-n,a(3'Ydqn-l,deiIJ,de-iIJ;q)j j

- (ad, (3d, 'Yd, q)nd L (d (3d d' ) 0 q , (7.6.6) j=O q, a, , 'Y , q J we find that

Ck,n = hk(a, b, e, dlq) 111 W(X; a, b, e, dlq)Pk(x; a, b, e, dlq)Pn(x; a, (3, 'Y, dlq) dx

_ . -n n (q-n,a(3'Ydqn-\q)j j 0

- hk(a, b, e, dlq)(ad, (3d, 'Yd, q)nd L ( d (3d d' ) 0 q bk,J j=O q, a, , 'Y , q J (7.6.7) and hence (7.6.3) follows from (7.5.16), (7.5.17) and (7.6.5). The 5rP4 series in (7.6.3) is balanced but, in general, cannot be transformed in a simple way, so one cannot hope to say much about the nonnegativity of ek,n unless the parameters are related in some way. One of the simplest cases is when the 5rP4 series reduces to a 3rP2 , which can be summed by the q-Saalschiitz formula. Thus for (3 = band 'Y = e we get n Pn(X; a, b, c, dlq) = L ek,npdx; a, b, e, dlq) (7.6.8) k=O with (a/a; q)n-k (abedqn-\ qh(q, be, bd, cd; q)nan-k (7.6.9) ek,n = (q, be, bd, cd; q)k(abcdqk-1; q)k(q, abedq2k; q)n-k' It is clear that Ck,n > 0 when 0 < a < a < 1, 0 < q < 1 and max (be, bd, cd, abed) < 1. Another simple case is when the 5rP4 series in (7.6.3) reduces to a summable 4rP3 series. For example, this happens when we set d = q!, c = -q! = 'Y, b = -a and (3 = -a. The 5rP4 series then reduces to

(7.6.10) which equals (a2qn+k+1 qk-n+1 a2qk+n+2 a2qk-n+2/a2.q2) ()/ ' , , , 00 ( 2 n+k) n-k 2 ( ) (q,a2q2k+1,a2q2k+2,a2q2/a2;q2)00 a q 7.6.11 by Andrews' formula (Ex. 2.8). Clearly, this vanishes when n - k is an odd integer and equals

2 2 2 2 2 (qa , q a ; q2)n_2Jo(q, 00 /a ; q2)Jo -----;-----;c----';;:~'____;;co__------"- (a2q2n+ 1-4j )j (7.6.12) (qa2, q2a2; q2)n_j 7.7 A difference equation and a Rodrigues-type formula 197 when n - k = 2j,j = 0, 1, ... , [~]. Since by (3.10.12) and (7.4.14)

. __ ~ ~I) _ (q2,qa2;q2)n (. 21 2) Pn (X, a, a, q ,q q - (2.) On X, a q , (7.6.13) a ,q n we obtain, after some simplification, Rogers' formula

o ( . 1 ) = [~l (3k("( j (3; qh(,,(; q)n-k(l - (3qn-2k) 0 (. (31 ) n X, "( q ~ ( . ) ((3 .) (1 _ (3) n-2k X, q (7.6.14) k=O q, q k q, q n-k after replacing 0;2, a2, q2 by ,,(, (3 and q, respectively. It is left as an exercise (Ex. 7.15) to show that formulas (7.6.2) and (7.6.14) are special cases of the q-analogue ofthe Fields-Wimp formula given in (3.7.9). For other applications of the connection coefficient formula (7.6.2), see Askey and Wilson [1985].

7.7 A difference equation and a Rodrigues-type formula for the Askey-Wilson polynomials

The polynomials Pn(x; a, b, c, dlq), unlike the Jacobi polynomials, do not satisfy a differential equation; but, as Askey and Wilson [1985] showed, they satisfy a second-order difference equation. Define

Eif(eiO ) = f (q±~eiO), (7.7.1)

6qf (e iO ) = (Et - E;;) f (e iO ) (7.7.2)

and Dqf(x) = 6q!(X) , X = cose. (7.7.3) uqx Clearly,

6q( cos e) = ~ (q~ - q-~) (e iO - e-iO ) = -iq- ~ (1 - q) sin e, (7.7.4) and 6q (aeiO , ae-iO ; q) n = 2aiq- ~ (1 - qn) sin e (aq~ eiO , aq~ e-iO ; q)n-l, (7.7.5) so that

"0 "0) 1"0 1 "0 ( 2a(1-qn) D q ae' , ae-'·'n q = - (1 _ q) (aq2 e' , aq2 e-' ,n-l,. q) (7.7.6) which implies that the divided difference operator Dq plays the same role for (aeiO,ae-iO;q)n as djdx does for xn. When q ---+ 1, formula (7.7.6) becomes

d~ (1- 2ax + a2)n = -2an(1- 2ax + a2t-1 .

Generally, for a differentiable function f (x) we have

. d hm Dqf(x) = -d f(x). q--+l X 198 Applications to orthogonal polynomials

Following Askey and Wilson [1985], we shall now use the operator Dq and the recurrence relation (7.5.3) to derive a Rodrigues-type formula for Pn(x; a, b, c, dlq). First note that by (7.5.2) and (7.7.5)

8qPn(x; a, b, c, dlq) 1 1 1 1 1 1 = -2iq2-n sine (1 - qn)(l - qn- abcd)Pn-l(X; aq2, bq2, cq2, dq2Iq). (7.7.7)

If we define

(7.7.8) and r (e ) = 4'" [q-n,abCdqn-l,aeiO,ae-iO. ] iO (7.7.9) n 'f'3 ab , ac , ad ' q, q , then the recurrence relation (7.5.3) can be written as q-n(1- qn)(l_ abcdqn-l)rn(eiO ) = A( -e) [rn(q-1eiO ) - rn(eiO )] + A(e) [rn(qeiO ) - rn(eiO )]. (7.7.10)

Also, setting

(7.7.11)

(7.7.12) we find that (7.7.10) can be expressed in the form q-n(1- qn)(l_ abcdqn-l)V(eiO)V(e-iO)Pn(x)

= 8q [{E:V(eiO )} {E';-V(e-iO )} {8qPn(xH]· (7.7.13) Combining (7.7.7) and (7.7.13) we have

- q-n/2V(eiO)V(e-iO)Pn(X; a, b, c, dlq)

= 8q [{ E:V(eiO )} {E.;-V(e-iO )} (e iO _ e-iO ) x Pn-l (X; aq! , bq! , cq! , dq! Iq)] . (7.7.14)

Since (e iO _ e-iO ) E:V(eiO)E';-V(e-iO )

(e iO - c iO ) (qe 2iO , qe-2iO ; q) 00 h( cos 61; aq! )h( cos 61; bq! )h( cos 61; cq! )h( cos 61; dq!) W (eiO;aq!,bq!,cq!,dq!lq)

eiO - e-iO Exercises 199

(7.7.14) can be written in a slightly better form

q-n/2W(eiO ; a, b, c, dlq)Pn(x; a, b, c, dlq)

= liq [(e iO - e-iO)-l W(eiO ; aq~, bq~, cq~ ,dq~ Iq)

1 1 1 1 ] X Pn-l (x; aq2 ,bq2 ,cq2, dq2lq) . (7.7.15) Observing that, by (6.3.1),

Vl- x2w(x; a, b, c, dlq) = W(eiO ; a, b, c, dlq) we find by iterating (7.7.15) that

w(x; a, b, c, dlq)Pn(x; a, b, c, dlq)

= (_I)k (1 ; q) k qnk/2-k(k+l)/4

x Dqk[ w(x; aq2,k bq2, k cq2, k dq2Iq)Pn_k(X; k aq'i,k bq'i, k cq'i, k dq'ilq) k ]

= (-It (1 ; q) n q(n 2 -n)/4 D~ [w(x; aqn/2, bqn/2, cqn/2, dqn/2Iq)] . (7.7.16) This gives a Rodrigues-type formula for the Askey-Wilson polynomials. By combining (7.7.7) and (7.7.15) it can be easily seen that the polynomials Pn(x) = Pn(x; a, b, c, dlq) satisfy the second-order difference equation

1 1 1 1 ] Dq [ w(x; aq2, bq2, cq2, dq 2 Iq)DqPn(x) + AnW(X; a, b, c, dlq)Pn(x) = 0, (7.7.17) where (7.7.18)

Exercises

7.1 If {Pn(x)} is an orthogonal system of polynomials on (-00, (0) with respect to a positive measure da(x) that has infinitely many points of support, prove that they satisfy a three-term recurrence relation of the form

XPn(X) = AnPn+l(X) + BnPn(x) + CnPn-l(X), n;::: 0,

with p-l(X) = 0, po(x) = 1, where An, Bn, Cn are real and AnCn+1 > °for n ;::: 0. 7.2 Let PO(X),Pl(X), ... ,PN(X) be a system of polynomials that satisfies a three-term recurrence relation 200 Applications to orthogonal polynomials

n = 0,1, ... , N, where P-l (x) = O,Po(x) = 1. Prove the Christoffel­ Darboux formula n (x - Y) LPj(x)pj(Y)Vj = Anvn [Pn+1 (x)Pn (y) - Pn(X)Pn+l(Y)] ' j=O °:S n :S N, where Vo = 1 and vnCn = vn-1An- 1, 1 :S n :S N. Deduce that n LP](x)Vj = Anvn [p~+1(X)Pn(X) - P~(X)Pn+l(X)l j=O and hence N LP](Xk)Vj = ANVNPN(Xk)p~+l(Xk) j=O

if x k is a zero of P N + 1 (x) . 7.3 Show that when n = N, the recurrence relation (7.2.1) reduces to (1 - q-j) (1 - cdqHl) PN(Xj) = CN [PN(Xj) - PN-l (Xj)] , where Pn(Xj) is given by (7.2.11), Xj by (7.2.9), and An and Cn by (7.2.5) and (7.2.6). Hence show that (7.2.1) holds with x = Xj,j = O,l, ... ,N. (Askey and Wilson [1979]) 7.4 If Pn(Xj), Vn and Wj are defined by (7.2.11), (7.2.3) and (7.2.15), respec­ tively, prove directly (i.e. without the use of Favard's theorem) that N N LPm(Xj)Pn(Xj)Wj = V;;:-l L Wj Dm,n j=O j=O and N N LPn(Xj)Pn(Xk)Vn = wjl L Wn Dj,k. n=O n=O [Hint: First transform one of the polynomials, say Pn(x), to be a multiple of the 4¢3 series on the left side of (7.2.14)]. 7.5 Let one of a, b, c, d be a nonnegative integer power of q-l and let

¢(a, b) = 4¢3 [ a~~~/~d; q, q] , where efg = abcdq. Prove the Askey and Wilson [1979] contiguous relation A¢(aq-l, bq) + B¢(a, b) + C¢(aq, bq-l) = 0, where A = b(l - b)(aq - b)(a - e)(a - f)(a - g), B = ab(a - bq)(b - a)(aq - b)(l - c)(l - d) - b(l - b)(aq - b)(a - e)(a - f)(a - g) + a(l - a)(a - bq)(e - b)(f - b)(g - b), C = -a(l - a)(a - bq)(e - b)(f - b)(g - b), Exercises 201

and derive the following limit case which was found independently by Las­ salle [1999]:

( (c-b)+ ab(1-c) _ aC(l-b)) 3(P2 (a,b,C;q, x y ) y x x,y abc _ (l-c)(y-a)(y-b)

0 for 0 ::; n ::; N - 1, where An and Cn are as defined in (7.2.5) and (7.2.6) and one of aq, cq, bdq is q-N, N a nonnegative integer. 7.7 Prove (7.2.22) directly by using the appropriate transformation and sum­ mation formulas derived in Chapters 1-3. Verify that

n

for x, y = 0,1, ... ,N, which is the dual of (7.2.18). 7.8 (i) Prove that the q-Kmwtchouk polynomials

Kn(x;a,N;q) = 3

N (-N ) "K ( -x. N')K ( -x. N') q ; q x ( )X ~ m q , a, , q n q , a, , q (.) -a x=o q, q x -1) N _(N+l) (q;q)n(1+a-1)(-a-1qN+\q)n = (-qa ; q a q 2 N (-a-I; q)n(1 + a-1q2n)(q-N; q)n X (_aqN+l )-nqn(n+l)bm,n, and find their three-term recurrence relation. (Stanton [1980b]) (ii) Let Kn(x;a,Nlq) = 2

~ -x -x (aq; q)N-x( _l)N-XqG) ~Km(q ;a,Nlq) Kn(q ;a,Nlq) (.) ( . ) x=o q,q x q,q N-x _ (q, aq; q)n(q; q)N-n (_l)n N (N+2n+l)_n(n+l);: - () a q U m n' q,q;q N ' 7.9 Prove that 202 Applications to orthogonal polynomials

where Pn(x) = Pn(x; a, b; q) are the little q-Jacobi polynomials and _qn (1- aqn+1)) (1- abqn+1) (1- qn) (1- bqn) (_aqn) An = C n = -:----=-:--:-:C-.,...... ,..,...... =.--'--:,:..-..".---=-:-:- (1 - abq2n+1) (1 - abq2n+2) , (1 - abq2n) (1 - abq2n+1) . 7.10 Prove that

where Pn(x) = Pn(x; a, b, c; q) are the big q-Jacobi polynomials and (1- aqn+1) (1- cqn+1) (1- abqn+1) A - --'------:,.,------':~---,--:;--,----;-,------''------:----;o---:-=-----'-- n - (1 _ abq2n+1) (1 _ abq2n+2) , (1 - qn) (1 - bqn) (1 - abc-1qn) n+1 Cn = (1 - abq 2 n ) ( 1 - abq 2 n+ 1) acq . 7.11 The affine q-Krawtchouk polynomials are defined by

-n K~ff (x; a, N; q) = 3¢2 [ q , x, -N0] ; q, q , 0 < aq < l. aq, q Prove that they satisfy the orthogonality relation

~ KAff (q-x' a N.q) KAff (q-x' a N.q) (aq,q-N;q)x x=o~m '" n '" ()q; q x qN-1)X (X) D x ( --- q-2 = m,n, m,n=0,1, ... ,N, a hn where = (aq, q-N; q)n (_1)n( )N-n Nn-m h n () aq q . q;q n (Delsarte [1976a,b], Dunkl [1977]) 7.12 The q-Meixner polynomials are defined by

Mn(x;a,c;q) = 2¢1 (q-n,x;aq;q,_qn+1jc), with 0 < aq < 1 and c > O. Show that they satisfy the orthogonality relation

~M ( -x )M ( -x ) (aq;q)x x G) Dm,n ~ m q ; a, c; q n q ; a, c; q ( . ) c q = h' x=o q, -acq, q x n where hn = (-acq; q)oo(aq; q)n qn (-c; q)oo(q, -qc-1; q)n .

(When a = q-r-1, the q-Meixner polynomials reduce to the q-Krawtchouk polynomials considered in Koornwinder [1989].) 7.13 The q-Charlier polynomials are defined by

cn(x;a;q) = 2¢1 (q-n,x;0;q,_qn+1ja). Exercises 203

Show that 00 aX (X) L cm(q-X; a; q)cn(q-X; a; q)-(-. -) q 2 x=o q, q X = (-a; q)oo(q, -qa-\ q)nq-nrSm,n'

7.14 Show that, for x = cos B, sin(n + I)B Cn(x; qlq) = . B = Un(x), n ~ 0 sm and . 1- qn hm ( iJ)Cn(x;iJlq)=cosnB=Tn(x), n~l, (3---71 2 1 - where Tn(x) and Un(x) are the Tchebichef polynomials of the first and second kind, respectively. 7.15 Verify that formulas (7.6.2) and (7.6.14) follow from the q-analogue of the Fields-Wimp formula (3.7.9).

7.16 Let x = cos B, It I < 1, and Iql < 1. Show that

fCn(x'iJlq) (>.;q)n tn = 2isinB (iJ,iJ;q)oo n=O ' (iJ2; q)n (1 - q) W(3 (xlq) (q, iJ2; q)oo

iO e - (quei() , que-iO , >.tu; q) 00 x l ("0 "0 ) dqu e iO iJue' ,iJue-' ,tu; q 00 and deduce that

00 t n (i) ~ Cn(x; iJlq) (iJ2; q)n

= (te-iO ; q)~} 2¢1 (iJ, iJe- 2iO ; iJ2; q, teiIJ ) .

00 (n) (iJt)n (ii) ~ Cn(x; iJlq)q 2 (iJ2; q)n

= (_te- iO ; q)oo 2¢1 (iJ, iJe 2iO ; iJ2; q, _te-iO ) . 7.17 Using (1.8.1), or otherwise, prove that if n is odd,

if n is even.

7.18 If -1 < q,iJ < 1, show that

7.19 Derive the recurrence relations 1 - qn+1 1 _ iJ2 qn-1 2xCn(x; iJlq) = 1 _ iJqn Cn+1(x; iJlq) + 1 _ iJqn Cn- 1(x; iJlq), 204 Applications to orthogonal polynomials

and 1 - j3 j3(1 - j3) Cn(x; j3lq) = 1 _ j3qn Cn(x; j3qlq) - 1 _ j3qn Cn- 2(X; j3qlq),

with C-2(x; j3lq) = C-1(x; j3lq) = 0, Co(x; j3lq) = 1, and n ~ 0. (Ismail and Zhang [1994]) 7.20 Prove that

r Cn(cos 0; j3lq) cos(n + 2k)e Wf3(cosOlq) dO = 7r/j3;~; ~)= j3k Jo q, ,q= (j3-\ q)k(j32; q)n(q; q)nH 1 - qn+2k n,k ~ 0, X (q; qh(q; q)n(j3q; q)nH 1 - qnH ' where Wf3(xlq) is defined in (7.4.6). 7.21 Using (7.4.15) and (7.6.14) prove that

h(x;"() ( I) ("t2,j3,j3q;q)= ~ (I ) h( . r.i)Cn x;j3 q = ( r.i2.) ~dk,nCn+2k x;"( q , X,/-, "(,,,(q,/-, ,q = k=O

where h(x; a) is as defined in (6.1.2) and k dk n = j3 b / j3; q)k(q; q)n+2k(j32; q)nb; q)nH(1 - "(qn+2k) . , (q; q)kb2; q)n+2k(q; q)n(j3q; q)nH(1- "() (Askey and Ismail [1983]) 7.22 Prove that the continuous q-Hermite polynomials defined in Ex. 1.28 sat­ isfy the orthogonality relation r ( I) ( I) 1(2iIJ ) 12 27r(q; q)n Jo Hm cosO q Hn cosO q e ;q = dO = (q;q)= Om,n. 7.23 Setting

in Ex. 7.19, show that 2x(1- j3qn)cn(x;j3lq) = (1- j32 qn)Cn+l(X;j3lq) + (1- qn)cn_l(x;j3lq),

for n ~ 0, with C-l (x; j3lq) = 0, co(x; j3lq) = 1. Now set j3 = s>.k and q = SWk, where Wk = exp(27ri/k) is a k-th root of unity, divide the above recurrence relation by 1-sWk and take the limit as S ---.. 1 to show that the limiting polynomials, c~(x; k), called the sieved ultraspherical polynomials of the first kind, satisfy the recurrence relation

2xc~(x; k) = C~+l (x; k) + C~_l (x; k), n -=I- mk, 2x(m + >')C~k(X; k) = (m + 2>')C~k+l (x; k) + mC~k_l (x; k) where k, m, n = 0,1, ... , cS(x; k) = 1 and cNx; k) = x. (Ai-Salam, Allaway and Askey [1984b]) Exercises 205

7.24 Rewrite the orthogonality relation (7.4.15) in terms of the sieved orthogo­ nal polynomial cn(x; ,8lq) defined in Ex. 7.23 and set ,8 = sAk and q = SWk. By carefully taking the limits of the q-shifted factorials prove that

1 c~(x; k)c~(x; k)w(x) dx = ~,n, 1-1 n where k-1 w(x) = 22A (k-1)(1_ x2)-! II Ix2 - cos2(7rjjk)IA j=O

and hn = ~(>'+1)1 (>.+lhn/kJ(2>')rn/kl r("2)r(>. +"2) (1) Ln/kJ (>')rn/kl

where the roof and floor functions are defined by

Ia l = smallest integer greater than or equal to a, l a J = largest integer less than or equal to a.

(AI-Salam, Allaway and Askey [1984b])

7.25 The sieved ultraspherical polynomials of the second kind are defined by

B~(x; k) = lim Cn(x; sAk+1WklsWk), Wk = exp(27rijk). 8---+1

Show that B~(x; k) satisfies the recurrence relation

2xB~(x; k) = B~+l (x; k) + B~_l (x; k), n + 1 -=I- mk, 2x(m + >')B~k-1 (x; k) = mB~k(x; k) + (m + 2>')B~k_2(X; k),

where B6(x; k) = 1; Bf(x; k) = 2x if k 2:: 2; Bf(x; 1) = 2(>' + l)x. Show also that B~(x; k) satisfies the orthogonality relation

1 B~(x; k)B~(x; k)w(x) dx = ~,n, 1-1 n where k-1 W(X) = 22A (k-1)(1_ x2)! II Ix2 - cos2(7rjjk)IA j=O

and _ 2r(>'+1) (l)Ln/kJ(>'+l)L~J hn - 1 1 ( ) ( ) . r("2)r(>. +"2) >. + 1 Ln/kJ 2>' + 1 Lnt 1 J

(AI-Salam, Allaway and Askey [1984b]) 206 Applications to orthogonal polynomials

7.26 Using (2.5.1) show that

( . b dl) _ (ab,ae,be,q;q)n PnX,a, ,e, q - (bd-l.)aeq ,qn n (abeq-1 eiO;q)k (1- abeeiO q2k-1) (aeiO,beiO,eeiO;q)k x {; (q; qh(l - abeq-1eiO ) (be, ae, ab; q)k

x (abedq-1; q)nH (de-iO ; q)n-k ei(n-2k)O (abee 20 ; q)nH (q; q)n-k Deduce that the polynomials Pn(x) = lim Pn(x;a,b,e,dlq) q-+O are given by

PO(X) = 1 = Uo(x),

P1(X) = (1- 84)U1(X) + (83 - 81)UO(X),

P2(X) = U2(x) - 81U1(X) + (82 - 84)UO(X), 4 Pn(X) = ~) -1)k8k Un_k(X), n ~ 3, k=O where

80 = 1, 81 = a + b + e + d, 82 = ab + ae + ad + be + bd + cd,

83 = abc + abd + aed + bcd, 84 = abed,

and Un (COS 0) = sin(n+1)ej sinO, U-1(x) = O. When max(lal, Ibl, lei, Idl) < 1 show that these polynomials satisfy the orthogonality relation 211 Pm(X)Pn(x)(l - x2)! dx 7r -1 (1- 2ax + a2 )(1 - 2bx + b2 )(1 - 2ex + e2 )(1 - 2dx + d2 ) 0, m#n, 1- abed = { (1 - ab)(l - ae)(l - ad)(l - be)(l - bd)(l _ cd)' m = n = 0, 1- abed, m = n = 1, 1, m = n ~ 2. (Askey and Wilson [1985]) 7.27 Prove that n+2. ) sin(n + 1)e (i) Pn (cos 0,· q, _ q, q ! , _!q Iq) -_ (q ,q n . 0 ' S1n (ii) Pn(cos 0; 1, -1, q!, -q! Iq) = 2(qn; q)n cos nO, n ~ 1,

· _ ! _! _ ( n+ 1. ) sin( n + ~)e (iii) Pn(cosO,q, 1,q, q Iq) - q ,q n sin(Oj2) ,

· _ ! _! _ ( n+1. ) cos(n + ~)e (iv) Pn(cosO,l, q,q , q Iq) - q ,q n cos(Oj2) . Exercises 207

7.28 Use the orthogonality relations (7.5.28) and (7.5.29) to prove the quadratic transformation formula (7.5.26). 7.29 Verify the orthogonality relations (7.5.28) and (7.5.29). 7.30 Verify formula (7.5.36). 7.31 Suppose that a,b,c,d are complex parameters with max (Ibl, Icl, Idl, Iql) < 1 < lal such that laqN+11 < 1 < laqNI, where N is a nonnegative integer. Use (6.6.12) to prove that

1 N [1 Pm(X)Pn(X)W(X; a, b, c, dlq) dx + 2n ~Pm(Xk)Pn(Xk)Wk

hn(a, b, c, dlq)' where Pn(x) = Pn(x;a,b,c,dlq), Xk = ~ (aqk +a-1 q-k) and Wk is given by (7.5.22). 7.32 Prove that (i) p~a,(3)(_x;q) = (-ltP~f3,a)(x;q), (ii) p~a,(3)(_xlq) = (_ltq(a-f3)n/2p~f3,a)(xlq). 7.33 Using (7.4.1), (7.4.7) and (2.11.2) prove that

(i) ~ (~;;qJ)n On(x; ,8lq)On(Y; ,8lq)tn

(t2,,8;q)= (,8t2, ,82; q)=

x W (~t2q-l. ~ teiiJ+i¢ te -iO-i¢ teiO - i¢ tei¢-iO. q ~) 8 7 jJ ,jJ, , , , , ,jJ , ~ (q; q)n 1 - ,8qn C ( ~I)O ( ~I) n (ii) f='o (,82;q)n 1-,8 n X;{J q n Y;{J q t

2 (t2,,8; q) = (,8teiiJ+i¢, q,8te iO - i¢; q) = (q,8t2, ,82; q)= (teiiJ+i¢, teiO - i¢; q)=

x SW7 (,8t2;,8, qteiiJ+i¢, qte-iO - i¢, teiO - i¢, tei¢-iO; q,,8) , where -1 < q,,8, t < 1 and x = cosB, y = cos¢. (Gasper and Rahman [1983a], Rahman and Verma [1986a]) 7.34 Show that Pn(cosB; a, b, c, dlq) = D-l(B)(ab, ac, bc; q)n

W qe- / d (dueiO,due- iO , abcdujq; q) (qju;q) (dU)n x l = n - du qeW /d (daujq, dbujq, dcujq; q)= (abcdujq; q)n q q, where -iq(l - q) D( B) = 2d (q, ab, ac, bc; q)=h( cos B; d)w( cos B; a, b, c, dlq). 208 Applications to orthogonal polynomials

Hence show that

00 ( '""" a 2c 2) ; q nPn (()cos ; a, aq 12 , C, cq 112 q ) n ~ 1 1 t n=O (q; q)n(a2q2, ac, acq2; q)n iIJ iIJ (at " ac2t -acte , -acte- ., q) 00 (-ct, -a2ct, teiIJ , te-iIJ ; q)oo

x 8 W7 ( -a2ctq-\ -ac, -a/c, -ctq-~, aeiIJ , ae-iIJ ; q, -ctq~) .

(Gasper and Rahman [1986]) 7.35 Show that Ex. 6.9 is equivalent to

1 w(y; a, b, p,eiIJ, J.Le-iIJlq)Pn(Y; a, b, c, dlq) dy 1-1 27r(abJ.L2; q)oo (ab; q)n n (q, ab, J.L2, aJ.LeiIJ , aJ.Le- iIJ , bJ.LeiIJ , bJ.Le- iIJ ; q)oo (abJ.L2; q)n J.L

x Pn(cos (); aJ.L, bJ.L, CJ.L- 1 , dJ.L- 1 Iq), where max(lal, Ibl, IJ.LI) < l. 7.36 Show that if for Iql < 1 we define

() (a; q)oo a; q v = ( v ) , aq ; q 00

where 1/ is a complex number and the principal value of qV is taken, then (7.7.6) extends to

D (ae iIJ ae-iIJ'q) =_2a(1-qV) (aq~eiIJ,aq~e-iIJ;q) . q , 'v 1 - q v-I

7.37 Let n = 1,2, ... ,r, x = cos(), and Un(x) = An,r (qv+leilJ , qv+le-ilJ ; q)n qn-r, qn+r+2v+2>-+I, qn+v+~ ,qn+2v, qn+v+leilJ, qn+v+le-ilJ . ] [ X 6¢5 q2n+2v+l, qn+v+>-+I, qn+2v+l, _qn+v+>-+I, _qn+v+~ ,q, q

Show that Un(x) satisfies the q-differential equation Dq[(qn+V+ 1eiIJ , qn+v+l e-ilJ; q)-nUn(x)]

1 - qn+2v 3 . 3. = - (qn+v+2 e,IJ, qn+v+2 e-,IJ; q) 1 - qn+l -2n-2v-l

X Dq[(q-n-v eiIJ , q-n-v e-iIJ ; q)n+2v+l Un+1 (x)]. (Gasper [1989b]) Exercises 209

7.38 Show that the discrete q-Hermite polynomials [n/2] H ( . ) = " (q; q)n (_l)k k(k-l) n-2k n X, q ~ (2 2) () q X k=O q; q k q; q n-2k

satisfy the recurrence relation

and the orthogonality relation

where 1jJ(x) is a step function with jumps Ixl (X 2q2, q; q2)= 2 (q2; q2)=

at the points x = ±qj, j = 0,1,2, .... (Ai-Salam and Garlitz [1965], Ai-Salam and Ismail [1988])

7.39 Let a < 0 and 0 < q < 1. Show that I: u$::)(x;q)u~a)(x;q) doJa) (x)

= (1- a)( _a)n(q; q) nqG)8m ,n, where

and oJa) (x) is a step function with jumps qk

at the points x = qk, k = 0,1, ... , and jumps -aqk

at the points x = aqk, k = 0,1,.... Verify that when a = -1 this orthogonality relation reduces to the orthogonality relation for the discrete q-Hermite polynomials in Ex. 7.38. (See Ai-Salam and Garlitz [1957, 1965], Ghihara [1978, (10.7)], and Ismail [1985b, p.590])

7.40 Show that if h ( . ) - ~ (q; q)n k n x, q - ~ ( ) ( ) x, k=O q; q k q; q n-k 210 Applications to orthogonal polynomials

then h~(x;q) - hn+l(x;q)hn-1(x;q) n qn-kxk = (1 - q)(q; q)n-l L h~_k(X; q) ( .) ,n ;::: 1. k=O q, q n-k

Deduce that the polynomials hn(x; q), which are called the Rogers-Szego polynomials, satisfy the Thnin-type inequality

for x ;::: 0 when 0 < q < 1 and n = 1,2, .... (Carlitz [1957b])

7.41 Derive the addition formula

Pn(qZ; 1, 1; q) py(qZ; qX, 0; q) = Pn(qx+y; 1, 1; q) Pn(qY; 1, 1; q) py(qZ; qX, 0; q) n (q. q) (q. q) qk(k+y-n) + L ' x+y+k , n+k k=l (q; q)x+y(q; q)n-k(q; q)~

X Pn_k(qx+y; l, qk; q) Pn-k(qY; l, l; q) Z x n (q; q)y(q; q)n+kqk(x+y-n+l) x Py+k(q ; q ,0; q) + L (.) (.) (. )2 k=l q,qy-kq,qn-kq,qk

X Pn_k(qx+y-k; l, l; q) Pn-k(qy-\ qk, qk; q) Py_k(qZ; qX, 0; q)

where x, y, z, n = 0,1, ... , and Pn(t; a, b; q) is the little q-Jacobi polynomial defined in Ex. 1.32. (Koornwinder [1991a])

7.42 Derive the product formula

00 Pn(qX; 1, 1; q) Pn(qy; 1, 1; q) = (1 - q) LPn(qZ; 1, 1; q) K(qX, qy, qZ; q)qZ, z=O where x, y, z, n = 0,1, ... , Pn(t; a, b; q) is the little q-Jacobi polynomial, and ( x+l y+l z+l. ) K( x y z. ) _ q , q , q ,q 00 xy+Xz+yz q ,q ,q ,q - (1 -)( .) q q q, q, q 00 x {3¢2(q-X,q-y,q-Z;0,0;q,q)}2.

(Koornwinder [1991a])

7.43 The q-Laguerre polynomials are defined by Exercises 211

Show that if a > -1 then these polynomials satisfy the orthogonality relation

(i)

and the discrete orthogonality relation

(ii)

where A = (q, -c(1 - q)qe<+1, -1/cqe«1 - q); q)oo . (q"'+1, -c(1 - q), -q/ c(1 - q); q)oo

(Moak [1981])

7.44 Let

v(y;a1,a2,a3,a4,a5Iq)

= h(y; 1, -1, q!, -q!, a1 a2a3a4a5) (1 _ y2)-! , y = cos cp. h(y;a1,a2,a3,a4,a5)

and

Show that

1 if) -if) (abC/-Leif),abC/-Le-if);q)n v(y; a, b, c, f-le , f-le Iq) (b 2"¢ b 2 "¢ ) 1 t t -1 a Cf-l e , a Cf-l e- ; q n

X Pn(y; a, b, c, dlq) dy

( if) -if) I ) (ab, ac, bc; q)n n = g a, b, c, f-le , f-le q ( b 2 2 b 2 ) f-l a f-l , aCf-l , Cf-l ; q n

X Pn (x; af-l, bf-l, cf-l, df-l-1Iq), x = cos (),

where Pn(x; a, b, c, dlq) are the Askey-Wilson polynomials defined in (7.5.2) and max(lal, Ibl, Icl, If-ll, Iql) < l. (Rahman [1988a])

7.45 Defining the q-ultrasphe'rical function of the second kind by

sin () h( cos 2(); (3) ~ Dn(x; (3lq) = 4 h( ()) L..,. b(k, n; (3) cos(n + 2k + l)e cos 2 ; 1 k=O 212 Applications to orthogonal polynomials

with b(k, n; (3) as in (7.4.9), prove that

C~ (cos 0; (3lq) + D~ (cos 0; (3lq)

= [4 sin 0 ((3, (3qn+1; q)ooh( cos 20; (3)] 2 (q, (32qn; q)ooh( cos 20; 1) x ((qn+1,(32qn,qe2i(),qe-2i();q)OO ((3qn, (3qn+1, (3e 2i(), (3e-2i(); q)oo

rI.. [ (3,q/(3,-ylq,-q .] x 5'1'4 q1-n /(3, (3qn+1, qe2i(), eq-2i() , q, q + ((3, q/(3, (3e 2i () qn+1, (3e- 2i () qn+\ q)oo ((3qn+1, q-n /(3, (3e 2i(), (3e- 2i(); q)oo qn+1,(32 qn,(3qn+1/2,_(3qn+1/2,_(3qn+1. ]) [ X 5¢4 (3qn+1, (32q2n+1, (3e2i() qn+1, (3e-2i() qn+1 , q, q .

(Rahman [1992a])

7.46 Let f(x) be continuous on [-1,1]. Show that

1 f(y) 1 Dq [/ K(x, y) ~dy = f(x), -1 y 1- y2

where

K( ) = (1 - q)(q, q, e2i¢, e-2i¢; q)ooh(x; _q1/4, _q3/4) X, Y 47rq1/4h(y; _q1/4, _q3/4, q1/2ei(), q1/2e-i())

(q, q, e2i¢, e-2i¢; q)oo

with x = cos 0, y = cos ¢. Note that this defines a formal inverse of the Askey-Wilson operator D q . (Ismail and Rahman [2002a,b])

7.47 Bustoz and Suslov's q-trigonometric functions are defined by

where x = cos o. (i) Show that

lim Cq(x; (1- q)w/2) = coswx and lim Sq(x; (1 - q)w/2) = sinwx. q--+1- q--+1- Notes 213

(ii) Prove the orthogonality relations

[11 W1/2(xlq)Cq(x; wm)Cq(x; wn)(1 - X2)-1/2dx

= [11 W1/2(xlq)Sq(x; Wm)Sq(X; wn)(1 - X2)-1/2dx (q1/2; q)2 [ 8 ] =7f ( . )2 00 Cq(7J; wn) -8 Sq(7J; w) 8m,n, q, q 00 W W=Wn where 7J = (q1/4 + q-1/4)/2, and Wm, Wn are two distinct roots of the equation Sq(7J;w) = O. (Bustoz and Suslov [1998], Suslov [2003])

Notes

§7.1 See also Atakishiyev and Suslov [1988a,b], Atakishiyev, Rahman and Suslov [1995], Nikiforov and Uvarov [1988], Nikiforov, Suslov and Uvarov [1991]' and Szego [1968, 1982]. For a classical polynomial system with complex weight function see Ismail, Masson and Rahman [1991]. The familiar connec­ tion between continued fractions and orthogonality was extended by Ismail and Masson [1995] to what they call R-fractions of type I and II, which lead to biorthogonal rational functions. See further work on continued fractions related to elliptic functions in Ismail and Masson [1999], Ismail, Valent and Yo on [2001], and Milne [2002]. In T.S. Chihara and Ismail [1993] extremal measures for a system of orthogonal polynomials in an indeterminate moment problem are examined. For orthogonal polynomials on the unit circle see Is­ mail and Ruedemann [1992]. Some classical orthogonal polynomials that can be represented by moments are discussed in Ismail and Stanton [1997, 1998]. §7.2 Andrews and Bressoud [1984] used the concept of a crossing number to provide a combinatorial interpretation of the q-Hahn polynomials. Koelink and Koornwinder [1989] showed that the q-Hahn and dual q-Hahn polynomials admit a quantum group theoretic interpretation, analogous to an interpreta­ tion of (dual) Hahn polynomials in terms of Clebsch-Gordan coefficients for SU(2). For how Clebsch-Gordan coefficients arise in quantum mechanics, see Biedenharn and Louck [1981a,b]. L. Chihara [1987] considered the locations of zeros of q-Racah polynomials and employed her results to prove non-existence of perfect codes and tight designs in the classical association schemes. The correspondence between q-Racah polynomials and Leonard pairs is outlined in Terwilliger [2003]. For the relationship between orthogonal polynomials and association schemes, see Bannai and Ito [1984], L. Chihara and Stanton [1986], Delsarte [1976b]' and Leonard [1982]. A multivariable extension of the q-Racah polynomials is considered in Gasper and Rahman [2003c], while a sys­ tem of multivariable biorthogonal polynomials is given in Gasper and Rahman [2003a], which are q-analogues of those found in Tratnik [1991b] and [1989], respectively. 214 Applications to orthogonal polynomials

§7.3 Ai-Salam and Ismail [1977] constructed a family of reproducing ker­ nels (bilinear formulas) for the little q-Jacobi polynomials. In Ai-Salam and Ismail [1983] they considered a related family of orthogonal polynomials associ­ ated with the Rogers-Ramanujan continued fraction. A biorthogonal extension of the little q-Jacobi polynomials is studied in Ai-Salam and Verma [1983a]. When b = 0 the little q-Jacobi polynomials reduce (after changing variables and renormalizing) to the Wall polynomials

n (j) . (n+1) [n] q 2 (_q-nx )1 Wn(x;b,q) = (-l)"(b;q)nq 2 L· (b. ). j=O J q ,q J and to the generalized Stieltjes-Wigert polynomials

which are q-analogues of the Laguerre polynomials that are different from those considered in Ex. 7.43. Since the Hamburger and Stieltjes moment problems corresponding to these polynomials are both indeterminate, there are infinitely many nonequivalent measures on [0, (0) for which these polynomials are or­ thogonal. See Chihara [1978, Chapter VI], [1968b, 1971, 1979, 1982, 1985], Ai-Salam and Verma [1982b]' L. Chihara and T.S. Chihara [1987], and Shohat and Tamarkin [1950]. §7.4 An integral of the product of two continuous q-ultraspherical polyno­ mials and a q-ultraspherical function of the second kind is evaluated in Askey, Koornwinder and Rahman [1986]. Ai-Salam, Allaway and Askey [1984a] gave a characterization of the continuous q-ultraspherical polynomials as orthogo­ nal polynomial solutions of certain integral equations. Askey [1989b] showed that the polynomials Cn(ix; jJlq), 0 :::; n :::; N, are orthogonal on the real line with respect to a positive measure when 0 < q < 1 and jJ > q-N. Is­ mail and Rahman [1991] showed that the associated Askey-Wilson polynomials r;::'(x; a, b, c, dlq) defined in Ex. 8.26 have an orthogonality property on [-1,1]. A survey of classical associated orthogonal polynomials is in Rahman [2001], and an integral representation is given in Rahman [1996b]. A projection for­ mula and a reproducing kernel for r;::'(x; a, b, c, dlq) is given in Rahman and Tariq Qazi [1997b]. Berg and Ismail [1996] showed how to generate one to four parameter orthogonal polynomials in the Askey-Wilson family by starting from the continuous q-Hermite polynomials. Also see Koelink [1995b] and Rahman and Verma [1987]. §7.5 Asymptotic formulas and generating functions for the Askey-Wilson polynomials and their special cases are derived in Ismail and Wilson [1982] and Ismail [1986c]. Kalnins and Miller [1989] employed symmetry techniques to give an elementary proof of the orthogonality relation for the Askey-Wilson polynomials. Following Hahn's approach to the classification of classical or­ thogonal polynomials N.M. Atakishiyev and Suslov [1992b] gave a generalized moment representation for the Askey-Wilson polynomials. Brown, Evans and Ismail [1996] showed that the Askey-Wilson polynomials are solutions of a Notes 215 q-Sturm-Liouville problem and gave an operator theoretic description of the Askey-Wilson operator V q . L. Chihara [1993] extended her work on q-Racah polynomials [1987] to the Askey-Wilson polynomials. Floreanini, LeTourneux and Vinet [1999] also employed symmetry techniques to study systems of con­ tinuous q-orthogonal polynomials. A multivariable extension of Askey-Wilson polynomials is given in Gasper and Rahman [2003b] as a q-analogue of Tratnik [1991a], see Ex. 8.29. Also see Gasper, Ismail, Koornwinder, Nevai and Stanton [2000], Spiridonov and Zhedanov [1995-1997], Stokman [1997a-2003a], Stok­ man and Koornwinder [1998], Vinet and Zhedanov [2001], and Wilson [1991]. §7.6 For additional results on connection coefficients (and the correspond­ ing projection formulas), see Andrews [1979a], Gasper [1974, 1975a]. Ex.7.5 A contiguous relation satisfied by a WP-balanced, generally non­ terminating 8¢7 series is given in Ismail and Rahman [1991]. For more results on contiguous relations and orthogonal polynomials, see Gupta, Ismail and Masson [1992, 1996], Gupta and Masson [1998] and Ismail and Libis [1989]. Ex. 7.7 Ismail [1995] gave a simple proof of (1.7.2) by making use of iterations of (7.7.6) and evaluating them at Xj = ~(aqj +q-j ja). An operator calculus for Dq is developed in Ismail [2001a]. Ex.7.8 Stanton [1981b] showed that the q-Krawtchouk polynomials Kn(x; a, N; q) are spherical functions for three different Chevalley groups over finite fields and derived three addition theorems for these polynomials by de­ composing the irreducible representations with respect to maximal parabolic subgroups. In Koornwinder [1989] it is shown that the orthogonality relation for the q-Krawtchouk polynomials Kn(x; a, Nlq) expresses the fact that the matrix representations of the quantum group Sp,U(2) are unitary. Ex.7.11 The affine q-Krawtchouk polynomials are the eigenvalues of the association schemes of bilinear, alternating, symmetric and hermitian forms over a finite field (see Carlitz and Hodges [1955], Delsarte [1978], Delsarte and Goethals [1975], and Stanton [1981a,b, 1984]). L. Chihara and Stanton [1987] showed that the zeros of the affine q-Krawtchouk polynomials are never zero at integral values of x, and they gave some interlacing theorems for the zeros of q-Krawtchouk polynomials. Ex. 7.12 and 7.43 N.M. Atakishiyev, M.N. Atakishiyev and Klimyk [2003] found the connection between big q-Laguerre and q-Meixner polynomials and representations of the group Uq(SUl,l). Ciccoli, Koelink and Koornwinder [1999] extended Moak's q-Laguerre polynomials to an orthogonal system for a doubly infinite Jacobi matrix originating from analysis on SUq (l, 1), and found the orthogonality and dual orthogonality relations for q-Bessel functions originating in Eq(2). Ex.7.22 Askey [1989b] proved that the polynomials Hn(ixlq) are orthog­ onal on the real line with respect to a positive measure when q > 1. Exercises 7.23-7.25 Other sieved orthogonal polynomials are considered in Ai-Salam and Chihara [1987], Askey [1984b], Charris and Ismail [1986, 1987, 1993], Charris, Ismail and Monslave [1994]' Ismail [1985a, 1986a,b], and Ismail and Li [1992]. Exercises 7.38-7.40 For additional material on q-analogues of Hermite 216 Applications to orthogonal polynomials polynomials, see Allaway [1980], Ai-Salam and Chihara [1976], Ai-Salam and Ismail [1988], Carlitz [1963b, 1972], Chihara [1968a, 1982, 1985], Dehesa [1979], Desarmenien [1982]' Hou, Lascoux and Mu [2003], Ismail [1985b], Ismail, Stan­ ton and Viennot [1987], Lubinsky and Saff [1987], and Szego [1926]. Ex. 7.41 Rahman [1989a] gave a simple proof for this addition formula. For derivations of the addition formula for Jacobi polynomials, see Koorn­ winder [1974a,b] and Laine [1982]. Ex.7.43 See also Cigler [1981] and Pastro [1985]. In view of the two different orthogonality relations for the q-Laguerre polynomials, it follows that there are infinitely many measures for which these polynomials are orthogonal. The Stieltjes-Wigert polynomials (see Chihara [1978, pp. 172-174], Szego [1975, p. 33] and the above Notes for §7.3) sn(x) = (_I)nq(2n+l)/4(q;q)~! t [~] qj2(_q!X)j, j=O J q which are orthogonal with respect to the log normal weight function

w(x) = k7r-! exp( -k21og2 x), 0 < x < 00, where q = exp[-(2k2)-1] and k > 0, are a limit case of the q-Laguerre poly­ nomials. Askey [1986] gave the orthogonality relation for these polynomials (with a slightly different definition) that follows as a limit case of the first orthogonality relation in this exercise. Ai-Salam and Verma [1983b,c] studied a pair of biorthogonal sets of polynomials, called the q-Konhauser polynomi­ als, which were suggested by the q-Laguerre polynomials. Ismail and Rahman [1998] studied two indeterminate Hamburger moment problems associated with L~(x; q), thus completing earlier work of Moak [1981]. For asymptotics of basic Bessel functions and q-Laguerre polynomials, see Chen, Ismail and Muttalib [1994]. Ex.7.46 A right inverse ofthe Askey-Wilson operator is derived in Brown and Ismail [1995]. Ex.7.47 The question of completeness of the q-trigonometric functions for use in a q-Fourier type analysis is dealt with in Ismail [200lb] and Suslov [200lc, 2003]. 8

FURTHER APPLICATIONS

8.1 Introduction

In this chapter we derive some formulas that are related to products of the q-orthogonal polynomials introduced in the previous chapter and use these formulas to obtain q-analogues of various product formulas, Poisson kernels and linearization formulas for ultraspherical and Jacobi polynomials. The method in Gasper and Rahman [1984] originates with the observation that since

j-I (q-X, aqX; q) j = II (1 - qk(q-x + aqX) + aq2k) k=O is a polynomial of degree j in powers of q-X +aqX, there must exist an expansion of the form Hk (q-X,aqX;q)j (q-X,aqX;q)k = L Am(j,k,a;q) (q-X,aqX;q)m· (8.1.1) m=O Since, for k ~ j, (q-X, aqX; q)J. 3¢2 (q-j, l-x, al+x; aqk, q1+k-j ; q, q) = --,------,-----,------,-"­ (q-k, aqk; q)j by the q-Saalschiitz formula (1.7.2), it is easy to verify that (q-X,aqX;q)j (q-X,aqX;q)k = (q;q)j(q;q)k(a;q)Hk j+k X L (8.1.2) m=max(j,k) This linearizes the product on the left side and forms the basis for the product formulas derived in the following section. Suppose {Bj } ~o and {Cj } ~o are arbitrary complex sequences and b, c are complex numbers such that (b; q)k, (c; q)k do not vanish for k = 1,2, ... Then, setting

Fn = (q-n,a.qn;q)j B (q-n,a.qn;qh , t j t Ck (8.1.3) j=O (q, b, q)j k=O (q, C, q)k we find by using (8.1.2) that

217 218 Further applications

k (q-k, aqm; q) j x L ( b m-k. ) B m- k+j (8.1.4) j=O q, q ,q j for n = 0,1,2,.... This formula does not extend to noninteger values of n because, in general, the triple sum on the right side does not converge.

8.2 A product formula for balanced 4¢3 polynomials

Since we are mainly interested in q-orthogonal polynomials which are express­ ible as balanced 4¢3 series or their limit cases, we shall now specialize (8.1.4) to such cases. Set B. - (b l , b2; q) j j C. _ ( CI , C2; q) j j () J - (b3, qabIb2jbb3; q)j q, J - (C3, qacIC2jcc3; q)j q , 8.2.1 where it is assumed that the parameters are such that no zero factors appear in the denominators. Then formula (8.1.4) gives n m (-n n) ( ) in=LL q ,aq;qm cI ,c2;qm-k qk2 -mk+m (q, b; q)k(q, C, C3, qacIC2jcc3; q)m-k m=Ok=O (b l ,b2;q)k ¢ [ qk-m,aqm,bIqk,b2qk ] X (b3,qabIb2jbb3;q)k 4 3 bqk,b3qk,abIb2qk+Ijbb3;q,q , (8.2.2) where in is the right side of (8.1.3) with B j and Cj as defined in (8.2.1). The crucial step in the next round of calculations is to convert the 4¢3 series in (8.2.2) into a very-well-poised 8¢7 series by Watson's formula (2.5.1), i.e., qk-m, aqm, bIqk, b2qk ] [ 4¢3 b k b k b b k+Ijbb ;q,q q , 3q ,a I 2q 3

(bb3qk-m jabl , bb3qk-m jab2; q)m-k (bb3q2k-m ja, bb3q-m jabIb2; q)m-k bb 2k-m-1 b k-m b k-m bb m-k) ( 3q 3q q k k k-m 3q X 8W7 ; ,--,bIq ,b2 q ,q ;q, b b . a a a I 2 (8.2.3) 8.2 A product formula for balanced 41>3 polynomials 219

Then, replacing j by j - k in the sum on the right side of (8.2.4), we obtain

q-n, aqn, bl , b2 ] [ q-n, aqn, CI, C2 ] 41>3 [ j ;q,q 41>3 j ;q,q b, b3, qabl b2 bb3 C, C3, qacIC2 CC3 ~ (q-n, aqn, CI, C2, qabdbb3, qab2jbb3; q)m m = ~ (q, C, C3, qajbb3, qacIC2jcC3, qabl b2 jbb3; q)m q

~ (bb3q-m-1 ja; q)j(l - bb3q2j-m-1 ja)(bl , b2 , bq-m ja, b3q-m ja, q-m; q)j X ~ (q; q)j(l - bb3q-m-1 ja)(bb3q-m jabl , bb3q-m jab2 , b3, b, bbJ/a; q)j

bb3qm )j [q-j, ql-m jc, ql-m j C3, bb3qj-m-1 ja, CC3q-m j acIC2. ] x ( -- 51>4 I I ' q, q . bl b2 q -m JCI, q -m j C2, bq-m ja, b3q-m ja (8.2.5)

Note that the 51>4 series in (8.2.5) is balanced and, in the special case C = aqjb and C3 = aqjb3, becomes a 31>2 which is summable by (1.7.2). Thus, we obtain the formula

bb3c2jaq, bq-m ja, b3q-m ja, q-m aq2 ] I ; q, ,(8.2.6) q -m j C2, b3, b, bb3ja, bl b2CIC2 where). = bb3q-m-1 ja. This formula is a q-analogue of Bailey's [1933] product formula

2FI ( -n, a + n; b; x) 2FI ( -n, a + n; 1 + a - b; y) = F4( -n, a + n; b, 1 + a - b; x(l - y), y(l - x)), (8.2.7) where F ( b d ) ~ ~ (a)m+n(b)m+n m n 4 a, ;C, ;x,y = ~~ m!n!(c)m(d)n x y . (8.2.8)

However, even though (8.2.6) is valid only when the series on both sides ter­ minate, (8.2.7) holds whether or not n is a nonnegative integer, subject to the absolute convergence of the two 2FI series on the left and the F4 series on the right. Application of Sears' transformation formula (2.10.4) enables us to trans­ form one or both of the 41>3 series on the left side of (8.2.6) and derive a number of equivalent forms. Two particularly interesting ones are 220 Further applications

(8.2.9)

bb3 C2/aq, ql-m /C2,

(8.2.10) b3 , b, where ,\ = bb3 q-m-l/a. Either of the formulas (8.2.9) and (8.2.10) may be regarded as a q-analogue of Watson's [1922] product formula for the Jacobi polynomials 2FI( -n, a + n; b; x) 2FI( -n, a + n; b; y) n(1+a-b)n ( ( )( )) = ( -1 ) (b)n F4 -n,a+n;b,l+a-b;xy, I-x l-y ,

(8.2.11) where n = 0,1, .... The special case in which the lOcP9 series in (8.2.6), (8.2.9) or (8.2.10) become balanced is also of interest in some applications. Thus, if we set C2 = aq/bIb2CI, then by using Bailey's transformation formula (2.10.8) we may express (8.2.9) in the form

q-n, aqn, bl , b2 ] [q-n, aqn, bb3 aq, bb3 /bl b2CI ] [ cd 4cP3 b b bb/bb ;q,q 4cP3 b b bb/bb ;q,q , 3, qa I 2 3 , 3, 3 I 2 = (aq/b,aq/b3 ;q)n (bb3 )n t (q-n,aqn,bICI,b2CI,aq/bIb2CI;q)mqm (b, b3 ; q)n aq m=O (q, aq/b, aq/b3 , bIb2CI, bb3 /bIb2; q)m

fJ' bIb2CI/b, bIb2CI/b3 , X lOcP9 [ b, b3 , 8.3 Product formulas for q-Racah and Askey-Wilson polynomials 221

bb3cI/aq, aqm, (8.2.12) qab1b2/bb3, b1b2c1q-m la, where /-l = b1b2c1q-1. This provides a q-analogue of Bateman's [1932, p. 392] product formula

2F1 ( -n, a + n; b; x) 2F1 ( -n, a + n; b; y) = (_I)n (1 + a - b)n:t (-n)k(a + n)k (1- x _ y)k (b)n k=O k!(1 + a - bh x 2F1 (-k, a + k; b; -xy/(1 - x - y)). (8.2.13)

8.3 Product formulas for q-Racah and Askey-Wilson polynomials

Let us replace the parameters a,b,b1,b2,b3,C1,C2 in (8.2.9) byabq,aq,q-X, cqx-N, bcq, c-1q-Y, qy-N, respectively, to obtain the following product formula for the q-Racah polynomials introduced in §7.2:

VVn(x;a,b,c,lV;q) VVn(y;a,b,c,lV;q) -1.) n (q-n abqn+1 qx-N qy-N c-1q-X c-1q-Y. q) (b q, qac ,q n n '"""' ' , , , , 'm m -N -N ) = (aq, bcq; q ) n c m=O~ (bq, q, qac -1 ,c-1 ,q , q ; q m q

,/,. [cq- m , q(cq-m)~, _q(cq-m)~, ca-1q-m, b-1q-m, q-m, q-X, X 10'1-'9 1 1 1 (cq-m)2, _(cq-m)2, aq, bcq, cq, cqx+ -m,

q -y , cq x-N , cq y-N b 2N +3 ] (8.3.1) cq y+1-m ,qN -x+1-m ,qN -y+1-m ; q, a q , where

VVn(x;a,b,c,lV;q) _ [q-n, abqn+1, q-X, cqx-N . ] - 41>3 -N b ,q,q (8.3.2) aq,q ,cq is the q-Racah polynomial defined in (7.2.17). This is a Watson-type formula. Two additional Watson-type formulas are given in Ex. 8.l. Letting c ----t 0 in (8.3.1) gives a product formula for the q-Hahn polyno­ mials defined in (7.2.21):

Qn(x; a, b, lV; q) Qn(Y; a, b, lV; q) (b ) n (-n b n+1 x-N y-N ) = (-aqtqG) q;q n L q ,a q ,q ,q ;q m (aqx+y)-m (aq;q)n m=O (q,bq,q-N,q-N;q)m

q-X, q-Y, b-1q-m, q-m b 2N+3] [ X 41>3 N-x+1-m N-y+1-m; q, a q . (8.3.3) aq,q ,q 222 Further applications

To obtain a Watson-type product formula for the Askey-Wilson polyno­ mials defined in (7.5.2) we replace a, b, bl , b2, b3 , Cl, C2 in (8.2.9) by abcdq-l, ab, aei(} , ae-i(}, ac, dei¢, de-i¢, respectively, where x = cos B, y = cos ¢. This gives Pn(x; a, b, c, dlq) Pn(Y; a, b, c, dlq) = (ab, ac, ad, ad, bd, cd; q)n(ad)-n n (q-n abcdqn-l dei() de-i(} dei¢ de-i¢. q) x L '( , , '1 ') 'm qm m=O q, ad, ad, bd, cd, da- ; q m

A. [aq- m /d, q(aq-m /d)~, -q(aq-m /d)~, ql-m /bd, ql-m lcd, q-m, X 10'1"9 1 1 (aq-m /d)'l, -(aq-m /d)'l, ab, ac, aq/d,

aei(} , ae-i(} , aei¢, ae-i¢ bcq ] ql-me-i() /d, ql-mei(} /d, ql-me-i¢ /d, ql-mei¢ /d; q, ad· (8.3.4)

When b = aq~ and d = cq~, the 1O¢9 series in (8.3.4) becomes balanced and hence can be transformed to another balanced 1O¢9 via (2.9.1). This leads to a Bateman-type product formula Pn(x; a, aq~, c, cq~ Iq) Pn(Y; a, aq~, c, cq~ Iq)

1 1 qv'l, -qv'l, aei¢, aq~ ei¢, cei¢, aei(} , X 1O¢9 [v' 1 1 1 v 'l , -v'l, acq'l, ac, a2q~ , acq~ ei¢-i(} , ae-i(} , a2c2qm, 1 .¢ 2 ~:mm+!; q, q] , (8.3.5) acq~ ei¢+i(} , q'l-me' /c, a ce q 2 where v = a2cei¢q-~. In fact, if we replace a and c by q(2o:+l)/4 and _q(2,6+l)/4, respectively, then this gives a Bateman-type product formula for the continuous q-Jacobi polynomials (7.5.24) which, on letting q ----t 1, gives Bateman's [1932] product formula for the Jacobi polynomials: PAOI ,,6)(x)PAOI ,,6)(y) = (_1)nC8 + 1)n t (-nh(n+a+,8+1)k (x+2 y)k PAOI ,,6) (l)PAOI ,,6) (1) (a+1)n k=o k!(,8+1h

x p(OI,,6) (1 + xY ) /p(a,,6)(l), (8.3.6) k x+y k which is equivalent to (8.2.13). For terminating series there is really no difference between the Watson formula (8.2.11) and the Bailey formula (8.2.7) since one can be transformed into the other in a trivial way. However, for the continuous q-ultraspherical polynomials given in (7.4.14), there is an interesting Bailey-type product for­ mula that can be obtained from (8.2.6) by replacing a, b, bl , b2, b3 , Cl, C2 by 8.4 A product formula 223 a4, a2q~ , aei() , ae-i() " -a2q~ aeiq, and ae-iq, , respectively'.

n (q-n a4qn aeiq, ae-iq, -aei() -ae-i(). q) _ '"' ' , , , , 'm m -~ ( 2 1 2 1 2 2 ) q m--0 q , a q2 , -a q2 ""-1 -a -a . q m _q-m, q( _q-m)~, _q(_q-m)~ , q~-m la2, _q~-mla2, q-m, [ X 1O¢9 ( -m)! -q 2, _(_q-m)~ , _a2q~ , a2q~, _q,

aei () , (8.3.7) _q1-me-i() la, For further information about product formulas see Rahman [1982] and Gasper and Rahman [1984].

8.4 A product formula in integral form for the continuous q-ultraspherical polynomials

As an application of the Bateman-type product formula (8.3.5) for the Askey­ Wilson polynomials we shall now derive a product formula for the continuous q-ultraspherical polynomials in the integral form Cn(X; ,8lq)Cn(y; ,8lq)

(r; ~)n ,8-n/2j1 K(x, y, Z; ,8lq)Cn(z; ,8lq) dz, (8.4.1) q, q n -1 where

K( . r-II ) = (q,,8,,8;q)=1 (,8e2i(},,8e2iq,;q)=12 x, y, Z, {J q 2n(,82; q)=

x W (z;,~eiO+iq",~e-i(}-iq",~ei(}-iq",~eiq,-i(}) (8.4.2) with w(z; a, b, c, d) defined as in (6.3.1) and x = cos e, y = cos ¢. First, we set c = -a in (8.3.5) and rewrite it in the form

! ! ! ! 1 + a2 qn ( _!)n rn(x; a, aq2, -a, -aq2Iq)rn(Y; a, aq2, -a, -aq2Iq) = 1 + a2 -q 2

~ (q-n, a4qn, -a2q~ ei(}+iq" -a2q~ eiq,-i() , -aq~ e-iq,; q) x ~ mqm m=O (q, a2q~, -a2q~, -a2q, -a3q~ eiq,; q) m

X lOW9 (-a3q-~eiq,;aeiq" -aeiq"aq~eiq"aei(},ae-i(},a4qm,q-m;q,q), (8.4.3) where 224 Further applications

rn(X; a, b, c, dlq) _ [q-n, abcdqn-1, aeil:l, ae-il:l . ] - 4¢3 b d ,q, q . (8.4.4) a ,ac,a The key step now is to use the d = -(aq)! case of Bailey's transformation formula (2.8.3) to transform the balanced 1O¢9 series in (8.4.3) to a balanced 4¢3 series:

(a2e-2i¢, -a3q! ei¢; q) m

(a4, -aq! e-i¢; q) m

a2e2i¢, _q! eil:l+i¢, _q! ei¢-il:l, q-m ] [ (8.4.5) X4¢3 21""'"1:1 2 1 "1:1 "'" 1 2"'" 2;q,q· -a q'2 eZ

! ! ! ! 1 + a2 q rn(X; a, aq2, -a, -aq2Iq) rn(Y; a, aq2, -a, -aq2Iq) = 2 n (_q-'2 l)n 1+a ~ (q-n, a4qn, a2e-2i¢, _a2q! eil:l+i¢, _a2q! ei¢-il:l; q) x ~ mqm m=O (q, a4, a2q!, -a2q!, -a2q; q) m a2e2i¢, _q! eil:l+i¢, _q! ei¢-il:l, q-m ] [ X 4¢3 2 1 ""'"1:1 2 1 "1:1+"'" 1 2"'" 2;q,q . (8.4.6) -a q'2 eZ

Transforming this 4¢3 series by Sears' transformation formula (2.10.4), we obtain a further reduction

(8.4.7)

Now observe that, by (6.1.1),

1 w (z. aei¢-il:l aeil:l-i¢ aeil:l+i¢ ae-il:l-i¢) (aei¢-il:l+i'I/J aei¢-il:l-i'I/J. q) dz j -1' , " " j

= j1 W (z; aqjei¢-il:l, aeil:l-i¢, aeil:l+i¢, ae-il:l-i¢) dz -1

(8.4.8) 8.4 A product formula 225 where lal < 1 and z = cos'ljJ. Hence q-m, a2, a2e2i¢, a2e-2iO ] [ 41>3 a,-aq2e4 2 1 i¢-iO ,-q2I-m e i¢-iO; q, q (q, a2, a2; q)oo I (a2e2i¢, a2e-2iO ; q)oo 12 2n(a4; q)oo

X 11 W (z', aei¢-iO , aeiO - i¢ , aeiO +i¢ , ae-iO - i¢) -1

(q, a2, a2; q)oo I (a2e2i¢, a2e-2iO ; q)oo 12 2n(a4 ; q)oo

x 11 W (z', aei¢-iO , aeiO - i¢ , aeiO +i¢ , ae-iO - i¢) -1 ( -aq~ ei'ljJ, -aq~ e-i'ljJ; q) X m dz. (8.4.9) ( -a2q~ ei¢-iO, -q~ eiO - i¢; q) m

Substituting (8.4.9) into (8.4.7) and using (2.10.4) we finally obtain

1 1 1 1 rn(X; a, aq'2, -a, -aq'2lq) rn(Y; a, aq'2, -a, -aq'2lq) = 11 K(x, y, z; a2lq) rn(Z; a, aq~, -a, -aq~ Iq) dz. (8.4.10) -1 This yields (8.4.1) if we replace a by f3~ and use (7.4.14). By setting f3 = qA in (8.4.1) and taking the limit q --+ 1, Rahman and Verma [1986b] showed that (8.4.1) tends to Gegenbauer's [1874] product formula O~(x)O~(y) O~(z) 11 (8.4.11) 0;(1)0;(1) = -1 K(x, y, z) 0;(1) dz, where

(8.4.12) according as 1 - x 2 - y2 - z2 + 2xyz is positive or negative. Rahman and Verma [1986b] were also able to derive an addition formula for the continuous q-ultraspherical polynomials corresponding to the product formula (8.4.1). This is left as an exercise (Ex. 8.11). 226 Further applications

8.5 Rogers' linearization formula for the continuous q-ultraspherical polynomials

Rogers [1895] used an induction argument to prove the linearization formula Cm(X; ,6lq)Cn(x; ,6lq) mi~,n) (q; q)m+n-2d,6; q)m-k(,6; q)n-k(,6; q)k(,62; q)m+n-k k=O (,62; q)m+n-2k(q; q)m-k(q; q)n-k(q; q)d,6q; q)m+n-k (1 - ,6qm+n-2k) X (1 _ ,6) Cm+n- 2k (X; ,6lq)· (8.5.1)

Different proofs of (8.5.1) have been given by Bressoud [1981d], Rahman [1981] and Gasper [1985]. We shall give Gasper's proof since it appears to be the simplest. We use (7.4.2) for Cn(x; ,6lq) and, via Heine's transformation formula (1.4.3) , 2iO c ( ',61 ) = (,6e- ; q) 00 (,6; q)m imO m X, q (q,6-1e-2iO; q)oo (q; q)m e

x 2cPl (q,6-1,,6-2ql-m;,6-1ql-m;q,,6e- 2iO ) , (8.5.2) where x = cos e. Then, temporarily assuming that Iql < 1,61 < 1, we have

(8.5.3) where

A _ (,6e- 2iO ; q) 00 (,6; q)m(,6; q)n i(m+n)O (8.5.4) m,n - (q,6-1e-2iO; q)oo (q; q)m(q; q)n e .

The crucial point here is that the 4cP3 series in (8.5.3) is balanced and so, by (2.5.1), 8.6 The Poisson kernel for Cn(x; ,8lq) 227

Substituting this into (8.5.3) and interchanging the order of summation, we obtain

min(m,n) X L k=O

X (,8-2 ql-m-n; q)2k (q,8-1 e-2iO)k (,8-1 ql-m-n; q)2k

A, ( (./-1 (./-2 1+2k-m-n. (./-1 1+2k-m-n. (./ -2iO) (856) x 2'/'1 ql-' ,I-' q ,I-' q ,q,l-'e, .. which gives (8.5.1) by using (8.5.2) and observing that both sides of (8.5.1) are polynomials in x. Notice that the linearization coefficients in (8.5.1) are nonnegative when -1 < ,8 < 1 and -1 < q < 1. For an extension of the linearization formula to the continuous q-Jacobi polynomials, see Ex. 8.24.

8.6 The Poisson kernel for Cn(x; ,8lq)

For a system of orthogonal polynomials {Pn(x)} which satisfies an orthogonal­ ity relation of the form (7.1.6), the bilinear generating function

00 (8.6.1) n=O is called a Poisson kernel for these polynomials provided that hn = CVn for some constant C > O. The Poisson kernel for the continuous q-ultraspherical polynomials is defined by

Kt(x, y; ,8lq) = ~ ~~;;;)~(~ ~~i Cn(x; ,8lq)Cn(y; ,8lq)tn, (8.6.2) where It I < 1. Gasper and Rahman [1983a] used (8.5.1) to show that

-qt,8! , qteiHi , qte-iO - i , -t,8! , ,8te-iO - i , ,8teiHi , teiO - i , tei-iO, (8.6.3) q,8tei-iO, q,8teiO - i , where x = cos(),y = cos¢ and max(lql, Itl, 1(81) < 1. They also computed a closely related kernel 228 Further applications

((3, t 2 ; q) 00 ((32, (3t2; q)oo

teiO+iq, , te-iO-iq, , (3te- iO - iq, , (3teiHiq, ,

(8.6.4)

Alternative derivations of (8.6.3) and (8.6.4) were given by Rahman and Verma [1986a]. In view of the product formula (8.4.1), however, one can now give a simpler proof. Let us assume, for the moment, that It(3-~ 1< 1 and 1(31 < l. Then, by (7.4.1) and (8.4.1), we find that, with z = cos'lj.;,

1 (t(3~eiW,t(3~e-iw;q) Lt(x, y; (3lq) = 1 K(x, y, z; (3lq) ( 1. 1. 00) dz -1 t(3-2 e2W , t(3-2 e-2W ; q 00 (q,(3,(3;q)ool ((3e2iO,(3e2iq,;q)00 12 27r((32; q)oo

By (6.3.9),

2 x 8 W 7 ((3t q-1., teiO+iq, , te-iO-iq, , teiO-iq, , teiq,-iO , (3. " q (3) . (8.6.6)

Formula (8.6.4) follows immediately from (8.6.5) and (8.6.6). It is slightly more complicated to compute (8.6.1). Consider the generating function

00 1 _ (3qn Gt(z) = L 1 _ (3 Cn(z; (3lq)tn n=O

((3te iW , (3te- iw ; q) 00 _ ~ ((3qte iW , (3qte- iw ; q) 00 (l-(3)(teiw ,te-iW ;q)oo 1-(3 (qteiW,qte-iW;q)oo

= 1 _ (3t2 ((3qte 2W , (3qte- iw ; q) 00 • (8.6.7) ( ) (te2w , te-2W ; q) 00 8.7 Poisson kernels for the q-Racah polynomials 229

Then Kt(x, Y; ,8lq)

= 11 K(x, y, Z; ,8lq) G 13_1 (z) dz -1 t 2 (1- t2)(q,,8,,8;q)= I (,8e2il1,,8e2i

7r h (cos"p; 1, -1, q~, -q~, qt,~ ) x 1 ( ) d"p. o h cos"p;,~ eil1+i

8.7 Poisson kernels for the q-Racah polynomials

For the q-Racah polynomials Wn(x; q) == Wn(x; a, b, c, N; q) we shall give conditions under which the Poisson kernel

N L hn(q)Wn(x; q)Wn(y; q)tn, 0::; t < 1, (8.7.1) n=O and the so-called discrete Poisson kernel Z (q-Z; q)n L ( -N. ) hn(q)Wn(x; q)Wn(y; q), (8.7.2) n=O q ,q n

Z = 0,1, ... , N, are nonnegative for x, y = 0,1, ... , N. Let us first consider a more general bilinear sum PAx, y) == PAx, y; a, b, c, a, ry, K, M, N; q) Z (q-Z; q) = L ( -K. rhn(a,b,c,N;q)Wn(x;a,b,c,N;q) n=O q ,q n

X Wn(y; a, aba-I, ry, M; q), (8.7.3) 230 Further applications where z = 0,1, ... ,min(K,N) and N:::::: M. If a = a,ry = c and M = N, then (8.7.3) reduces to (8.7.2) when K = N, and it has the Poisson kernel (8.7.1) as a limit case. From the product formula (8.2.5) it follows that Wn(x; a, b, c, N; q)Wn(y; a, aba-I, ry, M; q) (bq, aqlc; q)n cn ~ ~ (q-n, abqn+\ q)r+s (q-X, cqx-N, q-Y, ryqy-N; q)r (aq, b)c; q n r=O~ ~s=O (-N)q ; q r+s ( q, aq, q -M ,ab ryqa -1 ,cql-s ; q ) r

x-N -1 I-x ) 1 r s (q , c q ; q s - cq - r+s-rs A x (q, bq, ac-1q; q)s 1 _ cq-S q r,s, (8.7.4) where A = [ q-S, qr-y, ryqr+y-M, aqr+l, bcqr+l. ] (8.7.5) r,s 5¢4 aqr+l , q r-M ,ab rya -1 q r+l , cq l+r-s ,q, q . Using (8.7.5) in (8.7.4) and changing the order of summation, we find that

x (_1)r+sq(r+s)(2N-r-s+ll/2-rs A B T,S T,S' (8.7.6) where >. q>,~ _q>,~ qr+s-N qr+s-z ] B -[ " , , . N -r-s (8.7.7) r,s - 5¢4 >.~, _>.~, abqN+r+s+2, qr+s-K' q, q , with>' = abq2r+2s+1. We shall now show that when K = N, ( N-z ) q ; q z-r-s A, (qr+s-z ql+r+s-z. ql+N-z. q abqN+z+2) Br,s = ab N+r+s+2. ) 2'1"1 , , , , . ( q ,q z-r-s (8.7.8) To prove (8.7.8) it suffices to show that

1 1 a, qa2, -qa2 , b w] 4¢3 [ 1 1 ; q,- a2, -a2, w aq

(wblaq, wlb; q)oo A, (b b' bl' Ib) ( ) 2'1"1, q,w a,q,w (8.7.9) w I aq,w;q 00 whether or not b is a negative integer power of q, provided the series on both sides converge. Since 1 - aq2k = 1 - qk + qk(l - aqk), the left side of (8.7.9) equals w(l - b) 2¢I(aq,b;w;q,wla)+ ( ) 2¢I(aq,bq;wq;q,wlaq) aq 1-w (wbla,wlb;q)oo ( I I ) (I ) 2¢1 b, bq; wb a; q, w b w a,w;q 00 8.7 Poisson kernels for the q-Racah polynomials 231

w(l- b)(wbla, wlb; q)oo rI-. (b b· bl· Ib) + ( I ) 2'1-'1, q, W a, q, w aq w aq,w;q 00

(wblaq, wlb; q)oo rI-. (b b· bl· Ib) = 2'1-'1, q, W a, q, w (8.7.10) (w I aq,w;q) 00 by (1.4.5). Also, for 0: = a the 5¢>4 series in (8.7.5) reduces to a 4¢>3 series which, by (2.10.4), equals ( M+1-y-s b-1 -1 -y-s. ) q _' "( q ,q s (b"(qr+s+y-M) s (qr M, b"(qr+1; q)s q-S, qr-y , b-1q-S, C"(-lqM+1-y-s ] [ X 4¢>3 l+r-s M+1-y-s b- -1 -y-s; q, q . (8.7.11) cq ,q ,,,(1 q From (8.7.6), (8.7.8) and (8.7.11) it follows that JDz(x,y; a,b,c, a,,,(,lV, AI, lV; q) 20 (8.7.12) for x = 0,1, ... , lV, y = 0,1, ... , AI, z = 0,1, ... , lV when 0 < q < 1, 0 < aq < 1, 0 ::; bq < 1, 0 < c < aqN and cq ::; "( < qM -1 ::; qN -1. Hence the discrete Poisson kernel (8.7.2) is nonnegative for x, y, z = 0,1, ... ,lV when 0 < q < 1, o < aq < 1, 0 ::; bq < 1 and 0 < c < aqN. If in (8.7.3) we write the sum with lV as the upper limit of summation, replace (q-Z; q)n by (tq-K; q)n and let K ----t 00, it follows from (8.7.6) that

Lt(x, y; a, b, c, 0:, ,,(, AI, lV; q) N = L tnhn(a, b, c, lV; q)Wn(x; a, b, c, lV; q)Wn(y; 0:, abo:-1, ,,(, AI; q) n=O x N-x (b 2.) (-x x-N -y y-M.) (b q,aq I c;q ) N ~ ~ a q ,q 2r+2s q ,cq ,q ,,,(q ,q r - (abq2,c1;q)N ~ ~ (abqN+2;q)r+s (q,o:q,q-M, ab,,(qo:-1, cql-s; q)r ( x-N -1 -x ) (1 r-s) q ,c q ; q s - cq A C (_t)r+s (r+s)(2N -r-s+1)/2-rs X (bq, q,ac- 1)q;q s ( 1-cq-S ) rs" rs q , (8.7.13) for x = 0,1, ... , lV, y = 0, 1, ... , AI with Ar,s defined in (8.7.5) and

\ \ ! \! r+s-N ] C - [ A, qA2, -qA2, q . t N-r-s r,s - 4¢>3 \! \! b N+r+s+2 ,q, q , (8.7.14) A2,-A2,a q where A = abq2r+2s+1. However, by Ex. 2.2,

rI-. [A, qA~, -qA~, b-1 ] ; q, tb 4'1-'3 A~ , -A~ , Abq (t, Aq; q)oo ( ) (8.7.15) ( bAb.) 2¢>l b,tb;tq;q,Aq t, q, q 00 for max(ltbl, IAql) < 1. Use of this in (8.7.14) yields

Gr,s = (t, abq2r+2s+2; q)N-r-s 2¢>1 (qN-r-s, tqN-r-s; tq; q, abq2r+2s+2) , (8.7.16) 232 Further applications from which it is obvious that er,s 2: 0 for 0 :::; t < 1, r + s :::; N when o :::; abq2 < 1. Combining this with our previous observation that Ar,s equals the expression in (8.7.11) when a = a, it follows from (8.7.13) that Lt(x,y;a,b,c,a,,,!,M,N;q) > 0 (8.7.17) for x = 0,1, ... ,N, y = 0,1, ... ,M, 0 :::; t < 1 when 0 < q < 1, 0 < aq < 1, o :::; bq < 1, 0 < c < aqN and cq :::; "! < qM-l :::; qN-l. In particular, the Poisson kernel (8.7.1) is positive for x,y = O,l, ... ,N, 0:::; t < 1 when o < q < 1, 0 < aq < 1, 0 :::; bq < 1 and 0 < c < aqN. For further details on the nonnegative bilinear sums of discrete orthogonal polynomials, see Gasper and Rahman [1984] and Rahman [1982].

8.8 q-analogues of Clausen's formula

Clausen's [1828] formula

F [ 2a, 2b, a + b ] (8.8.1) {2Fl [a +ai ~ ! ;z] r 3 2 2a + 2b, a + b + ! ;z , where Izl < 1, provides a rare example ofthe square of a hypergeometric series that is expressible as a hypergeometric series. Ramanujan's [1927, pp. 23- 39] rapidly convergent series representations of l/n, which have been used to compute n to millions of decimal digits, are based on special cases of (8.8.1); see the Chudnovskys' [1988] survey paper. Clausen's formula was used in Askey and Gasper [1976] to prove that

F [-n,n+a+2,!(a+1).1-x] > 0 () 3 2 a+1,!(a+3) '2 - 8.8.2 when a > -2, -1 :::; x :::; 1, n = 0,1, ... , which was then used to prove the positivity of certain important kernels involving sums of Jacobi polynomials; see Askey [1975] and the extensions in Gasper [1975a, 1977]. The special cases a = 2,4,6, ... of (8.8.2) turned out to be the inequalities de Branges [1985] needed to complete the last step in his celebrated proof of the Bieberbach conjecture. In this section we consider q-analogues of (8.8.1). Jackson [1940, 1941] derived the product formula given in Ex. 3.11 and additional proofs of it have been given by Singh [1959], Nassrallah [1982]' and Jain and Srivastava [1986]. But, unfortunately, the left side ofit is not a square and so Jackson's formula cannot be used to write certain basic hypergeometric series as sums of squares as was done with Clausen's formula in Askey and Gasper [1976] to prove (8.8.2). In order to obtain a q-analogue of Clausen's formula which expressed the square of a basic hypergeometric series as a basic hypergeometric series, the authors derived the formula a,b,abz,ab/z . a2,b2,ab,abz,ab/z ] { [ ]}2 [ 4¢3 b! b! b,q,q 5¢4 a q2, -a q2,-a a 2b2 ,ab! q2,-a b!q2,-a b; q, q , (8.8.3) 8.8 q-analogues of Clausen's formula 233 which holds when the series terminate. See Gasper [1989b], where it was pointed out that there are several ways of proving (8.8.3), such as using the Rogers' linearization formula (8.5.1), the product formula in §8.2, or the Rah­ man and Verma integral (8.4.10). In this section we derive a nonterminating q-analogue of Clausen's formula which reduces to (8.8.3) when it terminates. The key to the discovery of this formula is the observation that the proof of Rogers' linearization formula given in §8.5 is independent of the fact that the parameter n in the 21>1 series in (7.4.2) is a nonnegative integer. In view of (7.4.2) let J(z) = 21>1(a,(3;aql(3;q,zql(3), (8.8.4) which reduces to the 21>1 series in (7.4.2) when a = q-n and z = e-2iIJ . Tem­ porarily assume that Iql < 1(31 < 1 and Izl ::::; 1. From Heine's transformation (1.4.3) , (8.8.5)

Hence, if we multiply the two 21>1 series in (8.8.4) and (8.8.5) and collect the coefficients of zj, we get

(8.8.6) where q-k,(3,(3q-kla,a ] [ (8.8.7) Ak = 41>3 (3q-k, (32 q-k la, aql (3; q, q is a terminating balanced series. As in (8.5.5) we now apply (2.5.1) to the 41>3 series in (8.8.7) to obtain that 2 A - (aql(3,a ql(32;q)00 w: (2/(3· (3 2 k+l/(32 -k. 1(3) ( ) k - ( 2 1(3 1(32.) 8 7 a ,a,a"a q ,q ,q,q . 8.8.8 a q , aq , q 00 Using (8.8.8) in (8.8.6) and changing the order of summation we get the formula J2(z) - ((3z;q)oo f= (1- a 2q2k l(3)(a2/(3,a,a,(3;q)k - (zql (3; q)oo k=O (1 - a 21 (3)(q, aql (3, aql (3, a 2ql(32; q)k

(a2ql(32;q)zk (Zq)k [a2q2k+l/(32,ql(3 ] x (a2ql(3; q)zk /3 21>1 a 2 q2k+l 1(3 ; q, (3z . (8.8.9)

Observe that since the 21>1 series in (8.8.9) is well-poised we may transform it by applying the quadratic transformation formula (3.4.7) to express it as an 81>7 series and then apply (2.10.10) to get the transformation formula (k+l) 1> [a2q2k+l/(32,ql(3 (3] (azql(3;q)oo(-al(3z)kq 2 2 1 a2q2k+l 1(3 ; q, z = ((3zla; q)oo(aql (3z, azql (3; q)k

aqk, aqk+! 1(3, -aqk+! 1(3, _ aqk+ 1 1(3 ] [ X 41>3 a2q2k+l 1(3, azqk+l 1(3, aqk+l 1(3z ; q, q 234 Further applications

1 1 rI-. [ /3z, zq"2 , -zq"2, -zq ] X 4'f'3 ; q, q . (8.8.10) qz2, o: zqk+l, /3 zql-k /0: We can now substitute (8.8.10) into (8.8.9) and change the orders of summation to find that J2(Z) = (/3z,o:zq//3;q)oo f (0:; q)m(0:2q//32; qhm (zq/ /3, /3z / 0:; q)oo m=O (q, o:zq/ /3, o:q/ /3z, 0:2 q/ /3, o:q/ /3; q)m

X qm 6W5(0:2//3;0:,/3,q-m;q,o:qm+l//32) (0:, 0:2q/ /3, zq, o:zq; q)oo f (/3z, zq!, -zq!, -zq; q)m m + (o:q/ /3, 0:2q/ /3, zq/ /3,0:/ /3z; q)oo m=O (q, qz2, o:zq, /3zq/o:; q)m q X6 W 5(0:2 //3; 0:, /3, o:q-m / /3z; q, zqm+l /(3). (8.8.11) Summing the above 6W5 series by means of (2.7.1), we obtain the formula

{ ( )}2 (/3z, o:zq/ /3; q)oo 2¢>1 0:, /3; o:q / /3; q, zq / /3 = ( //3 /3 / . ) zq , Z 0:, q 00

rI-. [0:, o:q/ /32, o:q! //3, -o:q! //3, -o:q/ /3 ] X 5'f'4 o:q/ /3, 0:2q/ /32, o:q/ /3z, o:zq/ /3 ; q, q (0:, o:q/ /32, zq, zq, o:zq/ /3; q)oo +~~~~~=-~~~~~- (o:q//3,o:q//3,zq//3,zq//3,o://3z;q)oo

rI-. [/3z,Zq//3,Zq!,-Zq!,-Zq ] x 5'f'4 2 ; q, q , (8.8.12) zq, z q, /3zq/o:, o:zq/ /3 which gives the square of a well-poised 2¢>1 series as the sum ofthe two balanced 5¢>4 series. By analytic continuation, (8.8.12) holds whenever Iql < 1 and Izq/ /31 < l. To derive (8.8.3) from (8.8.12), observe that if 0: = q-n, n = 0,1, ... , then (0:; q)oo = 0 and (8.8.12) gives J2(Z) (/3z, zql-n//3;q)oo ¢> [q-n,q!-n//3,ql-n//32,_q!-n//3,_ql-n//3 ] = (/3zqn,zq//3;q)oo 5 4 ql-n//3,ql-2n//32, zql-n//3,ql-n//3z ;q,q

_ (/32,/32;q)n (.:.)n rI-. [q-n,/32qn,/3,/3z,/3/z. ] - (/3 , /3.),q n /3 5'f'4 /32 , /3 q2,-1 /3 q2,-1 /3' q, q (8.8.13) by reversing the order of summation. Since 2 J( ) = (/3 ;q)n (.:.)n rI-. [q-n,/32qn, (/3z)!, (/3/z)! . 1 z 4'f'3 1 1 , q, q (8.8.14) (/3); q n /3 /3q"2, -/3q"2, -/3 by (7.4.14), it follows from (8.8.13) that

rI-. [q-n,/32qn,(/3z)!,(/3/Z)!. l}2 { 4'f'3 /3 q2,-1 /3 q2,-1 /3 ,q,q 8.8 q-analogues of Clausen's formula 235

(8.8.15) for n = 0,1, ... , which is formula (8.8.3) written in an equivalent form. Now note that

2¢1(a,~;aq/~;q,zq/~) (z(aq)!, -z(aq)!, zqa! I~, -zqa! I~; q)oo (zq!, -zq!, zq/~, -azq/~; q)oo

X 8 W 7 ( -az/~; a!, -a!, (aq)! I~, -(aq)! I~, -z; q, -zq), (8.8.16) by (3.4.7) and (2.10.1), and set a = a!,b = (aq)!/~ to obtain from (8.8.12) the following q-analogue of Clausen's formula: (a2z2q, b2z2q; q2)00 _1 }2 { (2 2b2 2 . 2) 8 W7( -abzq 2; a, -a, b, -b, -z; q, -zq) z q,a z q,q 00

_ (azq! Ib, bzq! la; q)oo ,j., [ a2, b2, ab, -ab, -abq! . ] - 1 1 5¥'4 1 1 1, q, q (zq'i lab, abzq'i; q)oo a2b2, abq'i, abzq'i, abq'i I z (zq, zq, a2, b2; q)oo + 1 1 1 1 (abq'i,abq'i,ablzq'i,abzq'i;q)oo

azq'i1 Ib, bzq'i1 la, zq'i,1 -zq'i, 1 -zq 1 X 5¢4 [ 2 1 3 ; q, q , (8.8.17) zq,z q,abzq'i,zq'ilab where Iql < 1 and Izql < 1. To see that (8.8.17) is a nonterminating q-analogue of Clausen's formula, it suffices to replace a by qa, b by qb and let q ----+ 1-; then the left side and the first term on the right side of (8.8.17) tend to the left and right sides of (8.8.1) with z replaced by -4z(l- Z)-2 and so, by (8.8.1), the second term on the right side of (8.8.17) must tend to zero. It is shown in Gasper and Rahman [1989] that the nonterminating exten­ sion (3.4.1) of the Sears-Carlitz quadratic transformation can be used in place of (8.8.10) to derive the product formula ( ) ( ) (az,abzlc;q)oo 2¢1 a,b;c;q,z 2¢1 a,aq I c;aq I b;q,z = ( b I ) z, z c; q 00

,j., [a, clb, (aclb)!, -(aclb)!, (acqlb)!, -(acqlb)! ] x 6¥'5 aq I b, c, ac I b, az, cq I bz ; q, q + (a,clb, az, bz,azqlc;q)oo (c,aqlb,z,z,clbz;q)oo

,j., [z, abzlc, z(ablc)!, -z(ablc)!, z(abqlc)!, -z(abqlc)! ] x 6¥'5 q, q , az, bz, azqI c, bzq I c, abz 21 c ; (8.8.18) where Izl < 1 and Iql < 1. This formula reduces to (8.8.12) when a = a, b = ~, c = aql ~ and z is replaced by zq/~. 236 Further applications

By applying various transformation formulas to the 2¢1 series in (8.8.12) and (8.8.18), these formulas can be written in many equivalent forms. For instance, by replacing b in (8.8.18) by c/b and applying (1.5.4) we obtain 2¢2(a, b; c, az; q, cz/b) 2¢2(a, b; abq/c, az; q, azq/c) _ (z,az/b;q)oo " [a,b,(ab)!,-(ab)!,(abq)!,-(abq)!. ] - ( /b) 6'1'5 / / ' q, q az, z ; q 00 abq c, c, ab, az, bq z + (a, b, cz/b, azq/c; q)oo (c, abq/c, az, b/ z; q)oo " [z, abz, z(a/b)!, -z(a/b)!, z(aq/b)!, -z(aq/b)! ] ( ) X 6'1'5 ; q, q , 8.8.19 az, cz/b, aqz/c, zq/b, az2/b where max(lql, lazq/cl, Icz/bl) < 1. If we replace a, b, z in (8.8.19) by qa, bb, z/(z - 1), respectively, and let q ----+ 1-, we obtain the Ramanujan [1957, Vol. 2] and Bailey [1933, 1935a] product formula 2Fl(a, b; c; z) 2Fl(a, b; a + b - c + 1; z) = 4F3(a, b, (a + b)/2, (a + b + 1)/2; c, a + b, a + b - c + 1; 4z(1 - z)), (8.8.20) where Izl < 1 and 14z(l- z)1 < 1. This is an extension of Clausen's formula in the sense that by replacing a, b, c, 4z(1 - z) in (8.8.20) by 2a, 2b, a + b + ~,z, respectively, and using the quadratic transformation (Erdelyi [1953, 2.11 (2)]) 1 1 2Fl(2a, 2b; a + b + "2; z) = 2Fl(a, b; a + b + "2; 4z(1 - z)), (8.8.21) we get (8.8.1). See Askey [1989d].

8.9 Nonnegative basic hypergeometric series

Our main aim in this section is to show how the terminating q-Clausen for­ mula (8.8.3) can be used to derive q-analogues of the Askey-Gasper inequalities (8.8.2) and of the nonnegative hypergeometric series in Gasper [1975a, Equa­ tions (8.19), (8.20), (8.22)]. As in Gasper [1989b], let us set

'Y = q2b,al = qb,a2 = qbeiO,a3 = qbe-iO,b1 = q2b,b2 = _qb, -n n+a d !(a+l) d H! 1 Cl = q ,C2 = q ,1 = q2 = - 2, el = q 2 = -e3, x = q, W = , in the r = 3, s = t = u = k = 2 case of (3.7.9) to obtain the expansion 8.9 Nonnegative basic hypergeometric series 237

q-j,qj+2b, qb,qbeiO, qbe-iO ] x 5rP4 [ 2b HI HI b ;q,q, (8.9.1) q ,q 2 , -q 2 , -q where, as throughout this section, n = 0,1, .... By Ex. 2.8 the 4rP3 series in (8.9.1) equals zero when n - j is odd and equals a-2b. 2) ( q, q ,q k 2k(n-2k+b+l/2) (q2n-4k+a+l, q2n-4k+2H2; q2)k q when n - j = 2k and k = 0,1, .... Hence, using (8.8.3) to write the 5rP4 as the square of a 4rP3 series, we have q-n, qn+a, qb, qbeiO, qbe-iO ] [ 5rP4 2b I (a+l) I (a+l) b ; q, q q ,q2 ,-q2 ,-q

[n/2] bIb 1 (_l)n(q-n,qn+a,q +"2,-q +"2;q)n-2k = ~ (q, q! (a+I), _q! (MI), qn-2k+2b; q)n-2k

(8.9.2)

(8.9.3) when a ~ 2b > -1 and 0 < q < 1. By setting x = cos e and letting q ----+ 1-, it follows from (8.9.3) that -n, n + a, b 1 - x] F [ . >0 -1 :s; x :s; 1, (8.9.4) 3 2 2b,~(a+1)'-2- - , when a ~ 2b > -1 which shows that (8.9.3) is a q-analogue of (8.9.4). When a = a + 2 and b = ~(a + 1), (8.9.4) reduces to (8.8.2). Special cases of (8.9.4) were used by de Branges [1986] in his work on coefficient estimates for Riemann mapping functions. Another q-analogue of (8.9.4) can be derived by using (8.9.1), (8.8.3) and Ex. 2.8 to obtain q-n, qn+a, qb, _qb, q!aeiO, q!ae-iO ] [ 6rP5 2b l(a+l) l(a+l) Ia Ia; q, q q ,q2 ,-q2 ,-q2,-q2

n + 1 1 ·0 1 "0 (q-n, qn a, q"2 a, q"2a e 2 ,q"2 ae-2 ; q)j (-l)jqj+(~) = .?= (q q!(a+l) _q!(a+l) _q!a qJ+a-l. q)" J=O' , " 'J 238 Further applications

( a-2b. 2) X q, q ,q k q2k 2 +k+2kb (qa, q2b+1; q2h q2k-n, qn+2k+a, qk+-iae!il:l, qk+-iae-!il:l ]}2 { [ (8.9.5) X 4¢3 q2k+!(a+l) , _q2k+!(a+1), _q2k+!a ; q, q , which shows that

(8.9.6) when a ~ 2b > -1 and 0 < q < 1. The expansions (8.12) and (8.17) in Gasper [1975a] are special cases of the q ----+ 1- limit cases of (8.9.2) and (8.9.5), respectively, when (7.4.14) and Gasper [1975a, (8.10)] are used. A q-analogue of the expansion (Gasper [1975a, 1989b])

F [-n,n+a+2,~(a+1)'(1_x2)(1_ 2)] 3 2 a+1,~(a+3)' y n n!(n+a+2).(0+2). _ '"' J 2 J (1 _ 2)j ~ 1). Y - j=O J."( n _ J.')' (0+3)2 j (.J + a + J

x { j!(n - j)! c!(n+1\X)CH !(n+2)(y)}2 (8.9.7) (a + 1)j(2j + a + 2)n-j J n-J is easily derived by employing (3.7.9), (8.8.3) and (7.4.14) to obtain ¢ [q-n, qn+n+2, q! (n+1), q! (n+1) e2il:l, q! (n+1) e-2il:l, 7 6 qn+1, q! (n+3), _q! (n+3) , _q! (n+2) ,

q! (n+2) e2ir, q! (n+2) e-2ir ] -q21(n+2) ,-q2 1(n+1) ; q, q

n (q-n qn+n+2 q! (n+2) q! (n+2) e2ir q! (n+2) e-2ir. q) . .. (j) = '"' ' , , , , J (-1)1 J+ 2 ~ (q q!(n+3) _q!(n+3) _q!(n+2) qj+n+l. q). q J=O ' , , , 'J

x { (q; q)j(q; q)n-j !(j+n+~) (qn+1; q)j(q2Hn+2; q)n-j q

x Cj (COS (); q! (n+1) Iq)Cn- j (COS T; qH! (n+2) Iq) } 2, (8.9.8) which is obviously nonnegative for real () and T when a > -2. If we proceed as in (8.9.5), but use the q-Saalschiitz summation formula (1.7.2) instead of Ex. 2.8, we find that 8.10 Applications in the theory of partitions of positive integers 239

.' (j) { [qj-n qj+n+a q!HiadiO q!Hiae-!iO ] ( I)J J(b+l)+ 2..!, " , • }2 x - q 4 '/-'3 J+1(a+l) J+1(a+l) J+1a ,q, q q 2 ,-q 2 ,-q 2 (8.9.9) and hence

(8.9.10) for real () when a ::::: c ::::: b, a + b > c > 0 and 0 < q < 1. By letting q ---+ 1 - in (8.9.10) it follows that

F [ -n, n + a, ~a, b .1 - x] > 0 (8.9.11) 4 3 ~ (a + 1), c, a + b - c' -2- - for -1 :::; x :::; 1 when a ::::: c ::::: band a+b > c > 0, which gives the nonnegativity of the 3F2 series in Gasper [1975a, (8.22)] when a = a + ~,b = ~a + £ and c = a + 1. Also see Gasper [1989d].

8.10 Applications in the theory of partitions of positive integers

In the applications given so far we have dealt almost exclusively with orthogo­ nal polynomials which are representable as basic hypergeometric series. These are important results and most of them have appeared in print during the last thirty years. They constitute the main bulk of applications as far as this book is concerned. However, our task would remain incomplete if we failed to mention some of the earliest examples where basic hypergeometric series played crucial roles. The simplest among these examples is Euler's [1748] enu­ meration p( n) of the partitions of a positive integer n, where a partition of a positive integer n is a finite monotone decreasing sequence of positive integers (called the parts ofthe partition) whose sum is n. To illustrate how p(n) arises in a q-series let us consider a formal series expansion of the infinite product (q; q)~} in powers of q:

00

k=O

00 (8.10.1) where 240 Further applications

n = kl . 1 + k2 . 2 + ... + kn . n, (8.10.2) p(O) = 1 and, for a positive integer n, p(n) is the number of partitions of n into parts::; n. In the partition (8.10.2) of n there are km m's and hence 0::; km ::; n/m, 1 ::; m ::; n. For small values of n, p(n) can be calculated quite easily, but the number increases very rapidly. For example, p(3) = 3, p( 4) = 5, p(5) = 7, but p(243) = 133978259344888. Hardy and Ramanujan [1918] found the following asymptotic formula for large n:

p(n) ~ 4n~ex+ C;) '] (8.10.3) Also of interest are the enumerations of partitions of a positive integer n into parts restricted in certain ways such as: (i) PN(n), the number of partitions of n into parts::; N, which is given by the generating function

00 (q; q)"i/ = 2::PN(n)qn, (8.10.4) n=O where n = kl . 1 + k2 ·2+ ... + kN . N has km m's ; (ii) Pe(n), the number of partitions of an even integer n into even parts, generated by

00 (q2; q2)~;;} = II (1 _ q2k+2)-1 k=O 00 = 2::Pe(n)qn; (8.10.5) n=O (iii) Pdist(n), the number of partitions of n into distinct positive integers, generated by

1 1 (-q; q)oo = 2:: 2:: ... qk 1 ·Hk2 ·2+ .. k1=O k2 =O 00 = 2:: Pdist (n )qn, (8.10.6) n=O where n = kl . 1 + k2 ·2+ ... + kn . n, 0 ::; ki ::; 1, 1 ::; i ::; n, and (iv) po(n), the number of partitions of n into odd parts, generated by

00 (q;q2)~1= II(1-q2k+l)-1 k=O 00 (8.10.7)

Euler's partition identity Pdist(n) = po(n) (8.10.8) 8.10 Applications in the theory of partitions of positive integers 241 follows from (8.10.6), (8.10.7) and the fact that 1 1)-1 (-q; q)OO = (q'l, -q'l; q 00 = (q; q2);;}. (8.10.9)

Other combinatorial identities of this type can be discovered from q-series identities similar to, but perhaps somewhat more complicated than, (8.10.9). For example, let us consider Euler's [1748] identity involving the pentagonal numbers n(3n ± 1)/2:

00

n=-oo 00 00

n=l n=l which is given in Ex. 2.18. A formal power series expansion gives

00

k=O 1 1 = L L ... (_I)kl+k2+··qkl·Hk2·2+ .. , (8.10.11)

k1=O k2 =O which differs from the multiple series in (8.10.6) only in the factor (-I)kl+k2+··. This factor is ±1 according as the partition has an even or odd number of parts. Denoting these numbers by Peven (n) and Podd (n), re­ spectively, we find that

00 (q; q)oo = 1 + L [Peven(n) - pOdd(n)] qn. (8.10.12) n=l From (8.10.10) and (8.10.12) it follows that

_{(-I)k forn=k(3k±I)/2, Peven (n) - Podd (n) - (8.10.13) o otherwise. Thus Euler's identity (8.10.10) expresses the important property that a positive integer n which is not a pentagonal number of the form k(3k ± 1)/2 can be partitioned as often into an even number of parts as into an odd number of parts. However, if n = k(3k ± 1) /2, k = 1,2, ... , then Peven (n) exceeds Podd (n) by (-I)k. See Hardy and Wright [1979], Rademacher [1973] and Andrews [1976, 1983] for related results. Two of the most celebrated identities in combinatorial analysis are the so­ called Rogers-Ramanujan identities (2.7.3) and (2.7.4) which, for the purposes of the present discussion, we rewrite in the following form:

00 n 2 1 1 + L -q- = ------(8.10.14) n=l (q;q)n fi (1- q5k+ 1) (1- q5k+4) k=O 242 Further applications

00 qn(n+l) 1 1 + L ( ) = (8.10.15) n=l q; q n TI (1 - q5k+2) (1 - q5k+3) k=O It is clear that the infinite product on the right side of (8.10.14) enumerates the partitions of a positive integer n into parts of the form 5k + 1 and 5k + 4, while that on the right side of (8.10.15) enumerates partitions of n into parts of the form 5k + 2 and 5k + 3, k = 0,1, .... Following Hardy and Wright [1979] we shall now give combinatorial inter­ pretations of the left sides of the above identities. Since k2 = 1 + 3 + 5 + ... + (2k - 1), we can exhibit this square in a graph of k rows of dots, each row having two more dots than the lower one. We then take any partition of n-k2 into at most k parts with the parts in descending order, marked with x's and placed at the ends of the rows of dots to obtain a partition of n into parts with minimal difference 2. For example, when k = 5 and n = 32 = 52 + 7 we add 4 x's to the top row, 1 each to the 2nd, 3rd and 4th rows, counted from above. This gives the partition 32 = 13 + 8 + 6 + 4 + 1 displayed in the graph below. • • • • • • • • x x x x • • • • • • x • • • • • x

• • x • The identity (8.10.14) states that the number of partitions of n in which the differences between parts are at least 2 is equal to the number of partitions of n into parts congruent to 1 or 4 (mod 5). Observing that k(k + 1) = 2 + 4 + 6 + ... + 2k, a similar interpretation can be given to the left side of (8.10.15). Since the first number in the above sum is 2, one deduces that the partitions of n into parts not less than 2 and with minimal difference 2 are equinumerous with the partitions of n into parts congruent to 2 or 3 (mod 5). For more applications of basic hypergeometric series to partition the­ ory, see Andrews [1976-1988], Fine [1948, 1988], Andrews and Askey [1977], and Andrews, Dyson and Hickerson [1988]. Additional results on Rogers­ Ramanujan type identities are given in Slater [1951, 1952a], Jain and Verma [1980-1982] and Andrews [1975a, 1984a,b,c,d].

8.11 Representations of positive integers as sums of squares One of the most interesting problems in number theory is the representations of positive integers as sums of squares of integers. Fermat proved that all 8.11 Representations of positive integers as sums of squares 243 primes of the form 4n + 1 can be uniquely expressed as the sum of two squares. Lagrange showed in 1770 that all positive integers can be represented by sums of four squares and that this number is minimal. Earlier in the same year Waring posed the general problem of representing a positive integer as a sum of a fixed number of nonnegative k-th powers of integers (positive, negative, or zero) with order taken into account and stated without proof that every integer is the sum of 4 squares, of 9 cubes, of 19 biquadrates, 'and so on'. More than 100 years later Hilbert [1909] proved that all positive integers are representable by s kth powers where s = s(k) depends only on k. For an historical account of the Waring problem, see Dickson [1920]' Grosswald [1985], and Hua [1982]. To illustrate the usefulness of basic hypergeometric series in the study of such representations we shall restrict ourselves to the simplest cases: sums of two and sums of four squares, where it is understood, for example, that n = xi + x§ = yi + y§ are two different representations of n as a sum of two squares if Xl -=1= Yl or X2 -=1= Y2· Let r2k (n) be the number of different representations of n as a sum of 2k squares, k = 1,2, .... We will show by basic hypergeometric series techniques that, for n ~ 1,

r2(n) = 4(dl (n) - d3 (n)), (8.11.1) r4(n) = 8 L d, (8.11.2) dln,4id where di(n), i = 1,3, is the number of (positive) divisors of n congruent to i (mod 4) and the summation in (8.11.2) indicates the sum over all divisors of n not divisible by 4. The numbers 4 and 8 in (8.11.1) and (8.11.2), respectively, reflect the fact that r2(1) = 4 since 1 = 02 + (±1)2 = (±1)2 +02, and r4(1) = 8 since 1 = 02 + 02 + 02 + (± 1)2 = 02 + 02 + (± 1) 2 + 02 = 02 + (± 1) 2 + 02 + 02 = (±1)2 + 02 + 02 + 02. Both of these results were proved by Jacobi by means of the theory of elliptic functions, but the proofs below are based, as in Andrews [1974a], on the formulas stated in Ex. 5.1, 5.2, and 5.3. Combining Ex. 5.1 and 5.2 we have

(8.11.3)

However, the bilateral sum on the left side is clearly a generating function of (-1 )nr2 (n) and so it suffices to prove that

00 00 (-1 )nqn(n+l)/2 "[dl(n) - d3 (n)](-qt = " (8.11.4) ~ ~ l+qn n=l n=l By splitting into odd and even parts and then by formal series manipulations, we find that 00 (-1 )nqn(n+l)/2 00 qr(2r+l) 00 q(m+l)(2m+l) L 1 + qn = L 1 + q2r - L 1 + q2m+l n=l r=l m=O 00 00 00 00

r=l m=r m=Or=m+l 244 Further applications

(Xl (Xl

m=Or=l

(Xl (Xl = L L(-lr (q(4m+l)r _ q(4m+3)r) m=Or=l

(Xl = L(-q)n(d 1 (n) - d3 (n)), (8.11.5) n=l which completes the proof of (8.11.4). To prove (8.11.2) we first replace q by -q in Ex. 5.1 and 5.3 and find that

(Xl nqn (8.11.6) = 1 + 8 ~ 1 + (_q)n ' where the last line is obtained from the previous one by expanding (1 + (_q)n)-2 and interchanging the order of summation. Now,

nqn nIl - (Xl --+ nq ( -----) - L 1 - qn L 1 + qn 1 _ qn n=l even n~2

(8.11. 7)

Thus,

(8.11.8) Exercises 245

Since T4(O) = 1, this immediately leads to (8.11.2). For a direct proof of (8.11.8) based only on Jacobi's triple product identity (1.6.1), see Hirschhorn [1987]. Ex. 5.1 and 5.4 can be employed in a similar manner to show that

T8(n) = 16( _l)n~) _1)dd3, n ~ 1. (8.11.9) din

See Andrews [1974a]. Some remarks on other applications are given in Notes §8.11.

Exercises

8.1 Prove for the little q-Jacobi polynomials Pn(x; a, b; q) defined in (7.3.1) that

(i)

(ii) Pn(x;a,b;q)Pn(y;a,b;q) n (b ) n (q-n abqn+1. q) m = (-bq)-nq-(2) q;qn L ' 'm(_byq2)m q(2) (aq; q)n m=O (q, bq; q)m

m (q-m abqm+1. q) X '""'~ ( ' ) 'k ( xq )k 2'/-'1A. ( q k-m ,xq;;b 0 q, b-1 y -1) . k=O q, aq; q k

8.2 Derive the following product formula for the big q-Jacobi polynomials defined in (7.3.10): 246 Further applications

8.3 Prove that

00 (abq,aq;q)n(1-abq2n+l) (t)n (b') (1 _ b) - Pn(qX; a, b; q)Pn(qY; a, b; q) n=OL q, q, q n a q aq

00 min(x,y) (-X -Y ) -s = (t, abq2; q)oo L L q ,q ; q sa q(x+y)(r+s) r=O s=O (q, bq; q)s (q, aq; q)r

X 2¢1 (0,0; qt; q, abq2r+2s+2) q-2rs-s2 tr+s, and that this gives the positivity of the Poisson kernel for the little q-Jacobi polynomials for x, y = 0,1, ... , 0:::; t < 1 when 0 < q < 1,0 < aq < 1 and O

X n (q-n, abqn+l, bqN -x+l, bqN -Y+\ q)m m fo (q, bq, abqN+2, bqN+l; q)m q

~ (b-1q-N-m-\ qh(1 - b-1q2k-N-m-l )(a-1b-1q-N-m-l, b-1q-m; q)k X 6 (q;q)k(1- b-1q-N-m-l)(aq,q-N;q)k

X (q-m, q-X, q-Y; q)k ( b)k (x+y+2m-k)k (b-1q-N, b-1qx-N-m, b-1qy-N-m; q)k a q ,

(ii) (bq, abqN+2; q) n (bqN+l)-n Qn(X; a, b, N; q)Qn(Y; a, b, N; q) = ( N) aq,q- ;q n n (q-n abqn+l bqN+l-x-y. q) m (x+l)m X L " , q m=O (q, bq, abqN+2; q)m

m (-m b m+l -x -y ) x'"""' q ,a q ,q ,q ;q k (bqN-X+2)k 6 (q, aq, q-N, bqN+l-x-y; qh qk-m,qk-x,bqN+1- X ] [ X 3¢2 0, bqN+l-x-y+k ; q, q .

8.5 Prove for the q-Racah polynomials that (i) Exercises 247

(ii)

(aqX+1, aqY+\ q) m m X q (aq, aq; q)m

-1 -N-m-1 -1 -m -lb-1 -N-m-1 X 10 W.9 ( ca q ;ca q ,a q , q-m, q-X, cqx-N, q-Y, cqy-N; q, qba-1) .

8.6 Show that (i) VVn(x;a,b,c,lV;q)VVn(y;a,b,c,A1;q) (bq,aqc-\q)ncn t (q-n,abqn+l,qx-N,qy-M;q)m (aq, bcq; q)n m=O (q, bq, aqc-l, c-1 ; q)m

x W. (cq-m. ca-1q-m b-1q-m q-m q-X cqx-N q-Y cqy-M. q ab M+N+3) 10 9 , , '" " , , q , where n = 0,1, ... , min(A1, lV), x = 0,1, ... ,lV, and Y = 0, 1, ... , A1; (ii) VVn(x;a,a,c,lV;q)VVn(y;a,a,c,lV;q) 2 2 (aqc-l,a qN+2;q)n (ac-1qN+1)-n ~ (q-n,a qn+l,aqx+\q)m (acq, q-N; q)n ~o (q, a2qN+2, aq; q)m

(aq1+y-x, ac-1qN-x-y+\ q)m m X q (aqc-1, aq1-x;q)m

x lO VV9 (aq-X;a2qm+l,q-m,aqN-x+l,c-1q-X,q-X,q-Y,cqy-N;q,q). 8.7 For the q-Hahn polynomials prove that (i) 248 Further applications

x Qn(x; a, b, N; q)Qn(Y; a, b, M; q)

Z z-r (-Z) (b 2) (-X -Y ) (x-N y-M ) = L L q ; q r+s a q ; q 2r+2s q , q ; q r q , q ; q s r=O s=O (q-M, q-N, abqN+2; q)r+s (q, aq; q)r(q, bq; q)s

( N-z ) X q ; q z-r-s (_1r+sa-Sq(r+s)(2N-r-s+1)/2-(X+Y+1)S (abqN+r+s+2; q)z-r-s

x 21>1 (qr+s-z, q1+r+s-z; q1+N -z; q, abqN +Z+2) 2 O.

for x = 0,1, ... ,N, Y = 0,1, ... ,M, z = 0,1, ... ,N, when 0 < q < 1, o < aq < 1, 0 :::; bq < 1 and N :::; M; (ii)

N (abq, aq, q-N; q)n(1- abq2n+1) ( t) n (n) " -aq qNn- 2 ~ (q, bq, abqN+2; q)n(1 - abq)

x Qn(X; a, b, N; q)Qn(Y; a, b, M; q) = t "'I: (abq2; qhr+2s (q-X, q-Y; q)r (qx-N, qy-M; qt r=O s=O (q-M,abqN+2;q)r+s (q,aq;q)r(q,bq;q)s

x (t, abq2r+2s+2; q)N-r-s 21>1 (qN-r-s, tqN-r-s; qt; q, abq2r+2s+2) x a-s q(r+s)(2N -r-s+l)/2-(x+y+1)s (-tr+s > 0,

for x = 0, 1, ... ,N, Y = 0,1, ... ,M, 0 :::; t < 1 when 0 < q < 1, 0 < aq < 1, 0 :::; bq < 1 and N :::; M. (Gasper and Rahman [1984]) 8.8 Prove that t (aq, abq, ~cq; q)n( -c)-n An(Z)qG) n=O (q, bq, aqc 1; q)n(abq; qhn

x Wn(x; a, b, c, N; q)Wn(y; a, aba-I, ,,(, M; q)

= tI: (q-Z;q)r+s (q-X,cqx-N, q-Y, "(qy-M; q)r r=Os=O (q-N;q)r+s (q,aq,q-M,ab,,(qa- 1, cq1-s; q)r

(qx-N, c-1q-X; q) s (1 - cqr-s) -rs

X (bq, q,aqc-1);q s (1- cq-S ) q Ar' s ILr+s, where z = 0,1, ... ,N, N:::; M,

An(z) = ~ (q-Z; q)n+k ILn+k ~ (q,abq2n+2;qh '

and {lLr} is an arbitrary complex sequence, Ar,s being the same as in (8.7.5). Exercises 249

8.9 Deduce from Ex. 8.8 that Z (abq, aq, d, e, bcq, q-Z; q)n (1 - abq2n+l) abqZ+2 ( )n ~ (q, bq, abq2ld, abq2 Ie, aqlc, abqz+2; q)n(1- abq) cde

x Wn(x; a, b, c, N; q)Wn(y; a, aba-I, ,,(, M; q)

= ( a b q 2 ,ab q 21d e; q) Z LZ Lz-r (-Zq , d ,e; q ) r+s (-xq , cq x-N ; q ) r (abq2 I d, abq2 I e; q) Z r=O s=o (q-N, deq-z-l lab; q)r+s (q, q-M; q)r (q-Y, "(qy-M; q)r (qx-N, c-1q-X; qt (1 - cqr-s) r+s-rs A x (aq, ab,,(qa-1, cql-s; q)r(q, bq, aqlc; q)s(1 _ cq-s) q r,s' 8.10 Use (7.4.14) to show that (8.8.3) is equivalent to the formula {Cn(cos8; fJlq)}2 _ (fJ2,fJ2;q)n -n [q-n, fJ2qn,fJ,fJe2i6, fJe-2i6 . ] - ( q,q,qn.) fJ 5¢4 fJ2 , fJ!q2,- fJ! q2,- fJ ' q, q , where n = 0,1, .... 8.11 In view of the product formula (8.4.10) and Gegenbauer's [1874, 1893] addition formula for ultraspherical polynomials (see also Erdelyi [1953, 10.9 (34)] and Szego [1975, p. 98]) it is natural to look for an expansion of the form Pn(z; a, aq~, -a, -aq~ Iq) n = L Ak,n( 8, ¢) Pk (z; aei6+i , ae-i6 - i, aei6 - i, aei-i6 Iq), k=O where Pn(z; a, b, c, dlq) is the Askey-Wilson polynomial. By multiplying both sides by W(Z', aei6+i , ae-i6 - i , aei6 - i , aei-i6)

and then integrating over Z from -1 to 1, show that (q; q)n (a4qn , a4q-l, a2q~, _a2q~, _a2; q) an-Tn A (8 ¢) - Tn Tn,n, - (q; q)Tn(q; q)n-Tn (a4q-l; q)2Tn (a2q~, -a2q~, -a2; qt x Pn-Tn (x; aqTn/2, aq(Tn+l)/2, _ aqTn/2, _aq(Tn+1)/2Iq)

X Pn-Tn (Y; aqTn/2, aq(Tn+l)/2, _ aqTn/2, _aq(Tn+l)/2Iq) .

(Rahman and Verma [1986b]) 8.12 Prove the inverse of the linearization formula (8.5.1), namely,

min(Tn,n) CTn+n(x; fJlq) = L b(k, m, n)CTn-k(x; fJlq)Cn-k(x; fJlq), k=O 250 Further applications

where b(k, m, n) = (q; q)m(q; q)n(f3; q)m+n ((3; q)m((3; q)n(q; q)m+n ((3-2 q-m-n,(3-1;q)k (1- (3-2 q2k-m-n) ( 2 _l)k X (q,(3-1 q1-m-n;q)d1 -(3-2q-m-n) (3q . 8.13 Give alternate derivations of (8.6.3) and (8.6.4) by using the q-integral representation (7.4.7) of Cn(x; (3lq) and the q-integral formula (2.10.19) for an 8(P7 series. 8.14 By equating the coefficients of ei (m+n-2k)£! on both sides of the lineariza­ tion formula (8.5.1) show that (i) (q-m-n, (3-1 q1-n; q) k (q-n, (3-1 q1-m-n; q)k

[(3, q-k, q-m, q-n, (3-1 q-m-n , X (3-1 q,1-k (3-1 q 1-m , (3-1 q1-n, (3-2 q1-m-n, qk-m-n: (3-2 q1-m-n, (3-2 q2-m-n, (3-1 q2-m-n .q q2. q] (3-1 q1+k-m-n : q1-m-n, q -m-n , " '(3 ,

for k = 0,1, ... , n, and (ii)

-m -n (3-1 -m-n, [(3, q-k, q , q , q x (3-1 q1-k, (3 -1 q 1-m , (3 -1 q 1-n , (3-2 q1-m-n, qk-m-n: (3-1 q1+k-m-n : for k = n, n + 1, ... ,n + m, where is the bibasic series defined in §3.9. (Gasper [1985]) 8.15 Show that, by analytic continuation, it follows from Ex. 8.14 that a, b, c, d q2] [ 4 rP3 bq/a , cq/a , dq/a ; q, a2 (a/d, bq/d, cq/d, abc/d; q)oo (q/d, ab/d, acid, bcq/d; q)oo

1 1 1 1 x 12WU(bc/d; (bcq/ad) 2 ,-(bcq/ad)2, q(bc/ad) 2 ,-q(bc/ad)2, ab/d, acid, a, b, c; q, q/a), Exercises 251

where at least one of a, b, e is of the form q-n, n = 0, 1, .... (Gasper [1985]) 8.16 From Ex. 8.15 deduce that

lOW9 (a; q!, -q!, -q, a2/b, a2/e, b, e; q, q/a)

_ (aq, a2/b, a2/ e, aq/be; q) 00 [a, b, e, bela - (a2, aq/b, aq/e, a2/be; q)oo 4cP3 bq/a, cq/a, beq/a2 where one of a, b, c is of the form q-n, n = 0,1, .... 8.17 Prove that a2, b2, abz, ab/z . { [ 2 2]}2 4cP3 a 2b2 q, -ab ,-a b' q q ,q _ [a2, b2, ab, abz, ab/z . ] - 5cP4 a 2b2 ,ab! q2, -a b!q2,-a b' q, q and

when the series terminate. (Gasper [1989b]) 8.18 With the notation of Ex. 7.37 prove that Dq [(q1-veili,q1-ve-ili;q)2vUn(cosB)] :s: ° when v > - ~ , ). ;::: 0, °< q < 1, B is real and n = 1, ... , r. (Gasper [1989b]) 8.19 Prove the expansion formulas

(i) £ (x'iw) = (q; q)oow-V q, (qV;q)oo(_qw2;q2)oo

00 x L in(1- qv+n)qn2 /4 J~~n(2w; q)Cm(x; qVlq), n=O 2 (ii) £ ( .' ) _ (-w ; q2)oo ~ n(n-2a-1)/4(' )n q x, zw - ( . . 1/2) ~ q ZW -ZW, q 00 n=O (_q(a+!3+ 1)/2; q1/2)n(qa+I3+\ q)n X ~----~~~~7-----~- (qa+!3+1; q)zn q(,6+n+1)/2 _q(a+n+1)/2 1/2.] (,6) X 2cP1 [ q(a+,6+2n+2)/2 ; q ,ZW Pnn, (xlq).

(See Ismail and Zhang [1994]) for (i), and Ismail, Rahman and Zhang [1996] for (ii).) 252 Further applications

8.20 Prove the following linearization formula Pm(X; a, q/a, -b, -q/blq)Pn(x; a, q/a, -b, -q/blq) 2min(m,n) L CkPm+n-k(X; a, q/a, -b, -q/blq), k=O

with a,b > O. (Koelink and Van der Jeugt [1998], after transforming their formula to a symmetric form in m and n) 8.21 Prove the following convolution identity for the Askey-Wilson polynomi­ als:

m+n bm- k [m+n] (b2;q)n(a2b2e2qm+n-\q)n(e2;qh ~ k q (e2, b2e2qk-l; q)k(b2e2q2k; q)m+n-k q-k b2e2qk-l q-n a2b2qn-l ] [ x 4¢3 b< a2b2e2q~+n-l, q-m-n ; q, q -1 -1 2 xPk ( W2 +2 W ;bWl,b/wl,es,e/slq ) Pm+n-k (WI +2 WI ;at,a/t,besqk ,beq k /slq )

-1 -1 WI WI ) (W2 W2 n n ) =Pn ( +2 ;at,a/t, bw2,b/W2Iq Pm +2 ;abtq ,abq /t,es,e/slq .

(Koelink and Van der Jeugt [1998]) 8.22 Show that

8 W7 ("(iibc/q; az, a/ z, ,,(a, "(b, "(c; q, qhd)

("(abc, be, bq/ d, eq/ d; q)oo

(abeq / d, q / "(d, qax"( / d, qa"( / xd; q) 00

00 (1 - abeq2n /d) (abe/d, ab, ae; q) (n+l) xL n(_ad)-nq 2 n=O (1 - abc/d) (q, eq/d, bq/d; q)n

,,(+,,(-1 _ _ _ _) (z+z-l ) x rn ( 2 ;a,b,e,q/dlq rn 2 ;a,b,e,q/dlq,

where a = yiabed/q, ab = ab, ac = ae, ad = ad. Exercises 253

(Stokman [2002])

8.23 Prove the following product formula for the continuous q-Jacobi polyno­ mials: rn(X; a, aq, -c, -cqlq2)rn(Y; a, aq, -c, _cqlq2)

= /1 K(x, y, z)rn(z; a, aq, -c, -cqlq2)dz, -1

where x = cos (), y = cos ¢, 0 ::::; (), ¢ ::::; 'if, 0 < q < 1, 0 ::::; a < c < 1, and

x (aei(}+i¢+i'I/J, aeilJ+i¢-i'I/J, aei'I/J-iIJ-i¢, ae-iIJ-i¢-i'I/J; q2)oo sin'ljJ X/I h( T; 1, -1, ql/2, _ql/2, -y'aC ei(IJ+¢)/2, _y'aCe-i (IJ+¢)/2) -1 h(T; vcei'I/J/2, VCe-i'I/J/2, - vcei'I/J/ 2, -VCe-i'I/J/2, ~ei(IJ-¢)/2, ~ei(¢-IJ)/2) dT X . VI - T2

(Rahman [1986d])

8.24 Let m, n be non-negative integers with n 2: m. Prove the following lin­ earization formula:

rn(x, ql/2, a, -b, _ql/2Iq)rn_m(x; ql/2, a, -b, _ql/2Iq) 2(n-m) = L Ck rm+k(X;ql/2,a,-b,-ql/2Iq), k=O where (ab; qhm(ab; qhn-2m(q, _ aql/2, bql/2, -ab; q)n Ck=~--~~~=-~~~~--~~~~~~~--~=------(abq; qhn(q, _ aql/2, bql/2, -ab; q)m( -q, aql/2, _bql/2, ab; q)n-m (1 - abq2m+2k)(abq2m _qm+l _bqm+l/2 ajb a2b2q2n q2m-2n. q)k X """ qn-m+k/2a-k (1- ab)(q, -abqm, _ aqm+l/2, b2q2m+l, q2m-2n+l jab, abq2n+l; qh

X lO W9(b2q2m; b2, b2q2n+1, q2m-2n+l ja2, abq2m+k, abq2m+k+l, ql-k, q-k; q2, q2).

(Rahman [1981])

8.25 Let the little q-Legendre function be defined by

-11 ab 11+1 ] P (qx'a b'q) = A, [ q , q 'q qx+l v E C, 11 '" 2'/"1 aq " , 254 Further applications

for x a nonnegative integer. Derive the following addition formula:

Pv(qZ;I,I;q) VVy(qZ;qX,q) = VVy(qZ; qX, q) Pv(qY, 1, 1; q) Pv(qx+y; 1, 1; q)

(Xl (q-V qV+l qX+Y+1. q)k (k+l) + L ' , '(_I)kqYk+ 2 k=1 (q,q;q)k x Pv-k(qY; l, qk; q) Pv_k(qX+Y; qk, qk; q) VVY+k(qZ; qX, q) Y ( ) (-V v+l ) + L q; q Y q ,q ; q k (_I)kq(X+Y+l)k-m k=1 (q,q;q)k(q;q)y-k x PV_k(qY-k;qk,qk;q) Pv_k(qX+Y-\qk,q\q) VVY+k(qZ;qX,q),

where VVn(x; a, q) is the Wall polynomial, defined in Notes §7.3. (Rahman and Tariq M. Qazi [1999])

8.26 The associated Askey-Wilson polynomials r~(x) = r~(x; a, b, c, dlq) satisfy the 3-term recurrence relation

where n = 0,1,2, ... , r=:1 (x) = 0, rO'(x) = 1, 0: is the association parame­ ter, and

AA = a-1(1 - abqA)(1 - acqA)(1 - adqA)(1 - abcdqA-l) (1- abcdq2A-l)(I- abcdq2A) , a(1 - bcqA-l) (1 - bdqA-l) (1 - cdqA-l) (1 - qA) C = ~----~~~--~~~~--~~~~~~ A (1 - abcdq2A-2) (1 - abcdq2A-l)

Verify that, with x = cos e,

n (q-n, abcdq2"'+n-l, abcdq2"'-I, aeiO , ae-iO ; q)k k r "'() x - L q n - k=O (q, abq'" , acq'" , adq"',abcdq",-I; qh

x 10 VVg (abcdq2",+k-2; q"', bcq",-I, bdq",-I, cdq",-I, qk+l, abcdq2",+n+k-l, qk-n; q, a2).

(Ismail and Rahman [1991]' Rahman [2001])

8.27 By applying Ex. 2.20 to the 10 VVg series in Ex. 8.26, show that Exercises 255

Hence show that <>() (abcdq2<>-1, q<>+\ q)n -n<> r x = q n (q,abcdq<>-l;q)n

x ill K(x,y)rn(y;aq<>/2,bq<>/2,cq<>/2,dq<>/2Iq)dy, where (q, q, q, abq<>-l, acq<>-l, adq<>-l, bcq<>-l, bdq<>-l, cdq<>-l, q<>; q)oo K (x Y)-~~~~~~~-7~~~~~~~--~~- , - 4n2(q<>+l, abcdq2<>-2; q)oo x (e 2i , e2i ; q)oo(1- y2)-1/2 h(y; q<>/2eiO , q<>/2e- iO ) x r (e2i'lj;,e-2i'lj;;q)ooh(cos'l/J;q~eiO,q~ciO) d Jo h(cos'l/J;aq"';;l ,bq"';;l ,cq",;;l ,dq"';;l ,ql/2ei,ql/2e-i O. (Rahman [1996b, 2001]) 8.28 The associated q-ultraspherical polynomials satisfy the 3-term recurrence relation 1 - aqn+l 1 - a(32 qn-l 2xC~(x;(3lq)= 1 - a (3 qn C~+l(x;(3lq)+ 1 - a (3 qn C~_1(x;(3lq),n20, with C:'\ (x; (3lq) = 0, Co(x; (3lq) = 1. Prove the following linearization formula min(m,n) C <> (x', (31 q)C<> (x', (3lq) "A(m,n)C<> (x' (3lq) m n = ~ k m+n-2k' , k=O where A(m,n) _ (q; q)m(q; q)n(aq; q)m+n-2k(a(3; q)m-k(a(3; q)n-k(a(3; q)k k - (aq; q)m(aq; q)n(a(32; q)m+n-2k(q; q)m-dq; q)n-k(q; q)k (a(32; q)m+n_k(qk-m-n ja; q)k k (1 - a(3qm+n-2k) x a (a(3q; q)m+n_k(qk-m-n; qh (1 - a(3) k-m-n-l k-m-nj (3 XlO W.9 ( q ;q a, qk-m-n ja(3, a, a(32 jq, qk-m, qk-n, q-k; q, q). (Qazi and Rahman [2003]) 8.29 A multivariable extension of the Askey-Wilson polynomials in (7.5.2) is given by Pn(xlq) = Pn(x; a, b, c, d, a2, a3,···, aslq) 256 Further applications

k where Xk = COS()k, X = (Xl, ... ,Xs), n = (n1, ... ,ns), Nk = L: nj, j=l

k Aj,k = II ai, A k+1,k = 1, Ak = A 1,k, 1 ~ j ~ k ~ s. i=j Prove that these polynomials satisfy the orthogonality relation

[11··· [11 Pn(xlq)Pm(xlq)p(xlq)dx1 ... dxs = An(q)6n,m

s with max(lql, lal, Ibl, lei, Idl, la21,···, lasl) < 1, 6n ,m = TI 6nj ,mj' j=l p(xlq) = p(x; a, b, e, d, a2, a3, ... ,as Iq) = (aei01 ae-i01 bei01 be-i01. q)-l , , , 'cx) S-l (e2iOk,e-2iOk;q)00(I_x~)-1/2 [ 1 x IT (ak+1 eiOk+ 1+iO k, ak+1 eiOk+1-iO k, ak+1 eiOk -iOk+1, ak+1 e-iOk+1-iO k; q)oo (e 2iOs, e-2iOa ; q)oo(1- X~)-1/2 X 7-~~~~~~~~~~ (eeiOs, ee-iOs, deiOs, de-iOs; q)oo '

An(q) = An(a, b, e, d, a2, a3,···, aslq) s (q A2 qNk+Nk-l-l. q) (A2 q2Nk. q) 1 (2 )S [ II ' k+1 , nk k+1 ,00 = 7r (q A2 qNk+Nk- 1 a2 qnk. q) k=l ' k 'k+1' 00

x (aeA2,sqNs, adA2,sqNs , beA2,sqNa, bdA2,sqNs ; q)~l,

where ai = ab and a~+l = cd. (Gasper and Rahman [2003b]) 8.30 Prove that (qk+l. q)oo r . . Pn (qk; q, 1; q) = 2~ Jo U2n (cos ()) (Hk( cos ()lq))2 (e 2•O, e-2•O; q)ood().

(Koelink [1996]) 8.31 Let max(lal, Ibl, lei, Idl) < 1, If I < min(lal, Ibl, lei, Idl) and let N be a non­ negative integer such that Ifq-NI < 1 < Ifq-N-1 1. Prove the biorthogo­ nality relation

1 () () ( ) 9o(a,b,e,d,l) , 1 Rm X Sn X V x; a, b, e, d, f dx = h ( b d I) um,n -1 n a, ,e, , for m = 0,1, ... , n = 0,1, ... , N, where v(x; a, b, e, d, I) is the weight function defined in (6.4.3), 9o(a, b, e, d, I) is the normalization constant given by (6.4.1),

h ( b d f) = 27r(1 - abedfq2n-1 ) (abed/q, ab, ae, ad, l/af, bedf /q; q)n n n a, ,e, , (/1 - abed q ) ( q, cd, bd, be, a 2 bedf, aq /)f; q n q , Notes 257

and Rm(x), Sn(x) are the biorthogonal rational functions defined by

Rm(x) = Rm(x; a, b, c, d, 1) (a 2 bcdj, bc, dj, dei () , de-i (); q)m (ad, dja, abcdjei () , abcdje-i(); q)m

X 10 W.9 ( aq -mjd ; q I-mjbd , q I-mjC, d 1jd';if ,a b c j ,aei() ,ae-i() , q -m ; q, q ) , Sn(x) = Sn(x; a, b, c, d, 1) (aqj j, qjaj, dei () , de-i (); q)n (ad, dja, qei () j j, qe-i () j j; q)n X 10 Wg(aq-n jd; ql-n jbd, ql-n jcd, qjdj, abcj jq, aei () , ae-i(), q-n; q, q) with x = cos (). (Rahman [1991])

Notes

§8.5 For additional nonnegativity results for the coefficients in the lineariza­ tion of the product of orthogonal polynomials and their applications to convo­ lution structures, Banach algebras, multiplier theory, heat and diffusion equa­ tions, maximal principles, stochastic processes, etc., see Askey and Gasper [1977], Gasper [1970, 1971, 1972, 1975a,b, 1976, 1983], Gasper and Rahman [1983b], Gasper and Trebels [1977-2000], Ismail and Mulla [1987], Koornwinder and Schwartz [1997], and Rahman [1986d]. §8.6 and 8.7 The nonnegativity of other Poisson kernels and their ap­ plications to probability theory and other fields are considered in Beckmann [1973] and Gasper [1973, 1975a,b, 1976, 1977]. A complicated formula for the Poisson kernel for the Askey-Wilson polynomials Pn(x; a, b, c, dlq) in the most general case is given in Rahman and Verma [1991] (a typo is corrected in Rahman and Suslov [1996b]; also see Askey, Rahman and Suslov [1996] for a nonsymmetric extension of this kernel). Rahman and Tariq M. Qazi [1997a] contains a Poisson kernel for the associated q-ultraspherical polynomials. §8.8 and 8.9 A historical summary of related inequalities is given in the survey paper Askey and Gasper [1986]. For additional material related to de Branges' proof of the Bieberbach conjecture, see de Branges [1968, 1985, 1986], and de Branges and Trutt [1978], Duren [1983], Gasper [1986], Koornwinder [1984, 1986], and Milin [1977]. Sums of squares are also used in Gasper [1994] to prove that certain entire functions have only real zeros. §8.10 Additional applications of q-series are given in A.K. Agarwal, Kalnins and Miller [1987], Alder [1969], Andrews, Dyson and Hickerson [1988], Andrews and Forrester [1986], Andrews and Onofri [1984]' Askey [1984a,b, 1988a, 1989a,e, 1992-1996]' Askey, Koornwinder and Schempp [1984]' Askey, Rahman and Suslov [1996], Askey and Suslov [1993a,b], M. N. At akishiyev , N. M. Atakishiyev and Klimyk [2003], Berndt [1988, 1989], Bhatnagar [1998, 1999], Bhatnagar and Milne [1997], Bhatnagar and Schlosser [1998], Bressoud and Goulden [1985, 1987], W. Chu [1998b], Chung, Kalnins and Miller [1999], 258 Further applications

Cohen [1988], Comtet [1974], Frenkel and Thraev [1995], Geronimo [1994], Ges­ sel and Krattenthaler [1997], Gustafson [1989-1994b]' Gustafson and Kratten­ thaler [1997], Gustafson and Milne [1986], Gustafson and Rakha [2000], Ismail [1990-2003b], Kadell [1987a-1998], Kirillov [1995], Kirillov and Noumi [1999], Kirillov and Reshetikhin [1989], Koelink [1994-2003]' Koelink and Rosengren [2001]' Koelink and Stokman [2001a-2003], Koelink and Swarttouw [1994], Koelink and Van der Jeugt [1998, 1999], Koornwinder [1989-1991a, 1992- 2003], Koornwinder and Swarttouw [1992]' Koornwinder and Touhami [2003], Krattenthaler [1984-2001], Krattenthaler and Rosengren [2003], Krattenthaler and Schlosser [1999], Leininger and Milne [1999a,b], Lilly and Milne [1993], Milne [1980a-2002], Milne and Bhatnagar [1998], Milne and Lilly [1992, 1995], Milne and Newcomb [1996], Milne and Schlosser [2002]' Rahman and Suslov [1996b], Rota and Goldman [1969], Rota and Mullin [1970], and Stokman [2002-2003c]. §8.11 Mordell [1917] considered the representation of numbers as the sum of 2r squares. Milne [1996, 2002] derived many infinite families of explicit exact formulas for sums of squares. Another proof of (8.11.1) is given in Hirschhorn [1985]. Also see Cooper [2001], Cooper and Lam [2002]' and Liu [2001]. Ex. 8.3 For more on classical biorthogonal rational functions, see Rah­ man and Suslov [1993] and Ismail and Rahman [1996]. Ex.8.11 A second addition formula for continuous q-ultraspherical poly­ nomials is given by Koornwinder [2003]. Addition formulas for q-Bessel func­ tions are given in Rahman [1988c] and Swarttouw [1992]. For other addition formulas, see Floris [1999], Koelink [1994, 1995a, 1997], Koelink and Swart­ touw [1995], and Van Assche and Koornwinder [1991]. Ex.8.19 A short elementary proof of the formula in (i) was found by Ismail and Stanton [2000], and is reproduced in Suslov [2003]. Other proofs of (ii) are given in Ismail, Rahman and Stanton [1999] and Suslov [2003, §4.8]. Ex. 8.20 The expression given for Ok is symmetric in m and n, but is equal to the one given in Koelink and Van der Jeugt [1998]. Ex. 8.29 This is a q-analogue of the orthogonality relation for the mul­ tivariable Wilson polynomials in Tratnik [1989]. Also see the multivariable orthogonal or biorthogonal systems in van Diejen [1996, 1997a, 1999], van Diejen and Stokman [1998, 1999], Gasper and Rahman [2003a,b,c], Rosengren [1999, 2001b], Stokman [1997a,b, 2000, 2001], and Tratnik [1991a,b]. 9

LINEAR AND BILINEAR GENERATING FUNCTIONS FOR BASIC ORTHOGONAL POLYNOMIALS

9.1 Introduction

Suppose that a function F(x, t) has a (formal) power series expansion in t of the form (Xl F(x, t) = L fn(x)tn. (9.1.1) n=O Then F(x, t) is called a generating function for the functions fn(x). A useful extension of (9.1.1) is the bilinear generating function

(Xl H(x, y, t) = L anfn(x)gn(y)tn. (9.1.2) n=O We saw some examples of generating functions in Chapters 7 and 8. Some im­ portant linear generating functions for the classical orthogonal polynomials are listed in Koekoek and Swarttouw [1998]. In this chapter we restrict ourselves entirely to generating and bilinear generating functions for basic hypergeo­ metric orthogonal polynomials. Of the many uses of generating functions the one that is most commonly applied is Darboux's method to find the asymp­ totic properties of the corresponding orthogonal polynomials which, in turn, are essential to determining their orthogonality measures. Darboux's method, as described in Ismail and Wilson [1982]' is as follows: If f(t) = 2::=0 fntn and g(t) = 2::=0 gntn are analytic in It I < rand f(t) - g(t) is continuous in It I :::; r, then fn = gn + O(r-n). In a slight generalization of this theorem Ismail and Wilson state, further, that if f(t) and g(t) depend on parameters al, ... , am and f(t) - g(t) is a continuous function of t, al, ... , am for It I :::; r and al, ... , am restricted to compact sets, then the conclusion holds uniformly with respect to the parameters. As far as the bilinear generating functions are concerned one of the most useful is the Poisson kernel K t (x, y) defined in (8.6.1). A related kernel, the so-called Christoffel-Darboux kernel, is defined for orthonormal polynomials Pn (x) by

~ () () _ ~Pn+1(x)Pn(Y) - Pn(X)Pn+l(Y) ~~x~y-k ' (9.1.3) k=O n+l x - y where kn is the coefficient of xn in Pn(x). (9.1.3) is a fundamental identity in the theory of general orthogonal polynomials, see Szeg6 [1975], Chihara [1978], Dunkl and Xu [2001], Nevai [1979], and Temme [1996]. Some evaluations of Poisson kernels were given in Chapter 8. Here we shall derive some important

259 260 Linear and bilinear generating functions bilinear generating functions for basic orthogonal polynomials and give some significant applications.

9.2 The little q-Jacohi polynomials

For the little q-Jacobi polynomials, defined in (7.3.1), we shall first prove that

00 ( ) L aq;q ntnqG)Pn(x;a,b;q) n=O (q;q)n

(-t; q)oo [-xt, bxq ] (9.2.1) = (-xt; q)oo 2¢3 -t, 0, 0 ; q, -aqt . Let G(x, t) denote the sum of the series in (9.2.1). Using (1.4.5) and (7.3.1) we have

Pn(x;a,b;q) = (~-\~)n(_xtq-G) 2¢I(q-n, bxq; xqI-n; q, aqn+I), (9.2.2) aq;q n which, when substituted into the left hand side of (9.2.1), gives G(x, t) = f t (bxq; qh(x-\ q)n-k (-xtt(aqn /x)k n=O k=O (q; q)k(q; q)n-k

= f (bxq; qh (_at)kqk2 f (x-\ q)n (-xtqkt k=O (q; qh n=O (q; q)n

= ~ (bxq; qh (_tqk; q)oo (_at)kqk2 (9.2.3) t='o (q; qh (-xtqk; q)oo ' from which (9.2.1) follows immediately. Note that while the q -+ 1- limit of

00 ( qa+ 1. q) (n) (1 - x ) L (. ') ntnq 2 Pn -2-;qa,qf3;q n=O q, q n is L~=o p;:,f3(x)tn, which has the value 2a+f3 R-I(l - t + R)-a(1 + t + R)-f3 with R = (1 - 2xt + t2)!, see Szego [1975, (4.4.5)], the last expression on the right side of (9.2.3) does not give this limit directly. Nevertheless, it can be used to derive an asymptotic formula for Pn(x; a, b; q) by applying Darboux's method. Observe that, as a function of t, the pole of G(x, t) that is closest to the origin is at -x-I, the next one at -(qx)-I, the next at _(q2 x )-I, and so on. Clearly,

lim (1 + xtqk)G(x, t) = (bxq; qh (x-I; q)oo (a/x)k, t->-(xqk)-l (q; qh (q; q)oo and so a suitable comparison function for asymptotic purposes is f (bxq; qh (x-\ q)oo (a/x)k k=O (q;q)k (q;q)oo l+xtqk

= (x-\ q)oo f (abq=+\ q)oo (-xt)=. (9.2.4) (q;q)oo ==0 (aq=/x;q)oo 9.2 The little q-Jacobi polynomials 261

Combining (9.2.4) and the left side of (9.2.1) we find, by Darboux's theorem, as extended by Ismail and Wilson [1982]' that (x-I. q) (n) Pn(x;a,b;q) "-' ( ') 00 (_x)nq- 2 , x # 0,I,q,q2, ... , aq; q 00 uniformly for x, a and b in compact sets. Now we shall reconsider the Poisson kernel for the little q-Jacobi poly­ nomials given in Ex. 8.3, which has the right positivity properties but suffers from the flaw that it does not directly lead to the known q ----+ 1- limit: f (2n + a + (3 + 1 ) (a + (3 + 1 )nn! tn p(a,(3) (cos 2(})p(a,(3) (cos 2c/J) n=O (a + (3 + 1)(a + l)n((3 + l)n n n 1 - t [a + (3 + 2 a+ (3 + 3 a2 b2 ] = (1 + t)a+(3+2 F4 2 ' 2 ; a + 1, (3 + 1; /1,2' /1,2 ' (9.2.5) where a = sin(}sinc/J, b = cos () cos c/J, /1, = Hi! +c~), 0:::; (},c/J:::; 7[, 0 < t < 1, see Bailey [1935, p. 102]. First, use the product formula Ex. 8.1(i) to obtain the following expression

00 1 - abq2n+l (abq, aq; q)n Kt(x, y) = L 1 _ b (b .) (t/aq)n Pn(x; a, b; q) Pn(Y; a, b; q) n=O a q, q, q n

00 min(x,y) ( b 2) (-X -y) = L L a q ;q 2r+2s q ,q str+sa-Sq(x+y)(r+s)-s2-2rs r=O s=O (q,aq;q)r(q,bq;q)s

x 00 1 - abq2(n+r+s)+1 (abq2r+2s+\ q)s (~) -t n ~ 1- abq2(r+s)+1 (q; q)n q ( ). (9.2.6)

What is needed to compute now is a series of the form

f 1 - aq2n (a: q)n qG) (_t)n == ht(a), say. (9.2.7) n=O I-a (q,q)n

Without the qG) factor the computation is trivial via the q-binomial theorem (1.3.2). But with this factor the best one can do seems to be to consider this as the b ----+ (Xl limit of the very-well-poised series

4 W3(a; b; q, t/b) which, by (3.4.8), transforms to

(t,aq2t2;q)00 ( ()1 ()1 1 1 / ) ( 2 2) SW7 aqt; aq 2,_ aq 2,qa2,-qa2,bqt;q,t b qt , atq ; q 00

= (1 _ t) (-qta~, -:q~ a~, t(aq)~ /~, tqa~ /b; q)oo (-tq'2, -tq, t/b, atq'2 /b; q)oo 1 1 1 1 1 1 1) X SW7 ( atq'2/b;tq'2,(aq)'2,qa'2,-a'2/b,-(aq)'2/b;q,tq'2 , (9.2.8) 262 Linear and bilinear generating functions from (2.10.1). Since 0 < t < 1, it follows by letting b ----; 00 that

ht(a) = (1 _ t) (-tqa~ 'l-tq~ d; q)oo (-tq'l, -tq; q)oo

X ,/-. [(aq)~,qa~,tq~.q tq~] (9.2.9) 3'1'2 -qta'l,-tq'la'l1 3 1" • So, if we replace a and b in (9.2.6) by qa and q(3, respectively, and use (1.6) and (1.25), then Kt(x, y) of (9.2.6) can be written in a more suggestive form (1- t) Kt(x,y) = 1 1 (-tq'l; q'l )a+(3+2

00 min(x,y) (q~(a+(3+2), q~(a+(3+3), _q~(a+(3+2), _q~(a+(3+3); q)r+s xL L (_ tq~(a+(3+3) _tq~(a+(3+4). q) r=O s=O " r+s (q-X, q-Y; q)s (tqx+y)r+s q-s2-2rs-2s x ~~~~~~~~~~----­ (q, qa+l; q)r(q, q(3+1; q)s qr+s+~ (a+(3+2) qr+s+~ (a+(3+3) tq~ [ 1] X 3¢2 _tqr+s+~(a+(3+3), _tqr+s+~(a+(3+4) ; q, tq2 . (9.2.10)

It is obvious that the q ----; 1- limit of (9.2.10) is indeed (9.2.5) and that the expression on the right side of (9.2.10) is nonnegative when 0 < t < 1 and a,j3 > -1.

9.3 A generating function for Askey-Wilson polynomials

There are many different generating functions for the Askey-Wilson polynomi­ als Tn(X; a, b, e, dlq) defined in (8.4.4), of which

~ (abed/q; q)n n ( I) I I Gt (x; a, b, e, d I)q = ~ (.) t Tn X; a, b, e, d q, t < 1, (9.3.1) n=O q, q n is one of the simplest. By Ex. 7.34, iO Gt(x;a,b,e,dlq) = D-1(B) re- /d (dueiO,due-iO , abedu/:; q)oo JqeiO /d (dau/q, dbu/q, deu/q, q)oo abed/q, be, q/u ] x 3¢2 [ ad, abedu/q ; q, adut/q dqu, (9.3.2) where D(B) is defined in Ex. 7.34. However, by (3.4.1), ,/-. [abed/q,be,q/u. adut/] = (abedt/q;q)oo 3'1'2 ad, abedu/q ,q, q (t; q)oo ¢ [(abCd/q)~, -(abed/q) ~, (abed) ~, -(abed) ~,adu/q . ] x 5 4 ad, abedu/q, abedt/q, q/t ,q, q (abed/q, adt, abcdut/q, adu/q; q)oo +~~~~~--~~--~~~ (ad, abedu/q, adut/q, l/t; q)oo 9.3 A generating function for Askey-Wilson polynomials 263

"21 , "21 , "21 , "21 ] '" [ t(abcdjq) -t(abcdjq) t(abcd) -t(abcd) ,adutjq . X 5'1'4 adt,abcdut j q,abcdt 2j q,qt ,q, q . (9.3.3) Substituting this into (9.3.2), using (2.10.18) and simplifying the coefficients, we find that G ( . b dl) = (abcdtjq; q)oo t x, a, ,c, q () t; q 00 ¢ [(abcdjq)~, -(abcdjq) ~, (abcd) ~, -(abcd) ~ ,aei(}, ae-i(} . ] x 6 5 ab, ac, ad, abcdtjq, qjt ,q, q (abcdj q, abt, act, adt, aei(} , ae-i(); q)oo + (ab, ac, ad, atei(} , ate-iO , t-I ; q)oo '" [t(abcdjq)~, -t(abcdjq)~, t(abcd)~, -t(abcd)~, ateiO , ate-iO . ] x 6'1'5 abt, act, adt, abcdt2 jq, qt ,q, q . (9.3.4) The generating function given in Ex. 7.34 is a special case of (9.3.4). A more difficult problem is to evaluate the sum of the series

Ht(x;a,b,c,dlq) = f ((a~q))ntn Tn(x;a,b,c,dlq), (9.3.5) n=O q, q n where a is an arbitrary parameter. For a = ab this would, in particular, give a q-analogue of the generating function of Jacobi polynomials given in the previous section. Note that Ht(x) == Ht(x; a, b, c, dlq)

(a; q)oo ~ (abcdjaq; q)m m ( I ) ( ) = ( b dj .) ~ (.) a Gtqrn x; a, b, c, d q. 9.3.6 a c q, q 00 m=O q, q m

So, use of (9.3.4) in (9.3.6) gives

Ht(x) = (a, abcdtjq; q)oo f (t, abcdjaq; q)m am (t, abcdjq; q)oo m=O (q, abcdtjq; q)m

X '" [(abcdjq)~, -(abcdjq)~, (abcd)~, -(abcd)~, aeiO , ae-iO . ] 6'1'5 a b,ac, a d ,a bc dt qm-I , q I-mjt ,q, q (a, abt, act, adt, aeiO , ae-iO ; q)oo f (abcdjaq; q)m + (ab, ac, ad, ateiO , ate-iO , t-I ; q)oo m=O (q; q)m . . (m+I) (ate,O,ate-,O;q)mq 2 (-at)m x ~--~~~~~--~~~~ (abt, act, adt, qt; q)m '" [tqm(abcdjq)~, -tqm(abcdjq)~, tqm(abcd)~, -tqm(abcd)~, x 6'1'5 abtqm, actqm, adtqm, tqm+ 1 ,

(9.3.7) 264 Linear and bilinear generating functions

Since, by (1.4.5) 2¢1(abed/aq, tq-n; abedtqn-\ q, aqn)

(atqn, abedqn-l; q)oo (-n / n-l n) = ( b d 1) 2¢1 q ,abed aq; abedq ; q, atq , (9.3.8) aqn, a e tqn- ; q 00 the first term on the right side of (9.3.7) becomes

(at; q)oo f (a, abed/ aq, aeiO , ae-iO ; q)nqG) (-at)n (t; q)oo n=O (q, ab, ae, ad, at, q/t; q)n

(9.3.9)

Interchange of the order of summation followed by the use of (1.4.5) and sim­ plification in the second term of (9.3.7) gives

(a, at2 , abedt/q, abt, act, adt, aeiO , ae-iO ; q)oo (ab, ae, ad, at, abedt2 /q, t-I, ateiO , ate-iO ; q)oo x f (abed/aq, r\ q)n (at2t n=O (q,abedt/q;q)n

rI. [at, t( abed/ q)! , -t( abed/ q)! , t( abed)! , -t( abed)! , ateiO , ate-iO . ] x 7'1'6 2 1 1 ,q, q . abt, act, adt, at ,abedtqn- ,tq -n (9.3.10)

In the special case b = aq!, d = eq!, e --+ -e and a = ab = a2q!, both 7¢6 series above become balanced 4¢3 's, which along with their coefficients, can be combined via (2.10.10). Denoting the sum of (9.3.9) and (9.3.10) in this combination by Gt(x; a, elq), we get

Gt(x; a, elq)

(a2t , a2tq! , a2eteiO , a2ete-iO ,(Xl. q) (aetq! , a3 etq! ,ateiO , ate-iO ; q)oo

00 ( ! 2 -! iO -iO ) ( 3 t! ) "'"' ae, aeq 2 ,e q 2, ae ,ae ; q n a e q 2 ; q 2n (2 2 !)n X L...J 1 1 . 1. a t q2 n=O (q, a2t, a2tq2, a2etq2 e'O, a2etq2 e-'O; q)n(a2e2; qhn

x 8 W7 ( a3 etq2n-!; atq! /e, aeqn, aeqn+!, aqneiO, aqne-iO; q, act). (9.3.11)

Unfortunately, neither the 8 W 7 series inside nor the outside series over n can be summed exactly except in the limit q --+ 1-. If we replace a and e by q!(o+!) and qH6+!), respectively, in (9.3.11), then

lim Gt (x;q!(o+!),q!(l3+!)lq) q-->l - 9.4 A bilinear sum for the Askey-Wilson polynomials I 265

= (1 _ t)i3+ 1R-a-i3-1 ~ ((3)n [2t2 (1 - x)] n ~ n! R2 n=O x F [n+~(a+(3+1),n+~(a+(3+2).2t(1-X)] ( ) 2 1 a + (3 + 1 + 2n 'R2· 9.3.12 By the summation formula Erdelyi [1953 Vol. 1. 2.8(6)] we can evaluate the above 2F1 series, and then do the summation over n in (9.3.12) by the bino­ mial theorem (1.3.1) to obtain the well-known generating function Szego [1975, (4.4.5)] for the Jacobi polynomials.

9.4 A bilinear sum for the Askey-Wilson polynomials I

We shall now compute the sum

F(x, ylq) := ~ (abed)~: !~~~/g; q)n (abed/ fg)nknTn(X)Tn(Y), (9.4.1) where Tn(X) is the Askey-Wilson polynomial given in (8.4.4),

kn = K,(a, b, e, dlq) (111 [Tn(xWw(x)dx) -1, (9.4.2) and f and 9 are arbitrary parameters such that the series in (9.4.1) has a convergent sum. By Ex. 7.34, Tn( cos B) = B-1 (B) lqe-i8/b (buei!i, bue-i!i, abedu/q; q)oo(ed, q/u; q)n qei8/b (bau/q, beu/q, bdu/q; q)oo(ab, abedu/q; q)n x (abu/q)ndqu, (9.4.3) qe - i1> /e ( i¢ -i¢ b d / . ) (bd /. ) Tn(COS¢) = C-1(¢) l eve ,eve ,a e v q,q 00 ,q v,q n qei1>/e (eav/q, ebv/q, edv/q; q)oo(ae, abedv/q; q)n x (aev/q)ndqv, (9.4.4) where iq(l - q) B(B) = - 2b (q,ae,ad,ed;q)ooh(cosB;b)w(cosB;a,b,e,dlq), (9.4.5) and iq(l - q) C(¢) = - (q,ab,ad,bd;q)ooh(cos¢;e)w(cos¢;a,b,c,dlq). (9.4.6) 2e Hence

F(x, ylq) = B-1(B)C-1 (¢) lqe-i8/b (buei!i, bue-i!i, abcdu/q; q)oo qeie /b (bau/q, beu/q, bdu/q; q)oo

qe - i1> /e (evei¢, eve-i¢, abedv/q; q)oo l X qei1>/e (eav/q,ebv/q,cdv/q;q)oo

x 8 W7(abcd/q; ad, f, g, q/u, q/v; q, ab2e2duv/ fgq2)dq udq v. (9.4.7) 266 Linear and bilinear generating functions

However, by (2.10.10) the above SW7 equals (abed, be/ j, be/ g, abed/ j g; q)oo (be, abed/ j, abed/ g, be/ jg; q)oo ad,j,g,abeduv/q2 ] [ x 4¢3 abedu/q, abedv/q, qjg/be; q, q (abed, ad, j, g, abeduv/q2, ab2e2du/ jgq, ab2e2dv/ jgq; q)oo +~~--~~~~--~~----~~~--~~~~=-~- (be, abed/j,abed/g, abedu/q, abedv/q, ab2e2duv/jgq2, jg/be; q)oo be/j,be/g,abed/jg,ab2e2duv/jgq2 ] [ x 4¢3 ab2e2du/ jgq, ab2e2dv/ jgq, beq/ jg ; q, q , (9.4.8) which, when substituted into (9.4.7), leads to the sum of two terms, say, F = Fl + F2 , where

F1(x, ylq) = (abed, be/ j, be/g, abed/ jg; q)oo B-l(B)C-1(¢) (be,abed/j,abed/g,be/jg;q)oo ~ (ad, j, g; q)n n x~ q n=O (q, qjg/be; q)n l qe - W /b (buei() , bue-i(} , abeduqn-l; q)oo X qeiO /b (bau/q, beu/q, bdu/q; q)oo qe - i¢ /e (evei , eve-i , abedvqn-l, abeduvq-2; q)oo x l (/ / / n-2) dqv dqu, qei¢ /e eav q, ebv q, edv q, abeduvq ; q 00 (9.4.9) and

F2(x,ylq) = (abed,ad,j,g;q)oo B-l(B)C-1(¢) (be, abed/ j, abed/ g, jg/be; q)oo ~ (be/ j, be/g, abed/ jg; q)n n x~ q n=O (q, beq/ jg; q)n iO X l qe - /b (buei() , bue-i() , ab2e2duqn-l / j g; q)oo qeiO /b (bau/ q, beu/ q, bdu/ q; q)oo i X l qe - ¢/e (evei , eve-i , ab2e2dvqn-l / jg, abeduv/q2; q)oo qei¢/e (eav/q, ebv/q, edv/q, ab2e2duvqn-2 / jg; q)oo x dqv dqu (9.4.10) with x = cos B, y = cos ¢. The q-integral over v on the right side of (9.4.9) can be expressed as a terminating S¢7 series via (2.10.19), which can then be transformed to a balanced 4¢3 series that can be transformed back into a different S¢7 series. The final expression for this q-integral turns out to be C(¢) (dei, de-i , bdu/q; q)n (ad, bd, d/a; q)n

X S W 7 (aq-n /d; aei , ae-i , ql-n /bd, q-n, abu/q; q, q2 / adu). (9.4.11) 9.4 A bilinear sum for the Askey-Wilson polynomials I 267

The q-integral over u in (9.4.9) then has the form l qe- iB /b (bue iO , bue-iO , abeduqn-\ q)oo d qeiB /b (bauqk-l, beu/q, bduqn-k-l; q)oo qU,

which, by (2.10.18), gives

B(()) (aeiO , ae-iO ; q)k(deiO , de-iO ; q)n-k. (9.4.12) (ad; q)n(ae; qh(ed; q)n-k Substitution of (9.4.11) and (9.4.12) into (9.4.9) then gives

F ( I) (abed, be/ f, be/ g, abed/ f g; q)oo 1 X, Y q = (be, abed/ f, abed/g, be/ fg; q)oo

00 (f 9 deiO de-iO dei de-i. q) x L " , , , ,n qn n=O (q, ad, bd, ed, d/a, qfg/be; q)n

x W. (aq-n/d· aeiO ae-iO aei ae-i ql-n/bd q1-n/ d q-n. q beq ) 10 9 \ ' , , , , , e " 'ad . (9.4.13)

The q-integral over v in (9.4.10) equals

C ¢ (abdue-i /q, ab2 edqne-i / fg; q)oo ( ) (ab 2 eduqn-1e-2 / fg, abde-2; q)oo

x 8 W 7 (abde-i /q; ae-i , be-i , de-i , q/u, fgq-n /be; q, ab2 eduqn-1ei / fg), (9.4.14)

which is a bit more troublesome than the previous case because the 8 W7 series is nonterminating unless fg/be is of the form q-k, k = 0,1, ... , which cannot be the case because of the factor (be/fg;q)oo in the denominator of F1(x,ylq) and of the factor (fg/be; q)oo in the denominator of F2 (x, ylq). So either we split up this 8 W7 series into a pair of balanced 4¢3 series via (2.10.10) and get bogged down in a long and tedious computation, or seek an alternative shorter method. In fact, by (6.3.9) the expression in (9.4.14) can be written as

C(rI.) (q aei bei dei ab2 cqn b2 cdqn abcdqn abu/q bdu/q adu/q· q) '1-', , , 'fg' fg , fg' , , ,00

assuming, for the time being, that

(9.4.16) 268 Linear and bilinear generating functions

If we make the further assumption that ad = be, then the q-integral over u reduces to j qe- iO /b (bue iO , bue-iO , ab2c2duqn-1 II g; q)oodqu qe iO /b (ab2eduqn-1ei Ilg, q-1 (abd) !uei'I/J-!i, q-1(abd)!ue-i'I/J-!i; q)oo'

which sums to

(9.4.17) x h(z; eiO-!i(adlb)!, e-iO-!i(adlb)!) .

So the q-integral in u over the expression in (9.4.14) equals

B(8)C( ¢)( q, aei, bei, ee-i , dei; q)oo 1(ae iO , eeiO , de iO ; q)oo 12 2n(ab,ae, ad, ad, bd, cd; q)oo (abedqn Ilg, ab2eqn Ig, b2edqn Ilg; q)oo x (abedqnei+iO Ilg, abedqnei-iO Ilg; q)oo 1 h(z;l,-l,q!,-q!) / x -1 h (z; (abl d)! d i, (adlb)! e!i, (bdl a)! e!i Ilg) h(z; (adlb)! eiO-!i, (adlb)! e-iO - !i, be( abd)! qne- !i I 1g) dz x v'f"=Z2' (9.4.18)

with ad = be. Since this integral is balanced we have, by (6.4.11), as its value the sum of multiples of two balanced very-well-poised 10 Wg series. Combining this result with (9.4.13), we then find the following bilinear summation formula

00 1 _ b2e2q2n-1 (b2e2q-1, ab, ae, I, g; q)n ( b2e2 ) n ; 1 - b2e2q-1 (q, be2a-1, b2ea-1, b2e21- 1, b2e2g-1; q)n Iga2

x T n(x;a,b,e,bca-1Iq) T n(y;a,b,e,bea-1Iq) _ (b2e2, bel I, bel g, b2e2II g; q)oo f (I, g; q)n 1(bca- 1eiO , bea-1ei; q)n 12 n - (be, b2e2I I, b2e2I g, bellg; q)oo n=O (q, be, be2a-1, b2ea-1, bea-2, 1gqlbe; q)n q 2 -n X W. (~. aeiO ae-iO aei ae-i ab-1e-2q1-n ae-1b-2q1-n q-n. q q) 10 9 be' , , " , " , + (b2e2,J, g, b2e2Ilg, b2e3 I alg; q)oo (ae, b2ea-1, be2a-1, be, cia, b2e2I I, b2e2lg, Iglbe, ab3 e2Ilg; q)oo

1(ee iO , bea-1ei, b3 e2eiO I 1g, ab2e2ei I 1g; q) 001 2 X ~------~~~

1(b2e2eiHi Ilg, b2e2eiO-i Ilg; q) 00 12 9.5 A bilinear sum for the Askey-Wilson polynomials II 269

x f: (bel f, belg, ab3e2I fg; q) J(b2e2eiHi¢ I fg, b2e2eilJ-i¢ I fg; q) ~12 qn n=O (q, b2c3lafg, beql fg; q) J(b 3e2eilJ I fg, ab2e2ei¢ I fg; q) J ab3e2qn-1 b3e3qn-1 b2e2qn ab2eqn. . . . ) ( . ,IJ -,IJ b '¢ b -'¢. x 10 w.9 fg , fg '---yg' ---yg,ae ,ae ,e, e ,q,q + (b2e2, f, g, b2e2 If g, ab2el f g; q)oo

(ab, ae, be, b2ea-1, ale, b2e2 If, b2e2lg, fglbe, ~3J;; q) 00

x I (aeilJ , bei¢, b3e3eiIJ lafg, b2e3ei¢ I fg; q) 001 2

Xf: (bel f, belg, b3e4/afg; q) J (b2e2eiHi¢ lafg, b2e2eilJ-i¢ I fg; qf J2 qn n=O (q, abe2 fg, beql fg; q) J(b 3e3 lafgeilJ , b2e3ei¢ I fg; q)nl b3e4qn-1 b2e2qn b3e3qn-1 b2e3qn. .. . ( ,IJ x 10 w.9 f '-f-'. f '-f-,ee ,ee-,IJ ,eeb '¢I a, bee -'¢/'a,q,q. ) ag 9 9 ag (9.4.19) Note that by analytic continuation we may now remove the restrictions in (9.4.16) and require only that no zero factors appear in the denominators on the right side of (9.4.19). Note also that we have tacitly assumed that a, b, e and the product f 9 are real, although it is not necessary. An application of this formula is given in Rahman [1999]. For a more complicated formula without the restriction ad = be, see Rahman [2000b].

9.5 A bilinear sum for the Askey-Wilson polynomials II

Let {wdk=O be an arbitrary complex sequence such that 2::%"=OIWkl < 00. Suppose 00 G1 (x, ylq) := L vnknTn(X)Tn(Y), (9.5.1) n=O where Tn(X) = Tn(X; a, b, c, dlq),

_ (be,ad;q)n( d)n (n)~(adqn,adqn;q)j Vn - ( ) -a q 2 ~ ( 2) wn+j, (9.5.2) abed; q 2n j=O q, abedq n; q j and kn is the normalization constant given in (9.4.2). We shall prove that

The proof is actually quite simple. Use (8.3.4) to write 270 Linear and bilinear generating functions

Tn(X)Tn(Y) = (bd, cd; q)n (~)n t (q-n, abedqn-\ qh (ae,ab;q)n d (q, ad; q)k k=O dei() de-i() dei de-i. q) x ( ' , , ,k qk (ad,bd,ed,dja;q)k x 10 Wg(aq-k jd; q1-k jbd, q1-k jed, q-k, aei() , ae-i(), aei , ae-i; q, beqjad). (9.5.4) Observe that f 1 - abedq2n-1 (abedq-l, ad; q)n (ad)-n . (be, ad; q)n (-ad)nqG) n=O 1 - abedq-1 (q, be; q)n (abed; qhn

X (q-n,abeqn-\q)k 00 (adqn,adqn;q)j ~..,...:.....,,----=:-----,--:....::..:..c.. '" W +. (ad, ad; qh ~ (q, abedq2n; q)j n J

00 1 b d 2k+2n-1 ( b d -1 ) = L -a e q a e q ;q 2k+n (-ltqG)-k n=O 1 - abedq-1 (q; q)n(ad, ad; qh

~ (ad, ad; q)k+n+j x ~ ( ) ( ) Wn+k+j j=O q; q j abed; q 2n+2k+j

_ (abed; qhk -k ~ (ad, ad; q)mwm W ( b d 2k-1. k-m. m-k) - (d d ) q ~ ( ) (b d) 4 3 a e q ,q ,q, q a ,a ; q k m=k q; q m-k a e ; q m+k

= wkq-k, by (2.3.4), which immediately gives (9.5.3).

9.6 A bilinear sum for the Askey-Wilson polynomials III

We shall now consider another bilinear sum, namely,

00 G2(X, ylq) = L 7rnknTn(X)Tn(Y), (9.6.1) n=O where 7r = (be, adj f; q)n ~ rrk n r (9.6.2) (ad, bef; q)n ~ (q, befqn, f q1-n jad; qh'

f i=- 0 being an arbitrary complex parameter and {rrdk=O an arbitrary complex sequence such that ~~=o Irrkl < 00. Changing the order of summation in (9.6.1), we can write it as

00 G2(x, ylq) = L ( b f rr; j d. ). j=O q, e ,q a, q J 9.6 A bilinear sum for the Askey-Wilson polynomials III 271

~ (be, adq-j I j; q)n (j j)nk () () x ~ (d b j j.) q nrn X rn Y . (9.6.3) n=O a, e q ,q n In the case ad = be, which is what we shall assume to be true, the sum over n in (9.6.3) is the special case of the left side of (9.4.19) with j and 9 replaced by be and beq-j I j, respectively. Note that in this special case the first term on the right side of (9.4.19) vanishes while the other two double series become single series. Thus,

(ae, be, be2a-l , b2ea-l , ea-I , bej, abj; q)oo 1 (jeiH , jeiO - i; q)0012 x (abj, bej; q)j 1(je iHi ,jeiO - i; q)j 12 (j, ej la; q)j 1(aje i , bjeiO ; q)j 12

. I . I . ajqj '0 '0 . . ) XIOW9 ( abjql- ;bcjql- ,jql,--,aez ,ae-z ,bez ,be-z ;q,q e

+ (b2e2, j, aj Ie; q)oo 1 (ae iO , bei , beja-Iei()., ejei; q)00.12

(ab, ae, be, b2ea-l , ae-I , bej, be2 ja-I; q)oo 1 (jezHz , jez(}-z; q)0012 x (bej, be2 ja-\ q)j 1(je iHi, jeiO - i; q)j 12 (j, aj I e; q)j 1(beja-Ie z(), ejez; q)j 12

2 I' I . I . ejqj x lOW9 ( be ja- ql- ;bejql- ,jql,-a-'

eei() ee-iO bea-Iei bea-Ie-i. q q) " , ". (9.6.4) Hence,

(ae, be, be2a-l , b2ea-l , cia, bej, abj; q)oo 1 (jeiHi , jeiO - i; q)0012

x 00 (abj; q)j 1(je iHi, jeiO - i; q)j 120j ~ (q, j, ejla, qj lac; q)jl(bjeiO , ajei; q)jl2

x 10 Wg(abjqj-\ bejqj-\ jqj, ajqj Ie, aeiO , ae-i(} , bei , be-i; q, q) + (b2e2, j, aj I c; q)oo 1(ae iO , bei , beja-IeiO , ejei; q)oo 12

(ab, ae, be, b2ea-1, ale, bej, be2 ja-I; q)oo 1 (jezHz , jez(}-z; q)0012

x 00 (bc2 ja-\ q)j 1 (jeiHi , jei(}~i; q)jl20j ~ (q, j, qjlac, aj Ie; q)jl(beja-IezO , ejez; q)jl2

x 10 Wg (bc2 ja-Iqj-\ bejqj-\ jqj, e~qj ,eeiO , ee-i , bca-Iei , bca-Ie-i; q, q). (9.6.5) 272 Linear and bilinear generating functions

Exercises

9.1 Show that

~ (cq; q)n p, ( b ) n ~ ( b') n x; a, ,c; q t n=O q, q, q n = 2(/J1(aq/x, 0; aq;q, qxt) l¢l(bx/c;bq;q,cqt),

when Pn(x; a, b, c; q) is the big q-Jacobi polynomial defined in (7.3.10). Prove also that

n ( b ) (aq/x, cq/x; q)oo n Tn x;a, ,c;q rv ( ) x aq, cq; q 00

as n ----t 00, for fixed x i=- 0, aqm+l, cqm+l, m = 0,1, ... , uniformly for x, a, band c in compact sets. (Ismail and Wilson [1982])

9.2 Prove that

Pn(qm; a, b; q) rv q(n-;m) (-aqt-m(bqm+\ q)oo/(aq; q)oo

as n ----t 00, m = 0,1, ... , uniformly for a and b in compact sets. 9.3 Show that

(i)

(ii)

9.4 For the q-Hahn polynomials defined in (7.2.21) derive the following gen­ erating functions: (i) Exercises 273

(See Koekoek and Swarttouw [1998])

9.5 For the dual q-Hahn polynomials defined in (7.2.23), prove that

N (-N ) (i) L q( . ;)q n Rn(J.l(x);b,e,N;q)tn n=O q, q n

= 2 (PI (q-X, e-1 q-X; bq; q, betqx+1 )(tq-N; q)N-x, ~ (q-N,bq;q)n R ( ( ) b N )tn (ii) ~ ( -1 -N.) n J.l X; ,e, ; q n=O q, e q ,q n

= 2¢1 (qx-N, bqx+\ e-1q-N; q, tq-X) (beqt; q)x.

(See Koekoek and Swarttouw [1998])

9.6 Al-Salam-Chihara polynomials Qn(x; a, blq) are the e = d = 0 special case of the Askey-Wilson polynomials defined in (7.5.2). Prove that

Qn(x; a, blq)tn (at, bt; q)oo , (i) ~ = f::o (q;q)n (te 2(},te- 2(};q)00 ~ (e; q)n Q ( bl) n (ii) ~ ( b.) n x; a, q t n=O q, a ,q n iO iO _ (at,et/a;q)oo A, [ab/e, ae , ae- . t/] - (i(} -i(}.) 3'1'2 b t ,q, e a , te ,te ,q 00 a ,a

where x = cosO, let/al < 1, It I < l. (See AI-Salam and Chihara [1976], Askey and Ismail [1984]' and Koekoek and Swarttouw [1998].)

9.7 Using the recurrence relation of C~(x;fJlq) in Ex. 8.28 show that

00

n=O

Deduce that

(Bustoz and Ismail [1982]) 274 Linear and bilinear generating functions

9.8 Prove that

where IAI < 1, It I < 1. Using Ex. 7.35 show that

00 L p~a,t3)(xlq)tn n=O (q, q(a+ t3+2)/2, q(a-t3 )/2, qt3; q) 00 (q(2a+3)/4 eiO, q2a+3)/4 e-iO; q1/2) 00

27r( qa+1, qa+ t3+1; q) 00 11 h(Y; 1, -1, q1/2, _q1/2, tq(5a+ t3+5)/4) X -1 h (Y; q(a+t3+1)/4, q(a+t3+3)/4, q(a-t3)/4eiO, q(a-t3)/4e-iO, tq(a+t3+1)/4) qa+1 tq(3a-t3+3)/4ei tq(3a-t3+3)/4e-i ] dy [ x 3CP2 tq(5a+ t3+5)/4ei, tq(5a+ t3+5)/4e-i ; q, qt3 -Y--=l=-==y=2'

where y = cos cP, provided 0 < (3 < a.

9.9 Show that f Pn(x; a, b, c, dlq) tn n=O (q,ab,cd;q)n

= 2CP1 (aj z, bj z; ab; q, zt) 2CP1 (cz, dz; cd; q, tj z),

where x = ~(z+z-1), Itzl < 1, Itz- 1 1< 1, and deduce the asymptotic formula (7.5.13). (Ismail and Wilson [1982])

9.10 (i) Prove that, for a(3 = ab,

(a2t, ateiHi j a, ateiO - i j a, atei-iO j a, ate-iO - i j a; q) 00 x sW7(a2tjq; aeiO , ae-iO , aei , ae-i , atj(3; q, btja), Ibtjal < 1.

(Askey, Rahman and Suslov [1996]; Koelink [1995a])

(ii) Show that the continuous q-Hermite polynomial defined in Ex. 1.28 is related to the Al-Salam-Chihara polynomial by Exercises 275

(iii) Using (i), (ii) and the summation formula Ex. 2.17(ii) deduce the q­ Mehler's formula

00 t n ~ Hn(xlq)Hn(Ylq) (q; q)n

(t 2 ; q)oo It I < 1, (teiO+ i , teiO-i , tei-iO , te-iO - i .,00 q) ,

where x = cos (), Y = cos ¢. (See Carlitz [1957a])

9.11 Use Ex. 7.33(ii) to prove that

00 n 1 m+ 1 L L 1 -=- ~n+; q~nCm(COS (); q~ Iq)Cm(cos ¢; q~ Iq) n=Om=O (qei(O+

(q~ ei(O+

(Ismail and Rahman [2002b])

9.12 Show that

f (~;';~~)) Cn(cos();(3lq)Cn(cos¢;(3lq)tn n=O ' ,q n ((3, ,,(, t 2, (3tei (O+-O) , (3te-i (O+-O) , te-i(O+

00 ((32/"1 tei(O+-O) te-i(O+-O) , (3te-i (O+

x 8 W7((3t2q2n-\ (3, tqnei(O+-O) , tqne-i(O+

(Ismail, Masson and Suslov [1997])

9.13 Prove that 276 Linear and bilinear generating functions

x ( - iq(1-v-n)/2e i(()+¢) , _iq(1-v-n)/2e -i(()+¢); q) n

X (iq(1+3V+n)/2e i(O-¢) , iq(1+3v+n)/2e i(¢-O); q) 00

(iq(1+v+n )/2e i(O-¢) , iq(1+v+n )/2e i(¢-O); q) 00

X 8 W 7 ( - qV; qV, iq(1+v+n)/2e i(O+¢) , iq(l+v+n)/2e-i (O+¢) ,

iq(1-v-n)/2e i(¢-O) , iq(1-v-n)/2e i(O-¢); q, qV).

(Ismail, Masson and Suslov [1997])

9.14 Show that

where Re v > 0, 0 < b < a, and 0 ::::; 'Ij! ::::; 7f. (Rahman [1988c])

9.15 If k = 0,1,2, ... , and af3 = ab, prove that

where 0 ::::; e ::::; Jr and 0 < ¢ < Jr. (Ismail and Stanton [1997]) Exercises 277

9.16 For k = 0,1,2, ... , show that

00

n=O

where 0 :::; ¢ :::; 7r and 0 < e < 7r. (Ismail and Stanton [1997])

9.17 (i) Use (9.5.1)-(9.5.3) to prove that t (q-N, O!(3"(Oq~ -1; q)n (1 - abcdq2n-~)1 (abcdq~l, ab, ac, .-ab; q)n n=O (0!(3, -,,(0, q)n (1 - abcdq )(q, bd, q)n(abcd, qhn x (nt1) dan [qn-N,0!(3"(oqn+N-1,abqn ,-cdqn . ] q ( I ) 4¢3 abcdq2n, O!(3qn, _"(Oqn , q, q x Tn(X; a, b, c, dlq) Tn(Y; a, b, c, dlq) = (-ab,,,(o:q)N (_l)N t (q-N,O!(3,,(oqN-l, -ab;q)ml(deil:/,~ei1>;q)mI2qm (ab, -,,(0, q)N m=O (q, ad, ad, bd, -0!(3, ,,(0, dla, q)m

x lOWg(aq-mld;q 1 -mlbd,q 1 -mlcd,q-m,ae'"1:/"1:/ ,ae-' ,ae'"1> ,ae-'"1> ;q, bcqad)'

(ii) From (i) deduce the projection formula

m p:::,b)(x;q) = Lg(n,m)p~o:,f3)(x;q), n=O

where p:::,b) (x; q) is the continuous q-Jacobi polynomial defined in (7.5.25), and

(Rahman [1985]) 278 Linear and bilinear generating functions

9.18 In (9.5.2) and (9.5.3) take ad = be and (ex,(3,"(,o;q)n Wn = --,------=------=--'------"-'----,---0----=----,----- (be, be, beql f, ex(3"(of Ib3e3; q)n'

Also, in (9.6.2) and (9.6.3) take ad = be and (exJlbe,(3J1be,,,(Jlbe,oflbe;q)n n an = q (ex(3"(oj2 Ib4 e4 ; q)n .

Choosing appropriate multiples of (9.5.1) and (9.6.1), show that f (1 - b2e2q2n-l) (b2e2q-l, ab, ae, ex, (3, ,,(, 0; q)n (ex(3"( f Ibe; qhn -0 (1 - b2e2q-l) (q be2a-1 b2ea-1 f3'Yf a'Yf af3f af3'Y8f. q) (b2e2. q)2 n- '" be ' be ' be ' b3 e3 , n ,n 2 2 ( ex(3"( f q2n-l ex(3"(f b e qn Of) (f)n X SW7 be ; b3e3 '-o-,exqn ,(3qn,,,(qn;q, be a2 x rn(x;a,b,e,bea-1Iq) rn(y;a,b,e,bea-1Iq)

(exf Ibe, (3 f Ibe, "( f Ibe, ex(3"( f Ibe; q)oo (ex(3f Ibe, ex"(f Ibe, (3"(f Ibe, f Ibe; q)oo

00 (ex, (3, ,,(, 0; q)nl(bea-1eil:l, bea-1ei¢; q)nl 2qn x ~ (q, be, be, b2ea-1, be2a-1, bea-2, ex(3"(of Ib3e3, beql f; q)n

X 10 Wg ( a2q-n Ibe; aql-n Ib2e, aql-n Ibe2, q-n, aeil:l, ae-il:l, aei¢, ae-i¢; q, q)

(ex, (3, ,,(, 0, ex(3"( f Ibe, ex(3"(oP Ib4 e4 , f, af Ie; q)oo + (ex(3 f Ibe, ex"( f Ibe, (3"( f Ibe, of Ibe, ab, ae, be, b2el a; q)oo

x 1 (aeil:l, bei¢, befeil:l la, efei¢; q)0012 . ( ale, be2 f I a, bel f, ex(3"(o f Ib3e3; q)oo 1(Je'CO+¢), fe,CI:I-¢); q)oo 12 2 2 X f (exJlbe, (3f Ibe, ,,(JIbe, of Ibe, be f la; q)n 1 (JeiCO+¢) ,jeiCO+¢); q)nl qn 4 n=O (q, f, qf Ibe, af Ie, ex(3"(oj2 Ib c4; q)n 1 (efei¢, befeil:l fa; q)nl 2

x 10 Wg(be2 fqn-l fa; fqn, befqn-l, efqn la, eeil:l, ee-il:l, beei¢ la, bee-i¢ la; q, q)

(ex, (3, ,,(, 0, ex(3"( f Ibe, f, ef I a, ex(3"(oP Ib4e4 ; q)oo +--,------,-----~-~--,------,-----~~--,-----~---,-----~~~~~~-- (ex(3f Ibe, ex"( f Ibe, (3"( f Ibe, of Ibe, ae, be, b2ea-1, be2a-1; q)oo 1(eeil:l, beei¢ I a, bfeil:l, afei¢; q)oo 12 x (cia, abf, bel f, ex(3"(o f Ib3e3; q)oo 1(JeiCO+¢), feiCI:I-¢); q)oo 12

~ (exf Ibe, (3f Ibe, "( f Ibe, of Ibe, abf; q)n 1 (JeiCO+¢), feiCI:I-¢); q)nl 2qn x ~ (q, f, qJlbe, eJla, ex(3"(oj2 Ib4 e4 ; q)n I(afei¢, bfeil:l; q)nl 2

x 10 Wg(abfqn-\ fqn, befqn-l, afqn Ie, aeil:l, ae-il:l, bei¢, be-i¢; q, q).

(Rahman [1985])

9.19 Show that the special case ex = be = -,,(, (3 = beq~ = -0, f = t of the formula in Ex. 9.18 gives the Poisson kernel for the special Askey-Wilson Exercises 279

polynomials Tn(X; a, b, e, bca-1Iq):

00 (1 b2 2 2n-1)(b2 2 -1 b .) " - e q e q ,a ,ae,q n (tla2)n f='o (1 - b2 e2 q-1 )(q, be2 a-1, b2 ca-1; q)n x Tn(X; a, b, e, bea-1Iq)(Tn(Y; a, b, e, bea-1Iq)) = (1 _ t 2) (beqt; q)oo f= (beq~, -be, -beq~; q)nl(bea-1ei(i, bea-1eiq,; q)nl2qn (tlbe; q)oo n=O (q, be2a-1, b2ea-1, be, bea-2, bcqt, beqlt; q)n

X 10 W g(a2q-n Ibe; aq1-n Ib2e, aq1-n Ibe2, q-n, aei(i, ae-i(i, aeiq" a-iq,; q, q) (t dtle b2e2. q) I(aei(i beiq, cteiq, bea-1tei(i. q) 12 + '" 00 , , '. '.00 1 2 1 2 1 (ab '"ae be ae- , b ea- , be a- t "CX>belt· q) I(te'(O+q,) , te,(q,-(i)·,ex> q) 12

00 (-t, tq~, -tq~, be2a-1t; q)nl (tei((iH) , tei((i-q,); q)nl2 n x ~ (q, qtlbe, qt2, atle; q)nl(eteiq" tbea-1ei(i; q)nl2 q x lOWg(bc2tqn-1 la; tqn, betqn-l, etqn la, beeiq, la, bee-iq, la, eei(i, ee-i(i; q, q)

(t ctla b2e2.q) I(eei(i bca-1eiq, ateiq, btei(i.q) 12 + '" 00, '.'.' 00 (ab, be, b2 ca-1, be2a-1, abt, belt; q)oo I (te'(O+q,) , te,((i-q,); q)oo 12 ~ (-t, tq~ , -tq~ ,abt; q)n I (tei((iH) , tei(q,-(i); q)n 12 n x f='o (q, qtlbe, qt2etla; q)nl(ateiq" btei(i; q)nl 2 q

X 10 W g( abtqn-1; tqn, betqn-1, atqn Ie, aei(i, ae-i(i, beiq" be-iq,; q, q),

where 0 < t < l. (Rahman [1985]; Gasper and Rahman [1986])

9.20 Prove that

00 L q-n 2(Pl (stei(i, ste-i(i; S2; q, -qnr 2h¢1 (sei(i It, se-i(i It; S2; q, _qn) n=-oo

(1 _ qt2) (q, q, _q2t 2 I s2, -s2 Iqt2; q) 00 (1 _ 2'j,t cos 0 + q;;2) (_qt2, _t-2, -1, -1, -q, -q, S2, S2, S-2, qs-2; q)oo

h(cos 0; st, -st, qst, sit, -qtls) x 1 1 h(cosO; q'2, -q'2, q, -q, qtls)

x 8 W 7 (qt2; q, stei(i, ste-i(i, qtei(i Is, qte-i(i Is; q, q),

where It I < 1, 0 ~ 0 ~ n, and appropriate analytic continuations of the 2 ¢1 series are used as necessary. (Koelink and Stokman [2003])

9.21 The continuous q-Hermite polynomial Hn(cosOlq) in base q-1 becomes 280 Linear and bilinear generating functions

Show that

(i)

(ii)

(Ismail and Masson [1994]) 9.22 (i) For 10J,81 < 1, prove that

2 C' ( 0.) _ (q, a 1,82; q)=(q,82; q2)= '-q cos ,a - (2. 2) qa ,q = x ~ r (e 2i,p,e-2i,p;q)= Eq(cos7jJ;,8)d7jJ. 2n io h(cosnl0/,.. Q!.eilJ(3 , Q!.e-ilJ)(3 (ii) Prove that Eq (cos 0; a )Eq (cos 0; ,8) ( 2. 2) = ( ,8-1 (1-n)/2. ) = ( ~;.q 2)= L -a t) ,q n ,8nqn2 /4cn( cos 0; I'lq) q ,q = n=O 1', q n _,8a-1q(n+1)/2, -1',8a-1q(n+1)/2 . 2] X 2CP1 [ n+1 ,q, a . I'q (See Bustoz and Suslov [1998] for (i), and Ismail and Stanton [2000, (5.8)] for an equivalent form of (ii).) 9.23 Prove that

L= (abedq-1"""n ae ad a' b' a' e' a'd'· q) (bb' edt·,2n_ q) (t) n n=O (q, cd, be't, b'et, bd't, b'dt; q)n(abedq-1; qhn aa'

x 8 W7 (bb' edtq2n-1; beqn, bdqn, b' e' qn, b' d' qn, btla'; q, a'tlb) x rn(x; a, b, e, dlq) rn(Y; a', b', e', d'lq) (bb' edt, dt I e', btei

(be't , b' et , bd't , b'dt , bb't " edt tei(9+

X 8 W 7 (bb'tlq; beilJ , be-ilJ , b' ei

x 8 W 7 (ee'tlq; eeilJ , ee-i(} , e' ei

where ab = a'b', cd = e'd', x = cosO, and Y = coscp. (Koelink and Van der Jeugt [1999]) Notes 281

Notes

§9.3 The special case a = ab, b = aq!, d = cq!, c ----; -c of (9.3.7), in which case the expression on the right side of (9.3.7) can be combined into a sum of SW7 series, was given by Ismail, Masson and Suslov [1997]. However, formula (3.6) in their paper does not directly lead to the generating function f: p~a,(3)(x)tn when one sets a = qa/2+1/4, c = qf3/2+1/4 and takes the limit n=O q ----; 1-. Ex.9.14 A product formula for Jackson's q-Bessel function JS 2) (x; q), which is obtainable from this formula, is used in Rahman [2000a] to evaluate a Weber-Schafheitlin type integral for these functions. Ex.9.18 This is a q-analogue of Feldheim's [1941] bilinear generating function for the Jacobi polynomials. Ex.9.20 Rosengren [2003e] computed a more general bilinear generating function in an elementary manner. Bustoz and Suslov [1998] gave a bilateral bilinear sum that generalizes the classical Poisson kernel for the Fourier series:

f tlnlein(x-y) - 1 - t2 0::; t < l. n=-oo - 1- 2tcos(x - y) + t 2 '

Ex. 9.21 The continuous q-Hermite polynomials in base q-l were intro­ duced by Askey [1989b] who also gave an orthogonality measure for those polynomials on (-00,00) and computed the orthogonality relation. Ismail and Masson [1994] gave a detailed account of the family of extremal measures for these polynomials. See also Carlitz [1963b], Ismail and Masson [1993], and Ismail [1993]. 10

q-SERIES IN TWO OR MORE VARIABLES

10.1 Introduction

The main objective of this chapter is to consider q-analogues of Appell's four well-known functions F l , F2, F3 and F4 . We start out with Jackson's [1942] (l), (2) , (3) and (4) functions, defined in terms of double hypergeometric series, which are q-analogues of the Appell functions. It turns out that not all of Jackson's q-Appell functions have the properties that enable them to have transformation and reduction formulas analogous to those for the Appell functions. Also, starting with a q-analogue of the function on one side of a hypergeometric transformation of a reduction formula may lead to a different q-analogue of the formula than starting with a q-analogue of the function on the other side of the formula. We find, further, that the alternative approach of using the q-integral representations of these q-Appell functions can be very fruitful. For example, it immediately leads to the fact that a general (1) series is indeed equal to a multiple of a 3¢>2 series (see (10.3.4) below). The q-integral approach can be used to derive q-analogues of the Appell functions that are quite different from the ones given by Jackson. In the last section we give a completely different q-analogue of F l , based on the so-called q-quadratic lattice, which has a representation in terms of an Askey-Wilson type integral. We do not attempt to consider Askey-Wilson type q-analogues of F2 and F3 because these are probably the least interesting of the four Appell functions and nothing seems to be known about these analogues. Instances of Askey­ Wilson type q-analogues of F4 have already occurred in Chapter 8 (product of two q-Jacobi polynomials) and then in Chapter 9 (§9.5, §9.6, Ex. 9.18, and Ex. 9.19). Since in this chapter we will be mainly concerned with deriving and applying q-integral representations of q-Appell functions, in many of the formulas it will be necessary to denote the parameters by powers of q. Once a formula has been derived via the q-integral techniques, if it does not contain any q-integrals or q-gamma functions, then the reader may simplify the formula by replacing the qa, qb, etc. powers of q by a, b, etc., as we did in the formulas in the exercises for this chapter.

10.2 q-Appell and other basic double hypergeometric series

In order to obtain q-analogues of the four Appell double hypergeometric series (see Appell and Kampe de Feriet [1926])

282 10.2 q-Appell and other basic double hypergeometric series 283

00 F1(a; b, b'; c; x, y) = L (10.2.1) m,n=O

D ( b b' , ) ~ (a)m+n(b)m(b')n m n E2a;, ;C,c;x,y = ~ "() (') x y, (10.2.2) m.n. C m C n m,n= O

D ( '. b b'.. ) _ ~ (a)m(a')n(b)m(b')n m n E3 a, a, , ,C, x, y - ~ "( ) x y , (10.2.3) m,n=O m. n. C m+n

F (b ' ) ~ (a)m+n(b)m+n m n 4 a, ; c, C ; x, y = ~ "() (') x y , (10.2.4) m.n. C m C n m,n= O Jackson [1942] replaced each shifted factorial by a corresponding q-shifted fac­ torial giving the functions

;r,.(ll( 'b b'· .. ) - ~ (a;q)m+n(b;q)m(b';q)n m n (1025) '±' a", c, q, x, y - ~ ( ) ( ) () x y , .. m,n=O q; q m q; q n c; q m+n

cI>(2 l(a;b,b';c,c';q;x,y) = f ((a;)q)(+nt~q))(b~;q)n) xmyn, (10.2.6) m,n=O q; q m q; q n c; q m c'; q n

;r,.(3l( '. b b'· .. ) - ~ (a; q)m(a'; q)n(b; q)m(b'; q)n m n (1027) '±' a, a, , ,C, q, x, y - ~ ( ) ( ) () x y, .. m,n=O q; q m q; q n c; q m+n

;r,.(4l( b' '.. ) _ ~ (a; q)m+n(b; q)m+n m n '±' a" c, C ,q, x, y - ~ (. ) (. ) ( . ) (" ) x y . (10.2.8) m,n=O q,qmq,qnc,qmc,qn

The series in (10.2.5)-(10.2.8) are absolutely convergent when Iql, lxi, Iyl < 1, by the comparison test with the series z=: n-O xmyn, and then their sums are called the q-Appell functions. Jackson also-considered q-analogues of the Appell series with yn replaced by ynqG) in each term of the series.

Similarly, one could obtain a q-analogue of the general double hypergeo­ metric series

(10.2.9) where we use the contracted notations defined in §3.7 by replacing each shifted factorial by a q-shifted factorial, such as in Exton [1977, p. 36] and Srivastava [1982, p. 278]. However, such a basic double series might not be of the same form as its confluent limit cases or as the double series obtained by inverting the base as in (1.2.24) or in Ex. 1.4(i). Therefore, with the same motivation as given on p. 5 for our definition of the reps series hypergeometric series, we 284 q-series in two or more variables define the q-analogue of (10.2.9) by

;r,.A:B·C [aA : bB; Cc ] '±'D'E"F., d D: eE; I F ;q;x,y = f (aA; q)m+n(bB; q)m(CC; q)n m,n=O (dD; q)m+n(q, eE; q)m(q, IF; q)n m+n)] D-A [ (m)] I+E-B [ (n)] I+F-C X [(_I)m+nq ( 2 (_I)mq 2 (-ltq 2 xmyn, (10.2.10) where q -=I=- 0 when min(D - A, 1 + E - B, 1 + F - C) < O. This double series converges absolutely for lxi, Iyl < 1 when min(D-A, I+E-B, I+F-C) ::::: 0 and Iql < 1. Also, confluent limit cases of (10.2.10) have the same form as the series in (10.2.10), and if one of the parameters b1 , ... ,bB , Cl, ... Cc equals 1, then the double series in (10.2.10) reduces to a single series of the form in (1.2.22). Since the series in (10.2.9) is called a Kampe de Feriet series when B = C and E = F, the series in (10.2.10) can be called a q-Kampe de Feriet series when B = C and E = F.

10.3 An integral representation for (1) (qa; qb, qb'; qC; q; x, y)

Since, by (1.11.7) and (1.10.13)

1 (q::q)m+n = rq(c) r ta+m+n- 1 (~~a~)oo dqt, (10.3.1) (q ,q)m+n r q(a)r q(c - a) 10 (tq, q)oo for 0 < Re a < Re c, we find that

(I)(qa. qb qb'.qc. q. x y) = rq(c) , , , '" rq(a)rq(c-a) 1 a-I (qt, xtq b ,ytqb' ; q)oo d ( ) X t ( c-a) qt, 10.3.2 1o xt, yt, tq ; q 00 which was given by Jackson [1942, p. 81]. This is a q-analogue of the well­ known integral representation of F 1 :

Fl(a; b, b' ; C; x, y) = r(a)~~; _ a) 11 ta- 1 (1 - t)c-a-l(1 - xt)-b(1 - yt)-b' dt, (10.3.3) o < Re a < Re c. It follows from (10.3.2) and the definition of Jackson's q-integral (1.11.3) that (qa xqb yq~'q) [ c-a X ] (I)(qa. qb qb'.qc. q. x y) = ' , ,00 A. q "y'q qa , , , " , ( C .) 3 '/'2 xqb yqb' " . q ,X,y,q 00 , (10.3.4) Andrews [1972] gave an independent derivation of (10.3.4). Some reduction formulas, analogous to the ones for Fl listed, for example, in Erdelyi [1953, Vol. I], can be derived from (10.3.4). 10.3 An integral representation for q,(1)(qa;qb,qb';qc;q;x,y) 285

Case I. Let y = xqb. Then (10.3.4) and (1.4.1) give q,(1)(qa; qb, qb'; qC; q; x, xl) _ (qa,xqb+b';q)oo [qc-a,~. a] - (cq,x,qoo .) 2cPl xqb+b, q, q _ [ qa, qb+b' . ] - 2cPl qC , q, X , (10.3.5) which is a q-analogue of the reduction formula

F1(a; b, b'; C; x, x) = 2Fl(a, b + b'; c; x). (10.3.6)

Case II. Let y = qc-a-b'. Then (10.3.4) and (1.4.1) give

(10.3.7)

(10.3.8)

Case III. Let c = b + b'. The 3cP2 series of (10.3.4) in this case becomes a series of type II, and hence by (3.2.7) qb+b'-a X y ] 3cP2 [ xqb, y~b,' ; q, qa

_ (qb+b',yqa-b;q)oo [qb+b'-a,qb,xqb / y . a-b] - (a b'.) 3cP2 xqb qb+b' ,q,yq . q ,yq ,q 00 ,

(10.3.9) provided Iyqa-bl < 1, which is a q-analogue of the reduction formula

F1(a; b, b'; b + b'; x, y) = (1 - x)-b(l - y)b-a2F1 (b, b + b' - a; b + b'; i =:). (10.3.10) 286 q-series in two or more variables

Integral representation. Since it follows from (10.2.6) that

(2) (qa; qb, l' ; qC, qC' ; q; x, y) OO(ab) =" q, q ; q m xm A, (qa+m qb'. qC' . q y) (10.4.1) 20/1 "'" m=O~ (q,q;q c) m and, by (1.11.9), a+m. ) a+m b'. c'. _ r q (c ') 11 (qu, uyq , q 00 b'-l 2¢1(q ,q ,q ,q,y)- r (b')r ('-b') ( c'-b'.) u dqu, q q c 0 uy, uq ,q 00 (10.4.2) when 0 < Re b' < Re c', we find that (2) (qa; qb, l'; qC, qC'; q; x, y) rq(C') r 1 (qu,uyqa;q)oo [qa,qb,o ] b'-l =rq(b')rq(c'-b')io (uy,uqC'-b';q)oo3¢2 qC,uyqa;q,x u dqu, (10.4.3) provided 0 < Re b' < Re c'. By (1.11.7)

(qb;q)n= rq(c) r1vb+n-1 (vq;q)oo dv (10.4.4) (qC;q)n rq(b)rq(c-b)io (vqc-b;q)oo q, when 0 < Re b < Re c. Thus, by combining (10.4.3) and (10.4.4) we obtain the following integral representation: (2) (qa; l, qb'; qC, qC'; q; x, y) = B-1(b,c- b)B-1(b',c' _ b') r 1 r 1 ub'-lvb-1 (qU,~v,~yqa;q)oo q q io io (uy,uqc-b,vqc-b;q)oo x 2¢1(qa,0;uyqa;q,xv)dqudqv, (10.4.5) when 0 < Re b < Re c and 0 < Re b' < Re c', which is a q-analogue of F2(a; b, b'; c, c'; x, y) = B-1(b, c - b)B-1(b', c' - b')

X fa1 fa1 Ub'-lVb-1(1_ u)c'-b'-l(l_ v)c-b-l(l_ uy - vx)-adudv,

(10.4.6) SInce

Transformation formulas. To derive q-analogues of the transforma­ tion formulas for F2 note that b" (yqa+m; q)oo + ' b' , + b' 2¢1(qa +m,q ;qC ;q,y) = ( ) 2¢2(qa m,qc - ;qC ,yqa m;q,yq ) y;q 00 (10.4.7) 287 by (1.5.4), and hence (10.4.1) and (10.4.7) give

(2) (qa; l, l'; qC, qC'; q; x, y) = (yqa;q)oo f f (qa;q)m+n(qb;q)m(qc'-b';q)n Xm(_ )n nb'+(~) (y; q)oo m=On=O (yqa; q)m+n(q, qC; q)m(q, qC'; q)n y q , (10.4.8) which is a q-analogue of Erdelyi [1953, Vol. I, 5.11 (7)]. Similarly, we get a q-analogue of Erdelyi [1953, Vol. 1., 5.11 (6)] by interchanging x +-+ y, b +-+ b', c+-+c'.

by (II1.2) and (II1.7) , and hence, with a bit of simplification, we find that

(2) (qa; qb, qb'; qC, qa; q; x, y)

( b') 00 00 (b) (-b' ) (b' ) = yq ;q 00 L L q;q m+n qa ,y;q,,:, q ;q nxm+nqmb'. (y; q)oo m=On=O (q; q)m(q; q)n(qC, yqb ; q)m+n (10.4.10) This can be written in the notation of (10.2.10) in the form

(2) (qa; qb, l'; qC, qa; q; x, y)

(yqb'; q)oo 2:2;1 [qb, 0 : qa-b', y; qb' b'] = , ·q·xq x (10.4.11) (y; q)oo 2:1;0 qC, yqb : 0; - " , , which is a q-analogue of Bailey [1935,9.5(6)]. If we specialize further by setting c = c' = a, then we can use the trans­ formation

(10.4.12) to get

(2) (qa; l, qb'; qa, qa; q; x, y) = (yqb';q)oo f (qb;q)m m f (q-m,qb';q)n (_ )n (a+m)n+G) (y; q)oo m=O (q; q)m X n=O (q, qa, yqb'; q)n y q = (yqb'; q)oo f f (qb; q)m+n(q~'; q)n(xy)nxm qn(n-1)+an (y;q)oo m=On=O (q,qa,yqb ;q)n(q;q)m 288 q-series in two or more variables

( b' ) 00 (b b' ) 00 (b+n ) _ yq ; q 00" q, q ; q n ( )n n(n-l)+an" q ; q m m - ( ) ~ ( b') xy q ~ ( ) x y;q 00 n=O q,qa,yq ;q n m=O q;q m

b b' ) 00 (b b' ) ( ) n _ ( xq, yq ; q 00" q, q ; q n xy n(n-l)+an - ( ) ~(a b' b) q x,y;q 00 n=O q,q ,yq ,xq;q n (xqb,yqb';q)oo [qb,qb' a] (10.4.13) = (x,y;q)oo 2¢3 qa,xqb,yqb' ;q,q xy , which is a q-analogue of Bailey [1935,9.5(7)].

Finally, let c = b. Then (10.2.6) gives

(2) (qa; l, l'; l, qC'; q; x, y)

00 (a b' ) 00 (a+n ) = L q ,q, ; q n L q ; q m xm n=O (q,qC ;q)n m=O (q;q)m (xqa;q)oo [qa,qb',o ] = (x; q)oo 3¢2 qC', xqa ; q, y , (10.4.14) which is a q-analogue of Bailey [1935,9.5(3)].

From (10.2.7) it follows that

(3) (qa, qa'; l, l'; qC; q; x, y)

00 (a b) = L 1'~ ;~ m Xm2¢1 (qa', l'; qc+m; q, y). (10.5.1) m=O q, q ; q m Using r1 ub+m- 1 (qu; q)oo d u = f q(b)f q(c - b) (qb; q)m(qc-b; q)n Jo (uqc-b+n;q)oo q fq(c) (qC;q)m+n' (10.5.2) for 0 < Re b < Re c, and r1 b'+n-l (qv;q)oo d _ fq(b')fq(c-b-b') (qb';q)n Jo v (vqc-b-b';q)oo qV- fq(c-b) (qc-b;q)n' (10.5.3) provided 0 < Re b' < Re (c-b), we obtain the following integral representation

(3) (qa, qa'; qb, l'; qC; q; x, y) fq(c) r1 r1 b-l b'-l (qu,qv,xuqa;q)oo =fq(b)fq(b')fq(c-b-b')Jo Jo u v (uqc-b,vqc-b-b',xu;q)oo

x 2¢1(qa', uqc-b;0;q,vy)dq udq v. (10.5.4) Since (10.5.5)

(10.5.4) is a q-analogue of Bailey [1935, 9.3(3)]. 289

The function <[>(3) (qa, qa'; qb, qb'; qC; q; x, y) does not seem to have any use­ ful transformation formulas, just as F3(a, a'; b, b'; c; x, y) does not, as was noted by Bailey [1935, §9.3]. Reduction formulas. Suppose a + a' = c. Then <[>(3) (qa, qc-a; l, l'; qC; q; x, y)

_ ~ (qa,qb;q)m m '" (c-a b'. c+m. ) - ~ ( c ) X 2'1'1 q ,q, q , y m=O q,q;q m

b' 00 00 b b' = (yq ; q)oo L L (qa; q)m+n(q ; q)m(q ; q)n m( _ t n(c-a)+G) (y; q)oo m=O n=O (qC; q)m+n(q; q)m(q, yqb'; q)n X y q , (10.5.6) by (1.5.4), which is a q-analogue of Bailey [1935, 9.5(4)] F1 (a; b, b', c; x, y) = (1 - y)-b' F3 (a, c - a; b, b'; c; x, -y-). (10.5.7) y-1 When a+a' = b+b' = c, the right hand side of (10.5.6) can be reduced to two 4¢3 series in the following way. Using f (t; q)m xm = (xt; q)oo , m=O (q; q)m (x; q)oo

it follows that <[>(3) (qa, qc-a; qb, qC-b; qC; q; x, y) q-b( qa, qb, qc-a, qC-b; q)oo

The limit of the right side of (10.5.8) as q ---+ 1- is

-a b-C{ r(c)r(b - a) ( 1 ) (I-x) (l-y) r(b)r(c_a)2F1 a,c-b;l+a-b;(l_x)(l_y) 290 q-series in two or more variables

r(c)r(a - b) a-b ( 1 )} +r(a)r(c-a) ((l-x)(l-y)) 2F1 b;c-a;l+b-a;(l_x)(l_y)

= (1 - y)a+b-c2F1(a, b; c; x + y - xy), (10.5.9) by Bailey [1935, 1.4(1)]. So we can regard (10.5.8) as a q-analogue of the formula F3(a, c - a; b, c - b; c; x, y) = (1 - y)a+b-c2F1(a, b; c; x + y - xy), (10.5.10) Bailey [1935, 9.5(5)]. If instead of starting with (3) (qa, qc-a; qb, qC-b; qC; q; x, y) we started with (1) (qa; qb, qC-b; qC; q; x, y) we would obtain a different q-analogue of (10.5.10). For,

(1) (qa; qb, qC-b; qC; q; x, y)

~ ~ (qa; q)m+n(qb; q)m(qc-b; q)n m n =~~ X Y m=O n=O (qC; q)m+n(q; q)m(q; q)n

00 (q b ) = L 1,qc.; ~ m xm21/JI(qa+m, qC-b; qc+m; q, y) m=O q, q ,q m = (yqC-b; q)oo f f (qa, qb; q)m(qc-a, qC-b~ q)n xm( _y)nqG)+(a+m)n (y;q)oo m=On=O (qC;q)m+n(q;q)m(q,yqC b;q)n (10.5.11) by (1.5.4), and (1) (qa; l, qC-b; qC; q; x, y)

_ (qa,xqb,yqc-b;q)oo A, [qc-a,x,y. a] - (c .) 3'1'2 xqb yqc-b, q, q q ,X,y,q 00 , _ (xqa+b-c,yqc-b;q)oo [qc-a,qc-b,yqC-b/x . a+b-C] - (. ) 3rP2 qC yqc-b ,q, xq , (10.5.12) x,y,q 00 , by (10.3.4) and (3.2.7), and hence f f (qa,qb;q)m(qc-a,qC-b~q)n xm(_ytqG)+(a+m)n m=O n=O (qC; q)m+n(q; q)m(q, yqC b; q)n _ (xqa+b-c;q)oo [qc-a,qc-b,yqC-b/x . a+b-c] - (x; q)oo 3rP2 qC, yqc-b ,q, xq . (10.5.13)

If we take the limit q ----+ 1- of (10.5.13) and replace y by y/(y - 1) we obtain (10.5.10). This is an example of a situation where one gets different q-analogues by starting with opposite sides of the same formula.

10.6 Formulas for a q-analogue of F4

Appell's F4 function is probably the most important of the four Appell func­ tions because of its applications in the theory of classical orthogonal polyno- 10.6 Formulas for a q-analogue of F4 291 mials. It has the integral representation F4(a, b; e, e'; x(l - y), y(l - x))

= f(e)f(e') 1111 ua-lvb- l (1 - u )c-a-l (1 - v)c'-b-l f(a)r(b)f(e - a)r(c' - b) 0 0 x (1 - ux)Ha-c-c' (1 - vy)Hb-c-c' (1 - ux - vy)c+c'-a-b-ldudv, (10.6.1) which was derived by Burchnall and Chaundy [1940] from their expansion F4(a, b; e, e'; x(l - y), y(l - x)) ~ (a)r(b)r(1+a+b-e-e')r ( ) = ~ I ( ) ( ') xr yr 2Fl a + T, b + T; e + T; X r=O T. ere r

X 2Fl (a + T, b + T; e' + T; y). (10.6.2)

Jackson's <1>(4), given by (10.2.8), does not seem to be useful in any applications that we have come across, and it does not have any formulas with its arguments x and y replaced by x(l - y) and y(l - x), respectively, so our approach will be to transform (10.6.1) and (10.6.2) into forms that we can find q-analogues of. First observe that, since

2Fl(a + T,b+ T; e+ T; x) = (1- x)-a-r2Fl (a + T,e - b; e+ T; _X_)x-I ,

2Fl(a + T, b + T; e' + T; y) = (1 - y)-b-r 2Fl (b + T, e' - a; e' + T; -y-) y-1 by (1.5.5), the expansion formula (10.6.2) transforms to F4(a, b; e, e'; x(l - y), y(l - x))

= (1 _ x)-a(1- y)-b ~ (a)r(b)r(1 + a + b - e - e')r (_x_)r (_y_)r ~ T!(e)r(c')r x-I y - 1

X 2Fl (a + T, e - b; e + T; X ~ 1) 2Fl (b + T, e' - a; e' + T; Y ~ 1)· (10.6.3)

Replacing x and y by x/(x - 1) and y/(y - 1), respectively, (10.6.3) becomes

F4 (a, b; e, e'; - (1 _ x ~1 - y) , - (1 - x ~(1 _ y)) =(1- )a(l_ )b~(a)r(b)r(1+a+b-e-e')r( )r X y r=O~ T.I( ere) ( ') r xy x2Fl(a+T,e-b;e+T;x) 2Fl(b+T,e'-a;e'+T';y). (10.6.4) Analogously, the integral representation (10.6.1) transforms to

(1- x)-a(1- y)-b F4( a, b; e, e'; - (1- x~(1- y)' - (1- x~(1- y)) f(e)f(e') f(a)f(b)f(e - a)f(e' - a) 292 q-series in two or more variables

1 1 X 10 10 u a- 1v b- 1(1_ u)c-a-1(1_ v)C'-b-1(1 _ ux)b-c(1- vy)a-c'

x (1 - uvxy)c+c'-a-b-1dudv, (10.6.5) where 0 < Re a < Re c and 0 < Re b < Re c'. Let M( ) = M( . b ,.) = ~ (a, b, abq/cc'; q)r (CC'xy)r X,y x,y,a, ,c,c,q ~ ( ') b r=O q, C, C ; q r a q x 2¢h(aqr,c/b; cqr;q,x) 2(/JI(bqr,c'/a;c'qr;q,y), (10.6.6) which, via (10.6.4), is a q-analogue of the left side of (10.6.5) on replacing a, b , c, c' b y q abc, q , q ,qc' ,respect· lve 1y. S·mce

(10.6.7) if 0 < Re a < Re c, and

'" (b+r c'-a. c'+r. ) _ rq(c') (qC';q)r 2'1-'1 q ,q ,q ,q,y - rq(b)rq(c' - b) (qb;q)r

1 ( c'-a) X Hr-1 qv, yvq ; q (Xl d (10.6.8) 1o v (yv, vqc' -b; q)(Xl qV, if 0 < Re b < Re c', we have the integral representation

M( ) _ rq(c)rq(c') rl rl a-I b-1 X,y - rq(a)rq(b)rq(c _ a)rc(c' _ b) io io u v c-b c'-a ) ( qu, qv, xuq ,yvq ,xyuv; q (Xl d d x (xu,yv,uqc-a,vqc'-b,xyuvqc+c'-a-b-1;q)(Xl qU qV, (10.6.9) which is an exact analogue of (10.6.5). However, (10.6.9) is, by the definition of q-integrals, the same as the double sum ( a b c-b c' -a ) M ( ) = q, q ,xq , yq , xy; q (Xl x, y (qC, qc' ,x, y, xyqc+c'-a-b-1; q)(Xl

(Xl (Xl ( c+c'-a-b-1) ( c-a) ( c'-b ) X L L xyq ;q m+n x,q ;q m,y,q ;q nqam+bn m=O n=O (xy; q)m+n(q, xqc-b; q)m(q, yqC -a; q)n (qa, qb, xqc-b, yqc'-a, xy; q)(Xl (qC, qC', x, y, xyqc+c'-a-b-1; q)(Xl xyqc+c' -a-b-1 : x, qc-a c'-b ] X c'p1:2;2 [ ; y, q .. a b 1:1;1 xy : xqc-b ; yqc'-a ,q, q ,q , (10.6.10) by (10.2.10). This double series looks more like the series in (10.2.5) for c'p(1) than the series (10.2.8) for c'p(4). Note that the term-by-term q ----+ 1- limit can 10.6 Formulas for a q-analogue of F4 293 be taken on the expression on the right side of (10.6.9) but not on its double sum in (10.6.10). However, it is in the form given by (10.6.6) that the function M(x, y) seems to be most useful. First of all, when cc' = abq the series in (10.6.6) becomes a product

2¢1(a, cjb; c; q, x) 2¢1(b, c' ja; c'; q, y) which corresponds to a q-analogue of the right hand side of the product formula

F4 (a, b; c, a + b + 1 - C; x(l - y), y(l - x)) = 2F1(a, b; C; x) 2F1(a, b; a + b + 1 - C; y) (10.6.11) in Bailey [1935, 9.6(1)]. We shall now manipulate the series in (10.6.5) in order to obtain an ex­ pression that closely resembles the F4 function. First, it follows by combining the terms in the obvious manner, that

M( ) ~ ~ (a, cjb; q)m(b, c' ja; q)n m n x, y = ~ ~ ( ) ( ') x y m=O n=O q, C; q m q, C ; q n q-m q-n abqjcc' ] [ x 3¢2 bq1-:n jc, ~q1-n jc'; q, q . (10.6.12)

Since the 3¢2 series above is balanced it can be summed by (1.7.2). A bit of simplification then leads to

M( ) _ ~ ~ (a, cq-n jb; q)m(b, c'q-m ja; q)n m n mn x, y - ~ ~ ( ) ( ') x y q , (10.6.13) m=O n=O q, C; q m q, C ; q n which has a straightforward q ----t 1- limit, but is not very useful because of the dependence of the terms (cq-n jb; q)m and (c'q-m ja; q)n on m and n. However, SInce

and (c' q-m j a; q)n 2'/'1A, (-nq ,aq;cm' ;q,cq, n-mj a,) (c';q)n by (1.5.2), we have

00 00 m n () (b) m n M(x,y) = L LLL a;q m+s ;q n+r X Y m=O n=O r=O s=o (q; q)m-r(q; q)n-s(q, C; q)r(q, c'; q)s

X (-lr+S(cjbr(c' ja)sqmn-ms-nr+(;)+G) = f f (a, b; q)r+s (_ cx)r (_ C'Y)S r=O s=o (q, C; q)r(q, c'; q)s b a

X q(;)+G)-rs",,r+s, (10.6.14) 294 q-series in two or more variables where ~ ~ (aqr; q)m(bqr; q)n m n mn 'Yr = ~ ~ x y q m=On=O (q; q)m(q; q)n (byqr;q)oo ,/, ( r b r ) = ( ) 2'/"1 aq ,y; yq ; q, x y;q 00 (axqr,byqr;q)oo ,/, ( r b r r b r ) = ( ) 2'/"2 aq , q ; axq , yq ; q, xy . (10.6.15) x,y;q 00 Since (xqa+r+s, yqb+r+s; q)oo/(x, y; q)oo is simply a q-analogue of the product (l_x)-a-r-s(l_y)-b-r-s and the above 2¢2 series approaches 1 when q ---+ 1-, in view of (10.6.6), we may introduce the following function f f (qa, qb; q)r+s( --:xqC- bt( _yqcl-a)s q(;)+G)-rs r=O s=O (q, qC; q)r(q, qC ; q)s(qax , qb y; q)r+s x 2¢2 (qa+r+s , qb+r+s; xqa+r+s, yqb+r+s; q, xy) as an appropriate q-analogue of

(1 - x )-a(1 - y) -b F4 (a, b; c, c'; - (1 _ x ~(1 _ y)' - (1 _ x ~(1 _ y)).

10.7 An Askey-Wilson-type integral representation for a q-analogue of F1

The q-integral representation for -~(l + cos7j1)'Y-O<-~

XX )-{3( y y )_{31 X ( 1 - "2 + "2 cos 7j1 1 - "2 + "2 cos 7j1 d7j1, (10.7.1) provided 0 < Re a < Re 'Y. This suggests the following integral representation

1r = A (ei ,p 1 (_ei,p -a l , e-i,p·q~)a, --- , _e-i,p.q~),~-! o 2 4 2 4 X (>..ei,p, >"e-i,p; q) -{3 (f-lei,p , f-le-i,p; q) _(31 d7j1, (10.7.2) where A is a normalizing constant, and >.., f-l are related to x and y by 4>.. 4f-l x = - (1- >..)2' y = - (1- f-l)2· (10.7.3) 10.7 Askey-Wilson-type integral representation 295

A more general form of the integral in (10.7.2), namely,

S(a,b,c,d,j,g;A,~)

= i7r h(cos1jJ; 1, -1, q~, -q~, A,~) dnl, () ( ) 'f/ 10.7.4 o h cos 1jJ; a, b, c, d, j, 9 was evaluated in Nassrallah and Rahman [1986]. It was found that

S(a,b,c,d,j,g;A,~) = ( b d) (q/abcd, aq/b, aq/c, aq/d, Aa, Va, ~a, ~/a; q)oo K, a, ,c, (qa2, q/bc, q/bd, q/cd, aj,J la, ag, g/a; q)oo

X lOWg(a2; ab, ac, ad, aj, ag, aq/A, aq/~; q, AM/abcdjg) + idem (a;j,g), (10.7.5) with IAM/abcdjgl < 1. The proof in Nassrallah and Rahman [1986] is long and tedious. We will give a shorter proof here. By (2.11.7) and (2.10.1)

h(Z;A,~) (Aa,A/a,~a,~/a;q)oo h(z;j,aq) (qa 2, ja,j/a,q; q)oo x 8W7(a2; aj, aq/A, aq/~, aei'lj;, ae-i'lj;; q, A~/aJ) + h(z;a,jq) (Aj,A/j,~j,~/j;q)oo h(z;j,aq) (qj2,aj,a/j,q;q)oo x 8 W7U2; aj, jq/A, jq/~, jei'lj;, je-i'lj;; q, A~/aJ), (10.7.6) where z = cos 1jJ. Substituting (10.7.6) into (10.7.4) gives

S(a,b,c,d,j,g;A,~) = (Aa,A/a,~a,~/a;q)oo f 1- a2q2n (a2,aj,aq/A,aq/~;q)n (A~)n (qa2,ja,f/a,q;q)oo n=O 1-a2 (q,aq/j,Aa,~a;q)n aj

1! ! X 7r h( cOS'f/,nl'·l ,- ,q2,-q2,aqn+l) d1jJ io h( cos 1jJ; b, c, d, g, aqn) + idem (a; J). (10.7.7) By (6.3.7) the integral displayed in (10.7.7) equals 2n(a2q2n+l, abcdqn, acdgqn, abdgqn, abcgqn; q)oo (bc,bd,bg, cd,cg, dg,abqn, acqn, adqn, agqn, a2bcdgq2n; q)oo x 8 W7(a2bcdgq2n-l; abqn, acqn, adqn, agqn, bcdgq-\ q, q) 2n(abcd,q/abcd,aq/b,aq/c,aq/d;q)oo (ab, ac, ad, be, bd, cd, ag,g/a, q/bc, q/bd, q/cd; q)oo (ab, ac, ad, ag; q)n (q)n X (aq/b,aq/c,aq/d,aq/g;q)n bcdg 2nq 2n(bcdg / q, q2 /bcdg, gq/b, gq/ c, gq/ d; q)oo - bcdg (q, be, bd, cd, bg, cg, dg, qg2; q)oo(1- agqn) (1 - ar)

X 8W7(g2;bg,cg,dg,agqn,gq-n/a;q,q/bcdg), (10.7.8) 296 q-series in two or more variables by (2.11.7) and (2.10.1). So we get S(a,b,e,d,j,g;A,M) = ( b d) (q/ abed, aq/b, aq/ e, aq/ d, Aa, AI a, Ma, M/ a; q)oo ~ a, ,e, (q/be,q/bd,q/ed,aj,j/a,ga,g/a,qa2;q)00 x lOWg(a2;ab,ae,ad,aj,ag,aq/A,aq/M;q, a~~g) + idem (a; f) 2nq (bedg/q,q2/bedg,gq/b,gq/e,gq/d;q)00 bedg (q,be, bd,ed, bg,eg, dg,qg2; q)oo f 1 - g2q2m (g2, bg, eg, dg, ag, t;:q; q)m (AM) m X m=O 1-g2 (q,9q/b,gq/e,gq/d,9q/:,¥;q)m abedjg x {(Aa'A/a'Ma'M/a,¥,~;q)oo (ja, j/a, ag, a/g, ~j,aAr;q)oo

x W7(A~a; AI j, M/ j, AM/q, agqm, aq-m /g; q, q) + idem (a; f)}. (10.7.9) Using (2.11.7) to evaluate the expression in { } above and simplifying, we obtain (10.7.5), which can be regarded as an extension of (10.3.4) of Askey­ Wilson type. In the special case AM = abedjg the integral in (10.7.4) was evaluated • 0: 1 a: + 1 ,,(-Ol 1 ,,{-ex + 1 m (6.4.11). If we replace a, b, e, d, j, 9 by q"2-"4, q"2 "4, -q-2--"4, -q-2- "4, Aq-f3 and Mq-f3', respectively, then this condition amounts to 'Y = {3 + {3', and therefore this case corresponds to (10.3.12) and (10.3.13). Note also that if A = 9 or M = j, which would imply A = M- f3 ' or M = Aq-f3, then the integral in (10.7.4) becomes an 8¢7 series (see §6.3) and hence an analogue of the Gaussian series 2F1. This, then, corresponds to (10.3.5) and (10.3.6). It can be shown that <[>1(0:; {3, (3'; 'Y; x, y) has a q-Appell type double series representation that corresponds to (10.2.5), see Ex. 10.16 for more on the general function S(a, b, e, d, j, g; A, M)'

Exercises

10.1 Show that

<[>(1)(b'/ . b b'. bb'.. ) = (b', bx, b'y/x; q)oo (i) x, , , , q, x, Y (bb' .) , ,x, y, q 00

(ii) <[>(1)( -q/y; b, qx/y2; -qbx/y; q; x, y) (_q, bx; q)00(xq2 /y2, x2q2 /y2; q2)00 (x, y, qx/y, -qbx/y; q)oo(x; q2)00 . (Andrews [1972]) Exercises 297

10.2 Derive the transformation formulas (i) (2) (a', b , b'· , c , c'· , q' , x , y) = (b, ax; q)oo f f (c/b, x; q)m(a, b'; q)n bmyn, (c, x; q)oo m=O n=O (ax; q)m+n(q; q)m(q, c'; q)n

(ii) (3) (a , a'·" b b'·, c , q' , x , y) (a, bx; q)oo ~ ~ (c/a; q)m+n(x; q)m(a', b'; q)n m n = ~~ ay, (c,x;q)oo m=On=O (q,bx;q)m(q,c/a;q)n and their special cases (iii) (2) (a', b, b'· , c, a' , q' , x , y)

-_ (b,ax;q)oo ( ) (3)( c/ b, 0, . x, b'. , ax,.. q, b, y ) , c,x;q 00 (3)(a a'· b b'· aa'· q' x y) (iv) , '" '" _ (a,bx;q)oo (2)('. '. .. ) - ( ) a ,x, b , bx, 0, q, a, y . aa', x; q 00 (Andrews [1972]) 10.3 Prove that 2 (l)(a'b =,j, a , aq , b 2] , , b'cq'x , , , , -x) 3 '1'2 [ C, cq ".q2 x .

Deduce the quadratic transformation formula 2 2 ,j, [c/a,x,-x. ] _ (c;q)00(x ;q2)00 ,j, [a,aq,b . 2 2] 3'1'2 bx -bx ,q, a - ( . ) (b2 2. 2) 3'1'2 C cq ,q, x . , a, q 00 x, q 00 , 10.4 Show that

(1) 2 X (bq!', b" b' b ". q' , y) 1 (xq'i;q)oo 1 1 1 = ( ) 2¢1((by/x)2,-(by/x)2;-b;q2,x). y;q 00 Hence, prove the quadratic transformation formula

3¢2 [ x, :;,b~;! ; q, bq! ]

(x,y,b2;q)00 (( /)1 ( /)1 1) -----'-----;-1-'------2¢1 by x 2, - by x 2; -b; q2 ,x . (bx, by, bq'i; q)oo 10.5 Derive the quadratic transformation formula (l)(a'b, , b'cx , , , y)

00 ( 2 1/ ) = L a, b ,xq 21 y; q n yn n=O (q, C, bq2; q)n q-n/2, _q-n/2, (by/x)!, -(by/x)! . [ ! !] X 4¢3 -b, ql/4-n/2(y/x)!, _ql/4-n/2(y/x)! ,q ,q . 298 q-series in two or more variables

10.6 Prove that (2) (a', b , b'· , c , c'· , q' , x , y) = f f (a; q)m+n(c/b; q)m(c' /b'; q)n (-bx)m( -b'y)nqG)+G) m=O n=O (ax; q)m+n(q, c; q)m(q, c'; q)n

X (~x;~)oo 2(/JI(aqm+n, O;axqm+n; q,y). x; q 00 Note that this gives a q-analogue of Bailey [1935, 9.4(8)].

10.7 Show that a general 3(P2 series with an arbitrary argument is a multiple of Jackson's (1) series, in particular,

a,b,c. ] _ (ax, b,c; q)oo (1)(. / / ... ) 3 x,d b,e c,ax,q,b,c. ,e x, ,e,q 00 10.8 Show that f f (a, b; q)m+n( -xc/b)m( -yb/a)n q(r;')+(~)-mn m=O n=O (q, c; q)m(q, b; q)n(ax, by; q)m+n x 2

_ (x,yb/a;q)oo (1)(. / /... / ) - ( b ) a,aq c,c b,c,q,cxy aq,x , ax, y; q 00 and that this is a q-analogue of Bailey's [1935, Ex. 20(ii), p. 102] reduction formula F4 (a, b; c, b; - (1 _ x~1 _ y) , - (1 _ x~(1 _ y)) = (1 - x) a (1 - y) aF1 (a; 1 + a - c, c - b; c; xy, x). 10.9 Show that f f (a, b; q)m+n( -xa/b)m( -yb/a)n q(r;')+G)-mn m=O n=O (q, a; q)m(q, b; q)n(ax, by; q)m+n

X 2

F4 (a, b; a, b; - (1 _ x~(1- y)' ---'---(l---X---':-~--'---(l---y---'--)) = (1 - xy)-l(l- x)b(l _ y)a. 10.10 Prove that f f (a, b; q)m+n( -aqx/b2 )m( -yb/a)n qG)+(~)-mn m=O n=O (q, aq/b; q)m(q, b; q)n(ax, by; q)m+n

X 2

_ (by/a; q)oo .t, [a, b, aq/by . /b] - (b ) 3'f'2 /b ,q,XY y;q 00 aX,aq Exercises 299

and that this is a q-analogue of Bailey's [1935, Ex. 20(v), p. 102] reduction formula

F4 (a, b; 1 + a - b, b; - (1 _ X~l _ y)' - (1 - x~(1- y))

a ( X(l- y)) = (1 - y) 2Fl a, b; 1 + a - b; - . I-x 10.11 Using (10.6.15) prove that f f (a, b; q)m+n( -cx/b)m( _qy/c)n qC;')+G)-mn m=O n=O (q, c; q)m(q, aq/c; q)n(ax, by; q)m+n x 2

= f f (a; q)m+n(b; q)m(b; q)n (_ c~ )m( _ yq~)n q(m-n)2j2, m=On=O (q, c, ax; q)m(q, aq/c, by; q)n bq'i C

where Icx/bl < 1 and Iy/cl < 1. This is a q-analogue of Bailey's [1935, Ex. 20(i), p. 102] reduction formula

F4 (a, b; c, 1 + a - c; - (1 _ x ~1 _ y) , - (1 - x ~(1 _ y)

= F2(a;b,b;c, 1 + a - c; _x_, -y-). x-I y-1

10.12 Show that if f(x, y) = (f>(l) (a; b, b'; c; q; x, y), then (i) (abx - c/q)f(q2x , qy) + (1 - bx)f(qx, y) + (c/q - a)f(qx, qy) + (x - l)f(x, y) = 0, (ii) (aby' - c/q)f(qx, q2y) + (1 - b'y)f(x, qy) + (c/q - a)f(qx, qy) + (y - l)f(x, y) = O.

10.13 Show that if f(x, y) = (f>(2) (a; b, b'; c, c'; q; x, y) then

ab'xf(q2x , qy) - ~ f(q2x, y) - axf(qx, qy) q + (1 + ~ - b' x ) f (qx, y) + (x - 1) f (x, y) = O.

10 .14 Show that if f(x " y) = (f>(3) (a a'·" b b'·, c· , q. , x , y) then abxf(q2x , y) - ~ f(q2x, qy) + ~ f(qx, qy) q q + (1 - ax - bx)f(qx, y) + (x - l)f(x, y) = O.

10.15 Show that if f(x,y) = (f>(4)(a,b;c,c';q;x,y) then abxf(q2x , q2y) - cf(q2x, y) - (a + b)xf(qx, qy) + (1 + c)f(qx, y) + (x - l)f(x, y) = O. 300 q-series in two or more variables

10.16 Using (6.3.3) twice show that

S(a,b,e,d,j,g;A,~) 2n(A/a,A/j,~/b,~/g;q)= ab(l - q)2(q, q, q, cd, fla, aq/ j, g/b, qb/g, aj, bg; q)= r (qu/a,qu/j,Au;q)= rb (qv/b,qv/g,~v,eduv;q)=d d x if (eu,du,Au/aj;q)= i g (ev,dv,uv,~v/bg;q)= qU qV, 2n(abed,A/a,Aa,~/b,~b;q)= (q, ab, ae, ad, be, bd, cd, j la, ja, g/b, gb; q)= x f f (ab;q)m+n(ae,ad,Alj;q)m(be,bd,~/g;q)n qm+n m=O n=O (abed; q)m+n(q, aq/ j, Aa; q)m(q, qb/g, ~b; q)n + idem (a; j) 2n(aedg,A/a,Aa,~/g,~g;q)= +-;-----'--.,--=-'----'=-....,....:-'c-:-.::..c....:....::-'-:--=:'--:-----:-- (q, ae,ag,ad,eg,dg, cd, j/a,aj,b/g, bg; q)= x f f (ag; q)m+n(ae, ad, AI j; q)m(eg, dg, ~/b; q)n qm+n m=O n=O (aedg; q)m+n(q, aq/ j, Aa; q)m(q, gq/b, ~g; q)n + idem (a; j). 10.17 Extend (10.3.4) to

where D is a q-Lauricella function. (Andrews [1972])

10.18 Prove that

1:3;3 [q-n : ale, e/d, e/e ; e/a, bid, b/e.. ] 1:2;2 be/de : a, q1-n /b ; b, q1-n /a ,q, q, q (e, able; q)n (a,b;q)n ' where n = 0,1, .... Deduce the summation formula f f (a/e, e/d: e/e; q)j(e~a, bid, b~e; qh b1akqjk j=O k=O (be/de, q)j+k(q, a, q)j(q, b, q)k (e, able; q)= (a, b; q)= .

(Gasper [2000]) Notes 301

10.19 Show that

j,k,m?O (,x,qz/"~y,,y,axz/,,~yz;q)oo (x, a~zh, ~,y, y, xz, yZ; q)oo

X 8 W7(~,yq-\~, I, ,y/a, 1/z, ~Y/X; q, axzh)· (Krattenthaler and Rosengren [2003]) 10.20 Extend (1.9.6) to

f f (blqm 1 , ... , brqmr; q)n+m(a, b; q)n(c, d; q)m n=Om=O (b l , ... , br ; q)m+n(q, bq; q)n(q, dq; q)m

X (a-lql-Mt(c-lql-M)m

(q, q, bq/ a, dq/ C; q)oo (bd db; q)ml ... (br /bd; q)mr (bd)M (bq, dq, q/a, q/c; q)oo(bl ; q)ml ... (br ; q)mr where ml, ... , mr are nonnegative integers, M = ml + ... + m r , and Iql-MI < min(lal, Icl)· (Denis [1988])

Notes

§1O.2 B. Srivastava [1995] introduced some elementary bibasic extensions of (10.2.5)-(10.2.8). For a quantum algebra approach to multivariable series, see Floreanini, Lapointe and Vinet [1994]. Sahai [1999] considered the q-Appell functions from the point of view of universal enveloping algebra of 8£(2). See also Jain [1980a,b] and, for connections of Appell functions with Ben root systems, Beerends [1992]. §§10.3-1O.6 Agarwal [1974] evaluated some q-integrals of the double se­ ries in (10.2.5)-(10.2.8). Also see Nassrallah [1990, 1991]. Exercises 10.1-10.2 Upadhyay [1973] gave some transformation and sum­ mation formulas for q-J ackson type double series. Ex. 10.3 This is a q-analogue of a quadratic transformation formula for Fl(a; b, b; C; x, -x) obtained by Ismail and Pitman [2000]. Exercises 10.12-10.15 Agarwal [1954] gave some 3-term contiguous rela­ tions satisfied by <[>(1), ... ,<[>(4), including formulas connecting them with the q-derivatives. Ex.1O.17 For an application of the q-Lauricella series, see Bressoud [1978]. Ex.1O.18 See Van der Jeugt, Pitre and Srinivasa Rao [1994] for more summation formulas of this type. For similar results when q ---+ 1, see Pitre and Van der Jeugt [1996]. 11

ELLIPTIC, MODULAR, AND THETA HYPERGEOMETRIC SERIES

11.1 Introduction

Since the series L Un is called a hypergeometric series if 9 (n) = U n +Ii Un is a rational function of n, and a q-(basic) hypergeometric series if g( n) is a rational function of qn, it is natural to call this series an elliptic hypergeometric series if g( n) is an elliptic (doubly periodic meromorphic) function of n with n consid­ ered as a complex variable. One motivation for considering these three classes of series is Weierstrass's theorem that a meromorphic function J(z) which sat­ isfies an algebraic addition theorem of the form P(f(U), J(v), J(u + v)) = 0 identically in u and v, where P(x, y, z) is a nonzero polynomial whose co­ efficients are independent of u and v, is either a rational function of z, a rational function of eAZ for some A, or an elliptic function (see, e.g., Erdelyi [1953, §13.11]' Rosengren [200la, 2003c], and, for a proof, Phragmen [1885]). Elliptic analogues of very-well-poised basic hypergeometric series were intro­ duced rather recently by Frenkel and Turaev [1997] in their work on elliptic 6j-symbols, which are elliptic solutions of the Yang-Baxter equation found by Baxter [1973], [1982] and Date et al. [1986-1988]. Frenkel and Turaev showed that the elliptic 6j-symbols are multiples of the very-well-poised 12Vl1 elliptic (modular) hypergeometric series defined in §11.3 (recall from §7.2 that the or­ dinary q-analogues of Racah's 6j-symbols are multiples of balanced 4¢3 series, which are transformable to 8¢7 series) and then used the tetrahedral symmetry of the elliptic 6j-symbols and the finite dimensionality of cusp forms to derive elliptic analogues of Bailey's transformation formula (2.9.1) for terminating 1O¢9 series and of Jackson's 8¢7 summation formula (2.6.2). This quickly led to a flurry of activity on elliptic hypergeometric series, resulting in several re­ lated papers (listed in alphabetical order) by van Diejen and Spiridonov [2000- 2003], Gasper and Schlosser [2003], Kajihara and Noumi [2003], Kajiwara, Masuda, Noumi, Ohta and Yamada [2003], Koelink, van Norden and Rosen­ gren [2003], Rosengren [200la-2003f], Rosengren and Schlosser [2003a,b], Spiri­ donov [2000-2003c], Spiridonov and Zhedanov [2000a-2003], Warnaar [2002b- 2003e], and others. In particular, general elliptic hypergeometric series and their extensions to theta hypergeometric series were introduced and studied by Spiridonov [2002a-2003a], who also considered theta hypergeometric integrals in Spiridonov [2003bj. Transformations of elliptic hypergeometric integrals are considered in Rains [2003bj. We start in §11.2 with the elliptic shifted factorials, Spiridonov's [2002aj "multiplicative" r+lEr theta hypergeometric series notation and its very-well-

302 11.2 Elliptic and theta hypergeometric series 303 poised r+1 Vr special case, and then point out some of their main properties. The "additive" forms of these series r+1er and r+1Vr are presented in §11.3, along with their modular properties, and the Frenkel and Turaev summation and transformation formulas. In §11.4 we present a simpler derivation of the elliptic analogue of Jackson's 8¢7 summation formula via mathematical induc­ tion, which is a modification of proofs that were kindly communicated to the authors by Rosengren (in a June 13, 2002, e-mail message), Spiridonov (in a Nov. 22, 2002, e-mail message) and Warnaar (see §6 in his [2002b] paper). Central to this method of proof and, in fact, to all of the formulas in this chapter is the theta function identity given in Ex. 2.16(i) which is simply the elliptic analogue of the trivial identity (1- xA)(I- xIA)(I- fLV)(I- fLlv) - (1- xv)(I- xlv)(I- AfL)(I- fLIA) = ~ (1 - XfL)(1 - xl fL)(1 - Av)(1 - Ajv). (11.1.1)

The elliptic analogue of Bailey's 1O¢9 transformation formula is derived in §11.5, along with some other transformation formulas. Theta hypergeometric extensions of some of the summation and transformation formulas in sections 3.6-3.8 are derived in §11.6. Some multidimensional elliptic hypergeometric series are considered in §11. 7, where we present Rosengren's [2003c] elliptic extension of Milne's [1985a] fundamental theorem and related formulas. Many additional formulas involving theta hypergeometric series and elliptic integrals are presented in the exercises at the end of the chapter.

11.2 Elliptic and theta hypergeometric series

Define a modified Jacobi theta function by

O(X;p) = (x,plx;p)oo, O(X1, ... , xm;p) = II O(Xk;P), (11.2.1) k=l where X,X1, ... ,Xm -=I- 0 and Ipi < 1. Since O(x;p) = (x;p)oo(plx;p)oo is the product of two infinite p-shifted factorials, analogous to the name of the triple product identity (1.6.1), we will call O(x;p) the double product theta function (with argument x and nome p). From (1.6.9) and (1.6.14) it follows that

(11.2.2) and O(qa. p) [a· (J T] = 'q(1-a)/2 (11.2.3) " O(q;p) with (11.2.4) where a, (J, T are complex numbers such that 1m (T) > 0 and (J -=I- m + nT for integer values of m and n. We set qa = e2niaa and pa = e 2nim . Following 304 Elliptic, modular and theta hypergeometric series

Warnaar [2002b], we define an elliptic (or theta) shifted factorial analogue of the q-shifted factorial by

rr~:~ B(aqk;p), n = 1,2, ... , (a; q,P)n = { 1, n=O, (11.2.5) n l 1/ rr-k=O- B( aq n+k.,p ) , n=-I,-2, ... , and let m (aI, a2,···, am; q,P)n = II (ak; q,P)n, (11.2.6) k=l where a, al, ... , am -I=- O. Analogous to the name q-shifted factorial for (a; q)n, we also call (a; q, P)n the q, P -shifted factorial in order to distinguish it from the a, T-shifted factorial defined in the next section. Notice that B(x; 0) = 1-x and thus (a; q, O)n = (a; q)n. Since q is called the base in (a; q)n and P is called the nome in B( a; p), we call q and P in (a; q, P)n the base and nome, respectively. Similarly, in order to distinguish the modular parameters a and T, we call a the base modular parameter and T the nome modular parameter. For the sake of simplicity, we decided not to use Spiridonov's [2002a,b] notation B(a;p; q)n for the elliptic shifted factorial. Corresponding to Spiridonov [2002a], we formally define an r+IEr theta hypergeometric series with base q and nome p by

(11.2.7) where, as elsewhere, it is assumed that the parameters are such that each term in the series is well-defined; in particular, the a's and b's are never zero. Un­ less stated otherwise, we assume that q and p are independent of each other, but we do not assume that the above series converges or that the numerator parameters aI, a2, ... , ar+l, denominator parameters b1, ... , bTl and the argu­ ment z in it are independent of each other or of q and p. Note that if ajpk is a nonpositive integer power of q for some integer k and a j E {I, 2, ... , r + I}, then the series in (11.2.7) terminates. Clearly, if z and the a's and b's are independent of p, then lim r+1Er(al, ... , ar+l; b1, ... , br; q,p; z) p--70 = r+1Er(al, ... , ar+l; b1, ... , br; q, 0; z) = r+lcPr(al, ... , ar+l; b1,···, br; q, z), (11.2.8) where the limit of the series is a termwise limit. As is customary, the notation r+1Er(al, a2, ... , ar+l; bl , ... , br; q,p; z) is also used to denote the sum of the series in (11.2.7) inside the circle of con­ vergence and its analytic continuation (called a theta hypergeometric function) outside the circle of convergence. Unlike nonterminating r+1Fr series and non­ terminating r+lcPr series with 0 < Iql < 1, which have radius of convergence R = 1 (as can be easily seen by applying the ratio test to (1.2.25) and (1.2.26)), 11.2 Elliptic and theta hypergeometric series 305 for 0 < Iql, Ipl < 1 and any R E [0,00] there is a nonterminating r+lEr series with radius of convergence R. In particular, there are nonterminating r+lEr series with 0 < Iql,lpl < 1 that converge to entire functions of z, which is not the case for nonterminating r+lFr and r+l¢r series with r = 0,1, ... and 0 < Iql < 1. For example, consider a nonterminating r+lEr series with 0< Iql, Ipi < 1 and (11.2.9) where ml, ... , mr+l are integers and br+l = q. Since the double product theta function identity ()(apm;p) = (-a)-mp-(';)()(a;p) implies that

(apm; q,P)n = (a; q,P)n( _a)-mnp-nG)q-m(~) for m, n = 0, ±1, ... , we find that

00

ml r 1'+1 E l' (al,···, ar+l,. alP , ... , arpm ..,q,p, Z ) - "(~ ZPr )n q MrG) , (11210).. n=O with ar+l = qp-mr +1 , Pr = rr~~i(-ak)mkp(~k), and Mr = ml + ... + mr+l. From (11.2.10) it is clear that this series converges to an entire function of Z when Mr > 0, converges only for Z = 0 when Mr < 0, and converges to 1/(1 - zPr) when Mr = 0 and IZPrl < 1. As in Spiridonov [2002a], we call a (unilateral or bilateral) series L Cn an elliptic hypergeometric series if g(n) = cn+1/cn is an elliptic function of n with n considered as a complex variable, i.e., g(x) is a doubly periodic meromorphic function of the complex variable x. For the r+lEr series in (11.2.7)

1'+1 ()( akq x ;p ) g ()x = Z II (11.2.11) k=l ()(bkqX;p) with br+l = q. Clearly g(x) is a meromorphic function of x. From (11.2.4) it is obvious that qx+a- 1 = qX and hence g(x + a-I) = g(x). Since, by (11.2.1),

()(aqx+ra- 1 ;p) = (_aqx)-l()(aqx;p), (11.2.12) it follows that (11.2.13)

Thus g(x + Ta-l ) = g(x) when

ala2··· ar+l = (b l b2 ··· br)q, (11.2.14) in which case g(x) is an elliptic (doubly periodic meromorphic) function of x with periods a-I and Ta-l . Therefore, we call (11.2.14) the elliptic bal­ ancing condition, and when (11.2.14) holds we say that r+lEr is elliptically balanced (E-balanced). In Spiridonov [2002a] an r+lEr series is called, simply, "balanced" when (11.2.14) holds, but here we need to distinguish between the different balancing conditions that arise. Notice that, unlike the requirement 306 Elliptic, modular and theta hypergeometric series that z = q in the definition of a balanced r+l ¢r series, no restrictions are placed on the argument z in the above definition of an E-balanced r+IEr series. If z = q, then, by the definition of a k-balanced r+l¢r series in §1.2, the r+l¢r series in (11.2.8) is (-1 )-balanced if and only if the elliptic balancing condition (11.2.14) holds. Analogous to the basic hypergeometric special case, we call the r+IEr series in (11.2.7) well-poised if qal = a2 bl = a3 b2 = ... = ar+lbr, (11.2.15) in which case the elliptic balancing condition (11.2.14) reduces to ala2···ar+12 2 2 = ()r+lalq . (11.2.16)

Via (11.2.5) and the n ---+ 00 limit case of Ex. 1.1(iv) with q replaced by p, we find that

2 1 111 11 B(aq n;p) = (qa"2,-qa"2,qa"2/p "2,-qa"2 p"2;q,P)n(_ )-n 1 1 1 1 1 1 q (11.2.17) B()a;p (a"2,-a"2,a"2p"2,-a"2/p"2;q,P)n is an elliptic analogue of the quotient 1 - aq2n (qd, -qd; q)n 1-a (d,-d;q)n ' which is the very-well-poised part of the r+l Wr series in (2.1.11) with al = a; see Ex. 11.3. Therefore the r+IEr series in (11.2.7) is called ve'T"y-well-poised if it is well-poised, 'T" ~ 4, and

1 1 ! 1 ! 1 a2 = qai, a3 = -qai, a4 = qai /p"2, a5 = -qai p"2. (11.2.18)

Corresponding to Spiridonov [2002b, (2.15)]' we define the r+l Vr very­ well-poised theta hypergeometric series by

r+I Vr(al;a6,a7, ... ,ar+l;q,p;Z)

= ~ B(alq2n; p) (aI, a6, a7, ... , ar+l; q, P)n (qz)n. ( ) n=O~ B( al;p ) ( q,alq / a6,alq / a7,···,alq / ar+l;q,P ) n 11.2.19

It follows that if (11.2.15) and (11.2.18) hold, then

r+I Vr(al;a6,a7, ... ,ar+l;q,p;Z)

= r+IEr(al, a2,···, ar+l; bl , ... , br ; q,p; -z), (11.2.20) and that r+l Vr is elliptically balanced if and only if

( a62 a7··· 2 ar+l2) q 2 = ( alq )r-5 . (11.2.21 )

As in Warnaar [2003c], when the argument z in the r+l Vr series equals 1 we suppress it and denote the series in (11.2.19) by the simpler notation r+l Vr(al; a6, a7, ... , ar+l; q, p). If aI, a6, a7, ... , ar+l are independent of p, then

lim r+l Vr(al; a6, a7,·· ., ar+l; q,p) p-+O = r-l Wr- 2(al; a6,···, ar+l; q, q), (11.2.22) 11.2 Elliptic and theta hypergeometric series 307 which shows that there is a shift r ----t r-2 when taking the P ----t 0 limit, and that the P ----t 0 limit of an r+l Vr(al; a6, a7,"" ar+l; q,p) series with aI, a6, a7,"" ar+l independent of p is an r-l W r - 2 series. As mentioned in the Introduction, Frenkel and Thraev showed that the elliptic 6j-symbols, which are elliptic solutions of the Yang-Baxter equation (formula (1.2. b) in their paper) found by Baxter [1973], [1982] and Date et al. [1986-1988] can be expressed as 12 Vu series (in the additive notation dis­ cussed in the next section). Then they employed the tetrahedral symmetry of the elliptic 6j-symbols, which is analogous to the symmetry of the classical, quantum and trigonometric 6j-symbols (see Frenkel and Turaev [1995, 1997]), and the finite dimensionality of cusp forms (see Eichler and Zagier [1985]) to derive (in their additive form) the following elliptic analogue of Bailey's 1O¢9 transformation formula (2.9.1)

12 Vu(a; b, c, d, e, I, Aaqn+l /el, q-n; q,p) (aq,aq/el,Aq/e, Aq/l;q,P)n (aq/e,aq/I,Aq/el,Aq;q,P)n x 12 VU(A; Ab/a, Ac/a, Ad/a, e, I, Aaqn+l/el, q-n; q,p) (11.2.23) for n = 0,1, ... , provided that the balancing condition bcdel(Aaqn+l/ef)q-nq = (aq)3, (11.2.24) which is equivalent to A = qa2/bcd, holds. Note that both of the series in (11.2.23) are E-balanced when (11.2.24) holds. Setting A = aid in (11.2.23) yields a summation formula for 10 Vg series that is an elliptic analogue of Jack­ son's 8¢7 summation formula (2.6.2) and of Dougall's 7F6 summation formula (2.1.6), which, after a change in parameters, can be written in the form:

TT ( • b d -no ) _ (aq, aq/bc, aq/bd, aq/cd; q,P)n ( ) lOv9 a, ,c, ,e,q ,q,p - ( aq /b ,aq/ c,aq /d ,aq/b c d ;q,p) n 11.2.25 for n = 0,1, ... , provided that the balancing condition bcde = a2qn+l, which can be written in the form (11.2.26) holds. Clearly, if a, b, c, d, e are independent of p, then (11.2.25) tends to Jack­ son's 8¢7 summation formula (2.6.2) as p ----t O. Unlike in the basic hyper­ geometric limit cases of (2.6.2) discussed in Chapters 1 and 2, one cannot take termwise limits of (11.2.25) to obtain a 3E2 analogue of the q-Saalschiitz formula (1.7.2), 2El analogues of the q-Vandermonde formulas (1.5.2) and (1.5.3), or even a lEo analogue of the terminating case of the q-binomial the­ orem (1.3.2) in Ex. 1.3(i). Therefore one cannot derive (11.2.25) by working up from sums at the lEo, 2El, and 3E2 levels as was done in Chapters 1 and 2, and so one is forced to employ a different approach, such as in the above­ mentioned Frenkel and Thraev derivation, or by some other method. Rather than repeating the Frenkel and Thraev [1997] derivation of (11.2.25), we will present in §11.4 a simpler derivation of (11.2.25) via mathematical induction, which is a modification of those discovered independently by Rosengren, Spiri­ donov, and Warnaar. We will then show in §11.5 that (11.2.23) follows from 308 Elliptic, modular and theta hypergeometric series

(11.2.25) in the same way as in our derivation of Bailey's 1O¢9 transformation formula (2.9.1) from Jackson's 8¢7 summation formula (2.6.2). Observe that if we set e = ±(aq)~, then (11.2.25) reduces to

V; ( . b d -no ) _ (aq, aqjbc, aqjbd, aqjcdi q,P)n ( ) 9 8 a, ,c, ,q ,q,p - ( jb j jd jb d ) 11.2.27 aq ,aq c,aq ,aq c iq,P n for n = 0, 1, ... , provided that the balancing condition (11.2.28) holds, which is equivalent to the elliptic balancing condition b2c2d2 = a3q2n+1. In view of (11.2.24), (11.2.26), (11.2.28), and of the required balancing con­ ditions for other significant special cases of the Frenkel and Turaev transforma­ tion formula (11.2.23) to hold, analogous to the definition of a VWP-balanced series given in §2.1 we call the series r+l Vr(ali a6, a7, ... , ar+li q,Pi z) a very­ well-poised-balanced (VWP-balanced) series when the very-well-poised balancing condition (11.2.29) holds. It follows that r+lVr(alia6,a7, ... ,ar+liq,P) is VWP-balanced if and only if (11.2.30) and that the summation formulas (11.2.25), (11.2.27) and the transformation formula (11.2.23) hold for n = 0,1, ... , when the series are VWP-balanced. Note that (11.2.30) reduces to (a6a7··· a2j+6)q = (alq)j when r = 2j + 5 is odd. It should also be noted that if either (11.2.30) or (a6··· ar+dq = -(±(alq)~t-5 holds, then r+l Vr(ali a6, a7, ... , ar+li q,p) is E-balanced, and hence an elliptic hypergeometric series. If r+l Vr(ali a6, a7, ... , ar+li q,p) is E-balanced, then it is VWP-balanced when r is even, but not necessarily when r is odd. In particular, the elliptic balancing condition (11.2.21) is not a sufficient condition for the Frenkel and Turaev transformation and summation formulas (11.2.23) and (11.2.25) to holdi the series in these formulas need to be VWP-balanced in order for these formulas to hold for n = 0,1, .... Since, if r+lEr is a well-poised series satisfying the relations in (11.2.15), r+lEr(al, a2,···, ar+li bl , ... , bri q,Pi z) 1. 1. 1. 1 1. 1 = r+9 Vr+8 (ali at , -at, at p2, -at jp2 ,a2, a3, ... ,ar+li q,Pi -z), (11.2.31) we find that the very-well-poised balancing condition for the above r+9 Vr+8 series is equivalent to the well-poised balancing condition !)r+l (ala2··· ar+dz = - ( ± (alq) 2 (11.2.32) for the r+lEr series in (11.2.31). Hence, a well-poised r+lEr series is called well-poised-balanced (WP-balanced) when (11.2.32) holds. In particular, the well-poised 4E3 series in the transformation formula in Ex. 11.6 is WP-balanced. Clearly, every VWP-balanced theta hypergeometric series is WP-balanced and, by the above observations, every WP-balanced theta hypergeometric series can be rewritten to be a VWP-balanced series of the form in (11.2.31). 11.2 Elliptic and theta hypergeometric series 309

Analogous to the bilateral r'¢r series, we follow Spiridonov [2002a] in defin­ ing a rGr bilateral theta hypergeometric series by

(11.2.33)

Note that rGr(al, ... ,ar;q,bl,b2, ... ,br-l;q,p;Z)

= rEr-l (al' ... ' ar; bl , ... , br- l ; q,p; z) (11.2.34) and, more generally, as in the bilateral basic hypergeometric case in (5.1.5), if the index of summation in an r+lEr series is shifted by an integer amount, then the resulting series is an r+l Gr + l series multiplied by a quotient of products of q,p-shifted factorials. Also note that corresponding to (11.2.9) we can consider nonterminating rGr series with 0 < Iql, Ipi < 1 and

where ml, ... , mr are integers. As in (11.2.10) we find that

00 rGr(al, ... ,ar;alpml, ... ,arpmT;q,p;z) = L (zur)nqNrG) (11.2.35) n=-oo with U r = I1~=l (_ak)mkp("';'k) and Nr = ml + .. ·+mr, which clearly converges for any z =I- 0 if and only if N r > O. If, as in Spiridonov [2002a], we replace r + 1 by r in the upper limit of the product in (11.2.11) and proceed as in the derivation of the condition (11.2.14) for an r+lEr series to be elliptically balanced, we find that the series rGr is elliptically balanced (E-balanced) if and only if

ala2··· ar = bl b2 ··· br · (11.2.36) If rGr is a well-poised series with (11.2.37) then the elliptic balancing condition (11.2.36) reduces to

al2 a2 2 ... a 2r = ( al b)r1 . (11.2.38) In view of (11.2.32), (11.2.34), and (5.1.7), it is consistent to call a well-poised rGr series well-poised-balanced (WP-balanced) when

(ala2··· ar)z = -( ± (albd!f. (11.2.39) In our consideration of modular series in the next section we are led to consider additive forms of special cases of the rather general power series

(11.2.40) 310 Elliptic, modular and theta hypergeometric series and the Laurent series rCis(al, ... ,ar;b1, ... ,bs;q,p;~,z) _ ~ (a1, ... ,ar;q,P)nA n - ~ n Z , (11.2.41) n=-oo (b1, ... ,bs;q,P)n where ~ = {An} is an arbitrary sequence of complex numbers. Some spe­ cial cases of these series with An = qcm(n-l)/2 or, more generally, with An = exp(aln+a2n2 +a3n3) are considered in Spiridonov [2002a, 2003bj. If An = 1 for all n, then we will suppress ~ from the left sides of (11.2.40) and (11.2.41). When we encounter series with more than one base or nome, such as in Ex. 11.25(i), Ex. 11.26 and in several of the formulas in §11.6, or multivari­ able formulas, such as the multivariable extension of the Frenkel and Thraev summation formula in §11. 7, we will write the series in terms of elliptic shifted factorials. To help keep the size of this book down we will not repeat the main elliptic identities and summation and transformation formulas in the appen­ dices. Nevertheless, for the convenience of the readers we collect below some of the most useful identities involving ()(a;p), the q,p-shifted factorials, and the q, P -binomial coefficients. ()(a;p) = ()(pja;p) = -a ()(lja;p) = -a()(ap;p), (11.2.42) ()(a 2;p2) = ()(a, -a;p), (11.2.43) ()(a;p) = ()(a, ap;p2) = ()(a~, -a~, (aq)~, -(aq)~ ;p), (11.2.44)

()(a;p) = ()(apn;p)( -a)npG), (11.2.45) (a; q,P)n()(aqn;p) = (a; q,P)n+l = ()(a;p)(aq; q,P)n, (11.2.46) (a; q,P)n+k = (a; q,P)n(aqn; q,ph, (11.2.47)

(a; q,P)n = (ql-n ja; q,P)n( -atqG), (11.2.48)

( . ) _ (a; q,P)n (_ ~)k (~)-nk (11.2.49) a,q,Pn-k-(l_nj.)q a,q,Pk a q ,

(aq-n;q,P)n = (qja;q,P)n( - ~rq-G), (11.2.50) -no ) _ (a; q,ph(qja; q,P)n -nk ( (11.2.51) aq ,q,p k - (l-kj.q a,q,Pn ) q , (a; q-l,P)n = (a-\ q,P)n( _a)nq-G), (11.2.52) 1 (_qja)n (n) (11.2.53) (a; q, p) -n = ( aq-n ;q,p ) n ( q j a;q,p) n q 2 , n ) (a; q,p)k(aqk; q,P)n (a; q,P)n+k (aq ;q,p k = =, (11.2.54) (a;q,P)n (a;q,P)n (a; q,P)n = (ap\ q,P)n( _a)nkpn(~)qkG), (11.2.55) (a 2; q2,p2)n = (a, -a; q,P)n, (11.2.56) (a; q,p2hn = (a~ , -a~, (aq) ~, -(aq) ~; q, P)n, (11.2.57) 11.2 Elliptic and theta hypergeometric series 311

(a; q,phn = (a, aq; q2,P)n, (11.2.58) (a; q,phn = (a, aq, aq2; q3,P)n, (11.2.59) (a; q,P)kn = (a, aq, ... , aqk-\ qk ,P)n' (11.2.60)

Corresponding to the q-binomial coefficient [~] q defined in Ex. 1.2, we define the q, P - binomial coefficient (or elliptic binomial coefficient) by (q;q,P)n (11.2.61 ) for k = 0, 1, ... ,n. Then

[ ~] = [n ~ k] = (q-~; q,ph (_qn)kq-(~). (11.2.62) q,p q,p (q, q,p)k For complex a we can employ (11.2.62) to define

[a] = (q-

(11.2.64)

(11.2.65)

(11.2.66)

Similarly, additional elliptic identities can be obtained by replacing each (a; q)n by (a; q,P)n and making any necessary changes in the other identities of Appendix I that do not contain any infinite products. Elliptic analogues of (1.6), (1.41), and of some other identities containing infinite shifted factorials and/or q-gamma functions can be obtained by using the Jackson [1905d] and Ruijsenaars [1997, 2001] elliptic gamma function defined by

00 1 _ z-lqj+lpk+1 f(z; q,p) = II 1 _ z j k ' (11.2.67) j,k=O q P where z, q, P are complex numbers and Iql, Ipi < 1, whose main properties are considered in Ex. 11.12. Since

. ) _ r(zqn; q,p) '71 ( (11.2.68) z, q,p n - f( z;q,p)' n E ~, and (11.2.69) 312 Elliptic, modular and theta hypergeometric series one can extend the definition of (z; q)cx in (1.6) by defining . ) _ r(zqcx; q,p) ( (11.2.70) z, q, p cx - r( z;q,p)' and extend the definition of [~] q in (1.41) and of [a] In (11.2.63) by k q,p defining (11.2.71) for Iql, Ipl < 1 and complex a and /3. The Felder and Varchenko [2003a] elliptic analogue of the Gauss multiplication formula (1.10.10) and of its q-analogue (1.10.11) is considered in Ex. 11.13.

11.3 Additive notations and modular series

In what follows it will be convenient to employ the standard notation C for the set of all complex numbers, and Z for the set of all integers. In terms of the elliptic number [a; a, T] defined in (1.6.14) the (additive) elliptic shifted factorials (or a, T-shifted factorials) are defined by

I1~:~[a + k; a, T], n = 1,2, ... , [a; a, T]n = { 1, n =0, (11.3.1) 1/ I1;;:~1 [a + n + k; a, TJ, n=-1,-2, ... , where a, a, TEe and the modular parameters a, T are such that that Im(T) > o and a rt Z+TZ with Z+TZ = {m+Tn: m,n E Z}. Let [n;a,T]! = [l;a,T]n and m [aI, a2,···, am; a, T]n = II [ak; a, T]n. (11.3.2) k=l

Then we can formally define the additive forms of the r+IEr and r+l Vr series to be the series

r+ler(al,a2, ... ,ar+l;bl , ... ,br ;a,T;z)

= [al,a2, ... ,ar+l;a,T]nzn, f (11.3.3) n=O [1, bl , ... , br ; a, T]n and

r+I Vr(al;a6,a7, ... ,ar+l;a,T;z) = f [al + 2n; a, T] [aI, a6, a7,···, ar+l; a, T]n n n=O [al;a,T] [1,1+al-a6,1+al-a7, ... ,1+al-ar+l;a,T]n z , (11.3.4) respectively, where, as usual, it is assumed that the parameters are such that each term in these series is well-defined. When the argument z in the r+ I vr 11.3 Additive notations and modular series 313 series equals 1 we suppress it and denote the series in (11.3.4) by the simpler notation r+lVr(al; a6, a7, ... , ar+l; a, T). Since, by (11.2.3) and (11.3.1),

[a; a, T]n = (qa; q,P)n qn(b-a /2 l (11.3.5) [b;a,T]n (qb;q,P)n and (11.3.6) we have that

r+ler(al, ... ,ar+l;b1, ... ,br;a,T;Z)

E (qa 1 qar+1 • qb 1 qbr . q p. zq(1+b 1 +·+br-(al +.+ar+dl/2) = r+l r , ... , " ... ,'" (11.3.7) and

(11.3.8)

From (11.2.4) and the multiplicative form of the elliptic balancing con­ dition in (11.2.14), it follows that r+ler is an elliptic hypergeometric series when

(11.3.9) which is the additive form of the elliptic balancing condition. Hence, r+ler is called E-balanced when (11.3.9) holds, and it follows that

(11.3.10) holds with the plus sign of the ± when a[al + a2 + ... + ar+l - (1 + b1 + b2 + ... + br )] is an even integer and the minus sign when it is an odd integer. Using (11.2.4) we find that the additive form of the multiplicative well-poised condition (11.2.15) is (11.3.11) in which case r+ler is called well-poised, and the elliptic balancing condition (11.3.9) reduces to

(11.3.12)

Note that (11.3.9) and (11.3.11) are not equivalent to the additive forms of the "balanced" and "well-poised" constraints in Spiridonov [2002a, (20) and (26)], respectively, which correspond to requiring that the expressions inside the square brackets in (11.3.9) and (11.3.11) are equal to zero. 314 Elliptic, modular and theta hypergeometric series

If r+ler is a well-poised series that can be written in the very-well-poised form (11.3.4), e.g., when, via (11.2.17) and (11.3.6), al 1 +T a5 = 1+-+-- 2 2IT' (11.3.13) then the elliptic balancing condition reduces to

r - 5 ] 2IT [ 1 + a6 + a7 + ... + ar+l - -2-(a1 + 1) EZ, (11.3.14) which is the condition for the r+lVr series in (11.3.4) to be elliptically balanced (E-balanced) and hence an elliptic hypergeometric series. Since e27ri(1'fl)/4 = ±1, it follows by setting z = qW that the additive form of the very-well-poised balancing condition in (11.2.29) is r-5 ] 1=f1 IT [W + 1 + a6 + a7 + ... + ar+l - -2-(a1 + 1) + -4-(r + 1) E Z, (11.3.15) which, via (11.3.8), is the condition for the series (11.3.16) to be very-well-poised-balanced (VWP-balanced) , where, as elsewhere, either both of the upper signs or both of the lower signs in =f1 and ±1 are used simultaneously. In particular, setting w = 0, it follows that if r is odd, then r+l vr(al; a6, a7,···, ar+l; IT, T) is VWP-balanced when r - 5 ] IT [ 1 + a6 + a7 + ... + ar+l - -2-(a1 + 1) E Z, (11.3.17) and if r is even, then r+lVr(al;a6,a7, ... ,ar+l;IT,T;±1) is VWP-balanced when r-5] 1=f1 IT [ 1+a6+a7+···+ar+1--2-(al+1) +-4- EZ. (11.3.18)

Thus, by replacing the parameters a, b, c, d, e in (11.2.25) by qa, qb, qC, qd, qe, respectively, and applying (11.3.8), we obtain the additive form of the Frenkel and Turaev summation formula: lOV9(a; b, c, d, e, -n; IT, T) [a + 1, a + 1 - b - c, a + 1 - b - d, a + 1 - c - d; IT, T]n (11.3.19) [a + 1 - b, a + 1 - C, a + 1 - d, a + 1 - b - C - d; IT, T]n for n = 0, 1, ... , provided that the series is VWP-balanced, i.e.,

IT[b + C + d + e - n - 2a - 1] E Z. (11.3.20) Similarly, it follows from (11.2.27), (11.2.28), and (11.3.8) that 9V8(a; b, c, d, -n; IT, T; ±1) [a + 1, a + 1 - b - c, a + 1 - b - d, a + 1 - c - d; IT, T]n (11.3.21 ) [a + 1 - b, a + 1 - c, a + 1 - d, a + 1 - b - c - d; IT, T]n 11.3 Additive notations and modular series 315 for n = 0,1, ... , provided that the series is VWP-balanced, i.e.,

a [ b + c + d - n - -a3 --1] + --1 =f 1 E Z. (11.3.22) 2 2 4 This formula also follows by setting e = (a+1)/2+k/2a for k = 0,1 in (11.3.19) and using the fact that [(a+1)/2+k/2a; a, T]n = (-1)k[(a+1)/2-k/2a; a, T]n. In addition, the Frenkel and Turaev transformation formula (11.2.23) has the additive form

12Vl1 (a; b, c, d, e, f, A + a + n + 1 - e - f, -n; a, T) [a + 1, a + 1 - e - f, A + 1 - e, A + 1 - f; a, T]n [a + 1 - e, a + 1- f, A + 1 - e - f, A + 1; a, T]n X 12Vl1 (A; A + b - a, A + c - a, A + d - a, e, f, A + a + n + 1 - e - f, -n; a, T) (11.3.23) for n = 0,1, ... , provided that the series is VWP-balanced, i.e., a[b + c + d + A - 2a - 1] E Z. (11.3.24) The summation formula (11.3.19) follows from (11.3.23) by setting A = a-d. Frenkel and Turaev [1997] used a slightly different notation for the z = 1 case of the series in (11.3.4) (with r-IWr-2 denoting an r+IVr series when z = 1) and assumed that the series terminates (to avoid the problem of convergence) and satisfies the balancing condition r-5 1 + (a6 + a7 + ... + ar+l) = -2-(al + 1). (11.3.25) They called the function that is the sum of their terminating series a modular hypergeometric function and justified the name "modular" by showing that if the series r+IVr(al; a6, a7,"" ar+l; a, T) terminates and (11.3.25) holds, then these functions are invariant under the natural action of 8L(2, Z) on the a and T variables, i.e., if aI, a6, a7, ... , ar+l E C, then a aT + r+IVr ( al; a6, a7,"" ar+l; --d' --db) CT+ CT+ = r+I Vr(al;a6,a7, ... ,ar+l;a,T) (11.3.26) for all (~ ~) E 8L(2, Z), where 8L(2, Z) is the modular group of 2 x 2 ma­ trices such that a, b, c, d E Z and ad - bc = 1. Therefore, we call (11.3.25) the modular balancing condition, and when it holds we say that the series r+IVr is M-balanced. Note that if the series r+IVr(al;a6,a7, ... ,ar+l;a,T) is M-balanced, then it is E-balanced and VWP-balanced, but if it is E-balanced or VWP-balanced then it is not necessarily M-balanced. Frenkel and Turaev also showed that if aI, a6, a7, ... , ar+l E Z, (11.3.25) holds, and the series r+l vr(al; a6, a7, ... ,ar+l; a, T) terminates, then the sum of the series is invari­ ant under the natural action of Z2 = {(m, n): m, n E Z} on the variable a, i.e.,

r+l vr(al; a6,···, ar+l; a + m + nT, T) = r+IVr(al;a6, ... ,ar+l;a,T) (11.3.27) 316 Elliptic, modular and theta hypergeometric series for all (m, n) E Z2, and hence is elliptic in a with periods 1 and T. A formal unilateral or bilateral series of the form Ln en (a, T) is called a modular series (or modular invariant) if it is invariant under the natural action of SL(2, Z) on the a and T variables, i.e.,

(11.3.28) for all (~ ~) E SL(2, Z), where, as usual, two formal series are considered to be equal if their corresponding nth terms are equal. Spiridonov [2002a] extended (11.3.26) by proving that the unilateral theta hypergeometrie series

(11.3.29) is modular if r = s + 1, (11.3.30a)

(11.3.30b) and ai + a~ + ... + a;+l = 1 + bi + b~ + ... + b;, (11.3.30e) and, more generally, that the bilateral theta hypergeometrie series

(11.3.31 ) is modular if r = s, (11.3.32a)

(11.3.32b) and 2 2 2 b2 b2 b2 a1 + a2 + ... + ar = 1 + 2 + ... + r' (11.3.32e) where it is assumed that z and the a's and b's are complex numbers such that the series are well-defined. Hence, res and rgs will be called M-balaneed when (11.3.30a)-(I1.3.30c) and (11.3.32a)-(I1.3.32c), respectively, hold. Notice that if rgr is well-poised with k = 1, ... ,r, (11.3.33) then 11.3 Additive notations and modular series 317 and so this well-poised rgr series is modular when the modular balancing con­ dition r al + a2 + ... + ar = "2 (a + 1) (11.3.35) holds. Similarly, a well-poised r+ler series with k = 1, .. . ,r, (11.3.36) is modular when the modular balancing condition r+1 al + a2 + ... + ar+l = -2-(a1 + 1) (11.3.37) holds, which reduces to (11.3.25) if (11.3.13) also holds. In order to prove the Frenkel and Turaev, and the Spiridonov modularity results, (11.3.27), and some more general results, we start by observing that, by (1.6.9) and Whittaker and Watson [1965, §21.11 and §21.5]' 'l9 1(x, e7riT) satisfies the modular symmetry relations (11.3.38) (11.3.39) where the square root of -iT is chosen so that its real part is positive, and the quasi-periodicity relations

'l9 1(X + 7f, e7riT ) = -'l91(x, e7riT ) , (11.3.40) 'l9 1(x + 7fT, e7riT ) = _e-7riT-2ix'l91(X, e7riT ). (11.3.41) Hence, by (1.6.14), the elliptic number [a; a, T] has the modular symmetries

[a; a, T + 1] = [a; a, T], (11.3.42)

[a; a IT, -liT] = e7ri (a2-1)0"2 /T [a; a, T], (11.3.43) and the quasi-periodicity relations

[k; a + 1, T] = (_l)k+l [k; a, T], (11.3.44) [k; a + T, T] = (-1 )k+le7ri(1-k 2 )(20"+T) [k; a, T], (11.3.45) where k E Z. See, e.g., van Diejen and Spiridonov [2000, 2001b], but note that (11.3.43) corrects a typo in both of these papers. Let us first consider when the bilateral series rgs is modular. Since the modular group 8L(2, Z) is generated by the two matrices

0 -1) (11.3.46) (10' which, respectively, correspond to the two transformations

(a, T) ----+ (a, T + 1), (11.3.47) it suffices to consider when each term

(11.3.48) 318 Elliptic, modular and theta hypergeometric series ofthe series in (11.3.31) is invariant under these transformations. From (11.3.1) and (11.3.42) it is clear that [a; u, T + l]n = [a; u, T]n for nEZ, and hence n E Z. (11.3.49) For the second transformation in (11.3.47) observe that from (11.3.1) and (11.3.43) it follows that

nEZ, (11.3.50) with vn(a) = n[a2 + a(n - 1) + (n - 1)(2n - 1)/6 - 1], and thus nEZ, (11.3.51) with '\n =n[ai + a~ + ... + a; - (bi + b~ + ... + b;)]

+ n(n - 1)[al + a2 + ... + ar - (b 1 + b2 + ... + bs)] + n(r - s)[(n - 1)(2n - 1)/6 - 1]. (11.3.52) Consequently, (11.3.53) if and only if (11.3.54) whenever cn(u, T) -I=- O. Obviously, (11.3.54) holds for n = 0 since we have '\0 = 0, and in order for it to hold for any n -I=- 0 with (u, T) replaced by (u/(CT+d),(aT+b)/(CT+d)) for (: ~) E 8L(2,Z), it is necessary and sufficient that '\n = O. Setting'\l = 0 in (11.3.52) gives

2 2 2 b2 b2 b2 a1 + a2 + ... + ar = 1 + 2 + ... + s + r - s, (11.3.55) which, with al = -1, b1 = 1, is a necessary and sufficient condition for the series

to be modular when ak -I=- 0 for k = 2, ... , r. Setting'\l = '\2 = 0 gives (11.3.55) and (11.3.56) which, with al = -2, b1 = 1, are necessary and sufficient for the series

to be modular when ak -I=- 0, -1 for k = 2, ... , r. Similarly, if Nand Mare nonnegative integers, then rgs( -N, a2, a3, ... , ar; M + 1, b2, b3, ... , bs; u, T; z) N =2: (11.3.57) 11.3 Additive notations and modular series 319 is modular for arbitrary values of the a's and b's if and only if An = 0, -M::::;n::::;N, (11.3.58) which, since An is a cubic polynomial in powers of n, is equivalent to (l1.3.32a)­ (l1.3.32c) if and only if N + M 2:: 3. In particular, when N + M < 3 the restriction r = 8 is not needed for the series in (11.3.57) to be modular. It follows from the above observations that the rgs series in (11.3.31) is modular for arbitrary values of its parameters if and only if (l1.3.32a)-(l1.3.32c) hold.

Replacing 8 by 8 + 1 in rgs and then setting bS +1 = 1 yields that the res series in (11.3.29) is modular for arbitrary values of its parameters if and only if (l1.3.30a)-(l1.3.30c) hold. It should be noted that, since the modularity of the above res and rgs series hold termwise under the stated conditions, when we extend res and rgs to

(11.3.59) and

(11.3.60) respectively, where A = {An} is an arbitrary sequence of complex numbers and aI, a2, ... , ar+l, Z E

rgs(al, ... ,ar;bl, ... ,bs;a+~+nT,T) = rgs(al, ... ,ar;bs, ... ,bs;a,T) (11.3.61 ) for all (~, n) E Z2, and hence is doubly periodic in a with periods 1 and T. Let al, ... , ar, bl , ... , bs E Z. Using (11.3.1) and (11.3.44), we find that cn(a + 1, T) = (-l)'Yncn (a, T) nEZ, (11.3.62) with

"In = n[al + ... + ar - (b l + ... + bs ) + (n + l)(r - 8)/2] (11.3.63) and cn(a, T) defined as in (11.3.48). So,

cn(a + 1, T) = cn(a, T) (11.3.64) if and only if "In E 2Z (11.3.65) 320 Elliptic, modular and theta hypergeometric series whenever cn(o-, T) -=I=- 0, where 2Z = {2m: mE Z} is the set of all even integers. Via (11.3.1) and (11.3.45),

k, nEZ, (11.3.66) with f-ln(k) = -n[k2 + k(n - 1) + (n - 1)(2n - 1)/6 - 1], and hence cn(o- + T, T) = (_1)"'ne-7riAn(2u+T)cn(0-, T) (11.3.67) for n E Z with An as defined in (11.3.52). Thus, if (11.3.65) holds, then (11.3.68) if and only if (11.3.69) when cn(o-, T) -=I=- O. Clearly, (11.3.69) holds for n = 0 since Ao = 0, and in order for it to hold for any n -=I=- 0 with 0- replaced by 0- + m + mT for (m, n) E Z2, it is necessary and sufficient that An = o. As in the modular case, it follows that if Nand M are nonnegative inte­ gers, then

rgs( -N, a2, a3, ... , ar; M + 1, b2, b3, ... , bs; 0-, T; z) N =2: (11.3.70) is invariant under the natural action of Z2 on the variable 0- for arbitrary values of the a's and b's if and only if

An = 0, "In E 2Z, -M:=::;n:=::;N, (11.3.71) which is equivalent to (l1.3.32a)-(l1.3.32c) when N + M ~ 3. Consequently, the rgs series in (11.3.31) is invariant under the natural action of Z2 on the variable 0- for arbitrary values of its parameters if and only if (l1.3.32a)­ (l1.3.32c) hold. Replacing s by s + 1 in rgs and then setting bs+1 = 1 yields that the res series in (11.3.29) is invariant under the natural action of Z2 on the variable 0- for arbitrary values of its parameters if and only if (l1.3.30a)­ (l1.3.30c) hold. In addition, since the invariance under the natural action of Z2 on the variable 0- in the above res and rgs series hold termwise under the stated conditions, it follows that the series in (11.3.59) and (11.3.60) are invariant under the natural action of Z2 on the variable 0- when (l1.3.30a)­ (l1.3.30c) and (l1.3.32a)-(l1.3.32c), respectively, hold. Spiridonov [2002a] called a (unilateral or bilateral) theta hypergeometric series L Cn totally elliptic if g( n) = cn+d Cn is an elliptic function of all free parameters entering it (except for z) with equal periods of double periodicity. He proved that the most general (in the sense of a maximal number of inde­ pendent free parameters) totally elliptic theta hypergeometric series coincide with the terminating well-poised M-balanced rer-l (in the unilateral case) and rgr (in the bilateral case) for r > 2. 11.4 Elliptic analogue of Jackson's 8¢7 summation formula 321

11.4 Elliptic analogue of Jackson's 8¢7 summation formula

Recall from (11.2.25) that the multiplicative form of the Frenkel and Thraev summation formula (11.3.19) is b d -no ) _ (aq, aq/be, aq/bd, aq/ed; q,P)n v; ( . (11.4.1) 10 9 a, ,e, ,e, q ,q,p - ( aq /b ,aq/ e,aq /d ,aq/b e d ;q,p )n with a2qn+l = bede and Ipi < 1. When p = 0 it clearly reduces to Jackson's summation formula (2.6.2). We will give a simple inductive proof of (11.4.1), which is a modification ofthose found independently by Rosengren, Spiridonov, and Warnaar. When n = 0 both sides of (11.4.1) are equal to 1, and for n = 1 it becomes 1 ()(aq2;p)(a, b, e, d, a2q2/bed, q-\ q,ph + ()(a;p)(q, aq/b, aq/e, aq/d, bed/aq,aq2; q,p)l q (aq, aq/be, aq/bd,aq/ed;q,p)l (11.4.2) (aq/b,aq/e,aq/d,aq/bed;q,p)l· By using (11.2.5) and (11.2.42), (11.4.2) can be rewritten as ()(aq/b, aq/e, aq/d, aq/bed;p) + ()(b, e, d, a2q2 /bed; p)aq/bed = ()(aq, aq/be, aq/bd, aq/ed;p). (11.4.3) However, (11.4.3) is equivalent to the identity in Ex. 2.16(i) if we replace x, A, /-l and v by aq/(be)!, (be)!, aq/d(be)! and (e/b)!, respectively. So (11.4.1) is true for n = 1. Let us now assume that (11.4.1) is true when n = m. We need to prove that

m+l ()(aq2k;p)(a, b, e, d, a2qm+2/bed, q-m-\ q,ph k ~ ()(a;p)(q, aq/b, aq/e, aq/d, bedq-m-l la, aqm+2; q,p)k q (aq, aq/be, aq/bd, aq/ed; q,P)m+l (11.4.4) (aq/b, aq/e, aq/d, aq/bed; q,P)m+l . Observe that, by (11.2.43), (a2qm+2/bed, q-m-\ q,ph (bedq-m-l/a, aqm+2; q,p)k (a2qm+l /bed, q-m; q,ph (bedq-m / a, aqm+l; q, ph x [()(a2qm+k+l/bed,bedqk-m-l/a,aqm+l,q-m-\p)] . ()(aqm+k+l, qk-m-l, a2qm+l /bed, bedq-m-l /a;p) (11.4.5) However, by Ex. 2.16(i) the ratio of the ()-terms inside the brackets in (11.4.5) is equal to 1 + bedq-m-l ()(aqk,qk,a/bed,a2q2m+2/bed;p) . a ()( aqm+k+l, qk-m-l, a2qm+l /bed, bedq-m-l / a; p) (11.4.6) When we multiply (11.4.6) by ()(aq2k;p)(a, b, e, d, a2qm+l /bed, q-m; q,p)k k ()(a;p)(q, aq/b, aq/e, aq/d, bedq-m la, aqm+l; q,P)k q 322 Elliptic, modular and theta hypergeometric series and sum over k from 0 to m+ 1 we find that the first term in (11.4.6) contributes a zero term at k = m + 1 because (q-m; q,P)m+l = 0, while the second term contributes a zero term at k = 0 because of the factor ()(qk;p). After some simplifications it follows that

10 Vg(a; b, e, d, a2qm+2/bed, q-m-l; q,p)

= 10 Vg(a; b, e, d, a2qm+l /bed, q-m; q,p) bedq-m (aq; q,ph()(b, e, d, a/bcd, a2q2m+2/bed;p) + _-=--_ -,-_-'------=,---=c-"--'--_'-----,---,--'------'-----,--_,------::--,------'--,------'--''--'------,-- a (aqm+l, bedq-m-l fa; q,ph()(aq/b, aq/e, aq/d;p) x 10 Vg(aq2; bq, eq, dq, a2qm+2 /bed, q-m; q,p). (11.4.7)

However, by the induction assumption both 10 Vg series on the right side of (11.4.7) can be summed. Thus, from (11.4.7) and the n = m case of (11.4.1),

10 Vg(a; b, e, d, a2qm+2/bed, q-m-l; q,p) (aq,aq/be, aq/bd,aq/ed; q,P)m (aq/b,aq/e,aq/d,aq/bed;q,p)m bedq-m (aq; q,ph()(b, e, d, a/bcd, a2q2m+2/bed;p) + --=--- -;---'--:-:~"::"':--~:--;---'------'----,---::-c,----=-;:------',-;-----'--'c..:--:-----,- a (aqm+l, bedq-m-l fa; q,ph()(aq/b, aq/e, aq/d;p) (aq3,aq/be,aq/bd,aq/ed;q,p)m x (aq2/b,aq2/e,aq2/d,a/bed;q,p)m (aq,aq/be,aq/bd,aq/ed;q,p)m (aq/b,aq/e,aq/d,aq/bed;q,p)m x ()(b, e, d, a2q2m+2 /bed; p) ] [1 _ (11.4.8) ()(aqm+l/b, aqm+l /e, aqm+l /d, bedq-m-l /a;p) , by use of the identities (11.2.42) and (11.2.43). Applying Ex. 2.16(i) once again we find that 1 _ ()(b, e, d, a2q2m+2 /bed; p) ()(aqm+l /b, aqm+l/e, aqm+l /d, bedq-m-l /a;p) ()( aqm+l, aqm+l /be, aqm+l /bd, aqm+l / cd; p) (11.4.9) ()(aqm+l/b, aqm+l /e, aqm+l /d, aqm+l /bed;p)' Substituting (11.4.9) into (11.4.8) and using (11.2.43) shows that (11.4.1) is true for n = m+ 1 whenever it is assumed to be true for n = m. This completes the proof of (11.4.1) by induction. Another proof of (11.4.1) by induction is indicated in Ex. 11.4. Note that by letting e ----+ aqn+l in (11.4.1) we obtain the summation formula for a truncated 8 V7 series t ()(aq2k;p) (a, b, e, a/be; q,ph qk k=O ()(a;p) (q, aq/b, aq/e, beq; q,P)k (aq,bq,eq,aq/be;q,p)n (11.4.10) (aq/b,aq/e,beq,q;q,p)n' which is an elliptic analogue of Ex. 2.5. 11.5 Elliptic analogue of Bailey's transformation formula 323

As alluded to in §11.2 the limit of the summation formula (11.4.1) as one of three parameters b, e, d tends to infinity does not exist when p -=I=- 0, even though these limits do exist when p = 0 yielding formula (2.4.2). Similarly, one cannot take the limit a ----7 0 after replacing d by aq/d in (11.4.1) to get an elliptic analogue of the q-Saalschiitz formula (1.7.2), as was done in the basic hypergeometric case in §2.7. The root cause of the nonexistence of these limits is the fact that lima--+o ()( a; p) and lima--+oo ()( a; p) do not exist in the extended complex plane when p -=I=- O. Nevertheless, one can obtain an elliptic analogue of the q-Saalschiitz formula (1.7.2) at the 10 Vg level by employing Schlosser's observation (see Warnaar [2003e]) that by making the simultaneous substitutions {a,e,d,e,p} ----7 {dp,dqp/c,a,ep,p2} in (11.4.1) we obtain the summation formula -n 2) (c/a, c/b,dqp, dqp/ab; q,p2)n V ( / (11.4.11) lO 9 dp;a,b,dqp c,ep,q ;q,p = (c, c / a b, d qp / a, d qp /b ; q, P2) n with cdqn = abe, Ipi < 1, and n = 0, 1, ... , which tends to (1.7.2) as p ----7 O.

11.5 Elliptic analogue of Bailey's transformation formula for a terminating 1O¢9 series

Recall from (11.2.23) that the multiplicative form of Frenkel and Turaev's transformation formula (11.3.23) is

12 Vl1 (a; b, c, d, e, j, Aaqn+1 /ej, q-n; q,p) (aq,aq/ej,Aq/e, Aq/j;q)n (aq/e,aq/j,Aq/ej,Aq;q)n x 12 Vl1 (A; Ab/a, Ac/a, Ad/a, e, j, Aaqn+1/ej, q-n; q,p) (11.5.1) with A = qa2/bcd. This formula reduces to the Frenkel and Turaev summation formula (11.4.1) when aq = be, bd, or ed, and it reduces to Bailey's transforma­ tion formula (2.9.1) when p = O. It can be proved by proceeding as in (2.9.2) and (2.9.3) with the q-shifted factorials and the S¢7 and 1O¢9 series replaced by q,p-shifted factorials and lOV9 and 12Vl1 series, respectively. In view ofthe importance of this transformation formula and for the sake of completeness, we decided to present its proof here. First observe that by (11.4.1)

10 Vg(A; Ab/ a, Ac/a, Ad/a, aqm, q-m; q,p) (Aq, b, c, d; q,P)m (11.5.2) (a/A, aq/b, aq/c, aq/d; q,P)m' and hence the 12Vl1 series on the left side of (11.5.1) equals

~ ()(aq2m;p) (a, e, j, Aaqn+l /ej, q-n; q,P)m (a/A; q,P)m m m=O~ ()( a; p) (q, aq/ e, aq/ j, ejq - n / a, aqn+l; q, P)m (Aq; q, P)m q f ()(Aq2j;p) (A, Ab/a, Ac/a, Ad/a, aqm, q-m; q,p)j j x j=O ()(A;p) (q, aq/b, aq/c, aq/d, Aql-m la, Aqm+l; q,p)j q 324 Elliptic, modular and theta hypergeometric series

= t f ()(aq2m.;p)()(~q2j;p) (a; q.'P)m+j(a/~; q,P)m-j qm m=O j=O ()(a, p)()(A,p) (Aq, q,P)m+j(q, q, P)m-j (e, j, Aaqn+l/ej, q-n; q,P)m(A, Ab/a, Ac/a, Ad/a; q,p)j (a)j x (aq/e, aq/ j, ejq-n / A, aqn+1; q, P)m(q, aq/b, aq/c, aq/d; q,p)j >: = t ()(Aq2j;p) (A, Ab/a, Ac/a, Ad/a, e, j, Aaqn+l /ej, q-n; q,p)j j=O ()(A;p) (q, aq/b, aq/c, aq/d, aq/e, aq/ j, ejq-n / A, aqn+l; q, p)j

(aq' q phj (aq)j 2' .. + '+1 . x (A : ' ) .""\ lQVg(aq J;a/A,eqJ,jqJ,Aaqn J /ej,qJ-n;q,p). q, q,p 2J A (11.5.3) However, by (11.4.1) 10Vg(aq2j; a/A,eqj, jqj, Aaqn+j+1/ej,qj-n;q,p) (aq2j+l,Aqj+l/e,Aqj+1/j,aq/ej;q,P)n_j (Aq2j+l,aqj+l/e,aqj+l/j,Aq/ej;q,P)n_j (aq,Aq/e,Aq/j,aq/ej;q,P)n (Aq,aq/e,aq/j,Aq/ej;q,P)n x (Aq;qhj(aq/e,aq/j,ejq=n/A;q)j (~)j. (11.5.4) (aq;qhj(Aq/e,Aq/j,ejq n/a;q)j a Substitution of (11.5.4) into (11.5.3) gives (11.5.1). Note that by applying the transformation formula (11.5.1) to the 12VU series on the right side of (11.5.1), keeping e, Ac/a, Ad/a and q-n unchanged, we obtain that 12 Vu(a; b, c, d, e, j, Aaqn+1 /ej, q-n; q,p) (aq,aq/ce, aq/de,aq/ej,Aq/j,b; q,P)n (aq/c, aq/d,aq/e, aq/j,Aq/ej,b/e;q,P)n x 12 Vu(eq-n /b; e, Ac/a, Ad/a, aq/bj, eq-n la, ejq-n /Ab, q-n; q,p), (11.5.5) which is the elliptic analogue of Ex. 2.19. Setting j = qa 2 /bcde in (11.5.1) gives the transformation formula for a truncated 10 Vg series

~ ()( aq2k;p)(a,b,c,d,e,qa2/bcde;q,ph k ()(a;p)(q, aq/b,aq/c, aq/d, aq/e, bcde/a; q,P)k q k=O~ (aq, bcd/a, a2q2/bcde,eq; q,P)n (aq/e,bcde/a, a2q2/bcd,q; q,P)n x 12 Vu (qa2/bcd; aq/bc, aq/bd, aq/cd, e, qa2/bcde, aqn+l, q-n; q,p) (11.5.6) in which the 12 Vu can be transformed via (11.5.1) and (11.5.5) yielding other 12 Vu series representations for the above truncated series. 11.6 Multibasic summation and transformation formulas 325

11.6 Multibasic summation and transformation formulas for theta hypergeometric series

First observe that by replacing a in (11.4.1) by a/q it follows that the n = 1 case of (11.4.1) is equivalent to the identity 1 _ B(b,c,d,a2 /bcd;p) B(a,a/bc,a/bd,a/cd;p) (11.6.1) B(a/b,a/c,a/d,bcd/a;p) B(a/bcd,a/d,a/c,a/b;p)· Next, corresponding to (3.6.12), define m:::;n, m = n+ 1, (11.6.2) m 2: n + 2, for n, m = 0, ±1, ±2, ... , and let

n-l ll(b d 2/b d ) U k, Ck, k, a kCk kiP II k (11.6.3) Un = k=O B(ak/bk,ak/Ck,ak/dk,bkckdk/ak;p)' for integer n, where it is assumed that the a's, b's, c's, d's are complex numbers such that Un is well defined for n = 0, ±1, ±2, .... Then, by using (11.6.1) with a, b, c, d replaced by ak, bk, Ck, dk, respectively, we obtain the indefinite summation formula n U-m - Un+l = L (Uk - Uk+l) k=-m

(11.6.4) for n, m = 0, ±1, ±2, .... Since Uo = 1 by (11.6.2), setting m = 0 in (11.6.4) gives the summation formula ~ B(ak,ak/bkck,ak/bkdk,ak/ckdk;p) k=O~ B(ak/bkckdk,ak/dk,ak/ck,ak/bk;p) x IT B(bj , Cj, dj, a;/bjcjdj;p) . j=O B(aj/bj , aj/cj, aj/dj , bjcjdj/aj,p)

= 1 _ B(bj , Cj, dj, a;/bjcjdj;p) IT (11.6.5) j=O B(aj/bj , aj/cj, aj/dj , bjcjdj/aj;p) for n = 0,1, ... , which is equivalent to formula (3.2) in Warnaar [2002bj. When p = 0 this formula reduces to a summation formula of Macdonald that was first published in Bhatnagar and Milne [1997, Theorem 2.27]' and it contains the summation formulas in W. Chu [1993, Theorems A, B, C] as special cases. Notice that in (11.6.1), (11.6.3), (11.6.4) and (11.6.5) we have arranged the components of each quotient of products of theta functions so that the well-poised property of these quotients is clearly displayed; e.g., in (11.6.5) 326 Elliptic, modular and theta hypergeometric series the quotients of the theta functions that depend on k are arranged so that each product of corresponding numerator and denominator parameters equals aVbkckdk, and each of the corresponding products that depend on j equals

If we set ak = ad(rst/q)k, bk = brk, Ck = csk, dk = ad2tk/bc, then Un reduces to _ (a;rst/q2,P)n(b;r,P)n(c;s,P)n(ad2/bc;t,P)n Un = ~--~~~~~~~~~~~~~~~~~~--~~- (dq;q,P)n(adst/bq;st/q,P)n(adrt/cq;rt/q,P)n(bcrs/dq;rs/q,P)n and it follows from (11.6.4) by applying (11.2.42), (11.2.43) and (11.2.49) that we have the Gasper and Schlosser [2003] indefinite multibasic theta hypergeo­ metric summation formula n O(ad(rst/q)k, brk/dqk, csk/dqk, adtk/bcqk;p) O(ad, bid, c/d, ad/bc;p) k=-mL (a; rst/q2 ,ph(b; r, p)k(C; s,ph(ad2/bc; t,ph k x (dq; q,p)k(adst/bq; st/q,p)dadrt/cq; rt/q,p)k(bcrs/dq; rs/q,p)k q O(a,b,c,ad2/bc;p) dO(ad, bid, c/d, ad/bc;p) (arst/q2;rst/q2,P)n(br;r,P)n(cs;s,P)n(ad2t/bc;t,P)n { x (dq; q,P)n(adst/bq; st/q,P)n(adrt/cq; rt/q,P)n(bcrs/dq; rs/q,P)n _ (c/ad; rt/q,P)m+l (d/bc; rs/ q,P)m+l (l/d; q, P)m+l (b/ad; st/q, P)m+l } (l/c; s, P)m+l (bc/ad2; t, P)m+l (l/a; rst/q2, P)m+l (l/b; r, P)m+l (11.6.6) for n, m = 0, ±1, ±2, . .. . Formula (3.6.13) follows from (11.6.6) by setting P = 0 and then setting r = P and s = t = q. If P = 0 and max(lql, Irl, lsi, Itl, Irs/ql, Irt/ql, Ist/ql, Irst/q21) < 1, then we can let n or m in (11.6.6) tend to infinity to obtain that this special case of (11.6.6) also holds with nand/or m replaced by 00, just as in the special case (3.6.14). Even though one cannot let n ---; 00 or m ---; 00 in (11.6.6) when P -I=- 0 to obtain summation formulas for nonterminating theta hypergeometric series since lima-->o O( a; p) does not exist when P -I=- 0, it is possible to let n ---; 00 or m ---; 00 in (11.6.4) to obtain summation formulas for nonterminating series containing products of certain theta functions. For example, if we let O(bk,ck,dk,a~/bkckdk;P) Z k = -:-;-----,::-'-c.:.:.....:.,.:---'--'-'----':''':--'-':--''--'-'-:--=--'; __---,- O(ak/bk,ak/ck,ak/dk,bkckdk/ak;p) denote the kth factor in the product representation (11.6.3) for Un and observe that when a is not an integer power of P

O(b;p) = 1 lim < 1, 1 Ipi b ..... a'l O(a/b;p) , 11.6 Multibasic summation and transformation formulas 327 then it is clear that there exist bilateral sequences of the a's, b's, c's, and d's in (11.6.4) such that Re Zk > 0 for integer k and the series

00 L log Zk converges, (11.6.7) k=-oo where log Zk is the principal branch of the logarithm (take, e.g., bk, Ck, and 1 d k so close to af, that Ilogzkl < l/k2 for k = ±1,±2, ... ). Then limn---> 00 Un and limm--->oo U-m exist, and we have the Gasper and Schlosser [2003] bilateral summation formula

provided that (11.6.7) holds. However, such bilateral sums do not appear to be particularly useful. It is more useful to use the m = 0 case of (11.6.6) in the form n ()(ad(rst/q)k, brk /dqk, csk /dqk, adtk /bcqk;p) ()(ad, bid, e/d, ad/be;p) Lk=O (a;rst/q2,p)k(b;r,p)k(e;s,p)k(ad2/be;t,p)k k x (dq;q,p)k(adst/bq;st/q,p)k(adrt/eq;rt/q,p)k(bers/dq;rs/q,p)k q ()(a,b,e,ad2/be;p) d ()(ad, bid, e/d, ad/be; p) (arst/q2;rst/q2,P)n(br;r,P)n(es;s,P)n(ad2t/bc;t,P)n x ~--~~~-=~~~~~~~--~~~~~--~~- (dq;q,P)n(adst/bq;st/q,P)n(adrt/eq;rt/q,P)n(bers/dq;rs/q,P)n ()(d,ad/b,ad/c,be/d;p) (11.6.9) d ()(ad, b/ d, e/d, ad/be; p) , its e = s-n special case n ()(ad(rst/q)k, brk/dqk, sk-n /dqk, adsntk /bqk;p) ()(ad,b/d,s-n/d,adsn/b;p) k=OL (a; rst/q2,p)k(b; r,p)k(S-n; s,p)k(ad2sn /b; t,p)k k x q (dq;q,p)k(adst/bq;st/q,p)k(adsnrt/q;rt/q,p)k(brs1-n/dq;rs/q,p)k ()(d,ad/b,adsn,dsn/b;p) (11.6.10) ()(ad,d/b,dsn,adsn/b;p) , 328 Elliptic, modular and theta hypergeometric series and the d ----t 1 limit case of (11.6.10) n ()(a(rst/q)k, brk /qk, sk-n /qk, asntk /bqk;p) ()(a b s-n asn/b'p) k=OL "" (a; rst/q2 ,ph(b; r, p)k(S-n; s,p)k(asn /b; t, P)k k X (q; q, p)k(ast/bq; st/q, p)k(asnrt/q; rt/q, p)k(brs1- n /q; rs/q,p)k q = on,O, (11.6.11) where n = 0,1, ... , which are generalizations of (3.6.15), (3.6.16), and (3.6.17), respectively. In particular, replacing n, a, b, and k in the s = t = q case of (11.6.11) by n - m, armqm, brmq-m, and j - m, respectively, gives the orthogonality relation n L anjbjm = on,m (11.6.12) j=m with

(11.6.13)

(11.6.14) which shows that the triangular matrix A = (anj) is inverse to the triangular matrix B = (b jm ), and gives a theta hypergeometric analogue of (3.6.18)­ (3.6.20). Proceeding as in the derivation of (3.6.22), it follows that (3.6.22) extends to the bibasic theta hypergeometric summation formula

()(a/r, b/r;p) t (aqk, bq-k; r'P)n_1()~aq2k /b;p) (_l)kq(~) = On 0 k=O (q; q,ph(q; q,P)n-k(aq /b; q,P)n+1 ' (11.6.15) for n = 0, 1, ... , which reduces to

8V7(a/b;q/b,aqn-l,q-n;q,p) = on,O when r = q. The summation formula (11.6.10) and the argument in §3.8 can be em­ ployed to extend (3.8.14) and (3.8.15) to the quadratic theta hypergeometric transformation formulas n ()( acq3k; p) (a, b, cq/b; q,P)k(f, a2c2q2n+1 / f, q-2n; q2,P)k k {; ()(ac;p) (cq2, acq2/b, abq; q2,p)k(acq/ f, f / acq2n, acq2n+1; q,p)k q (acq;q,P)2n(ac2q2/bf,abq/f;q2,P)n (acq/f;q'P)2n(abq,ac2q2/b;q2,P)n x 12 Vn (ac2/b; f, ac/b, c, cq/b, cq2/b, a2c2q2n+1 / f, q-2n; q2,p) (11.6.16) and 2n ()( acq3k; p) (d, f, a2c2q/ df; q2 ,ph(a, cq2n+1, q-2n; q,P)k k {; ()(ac;p) (acq/d, acq/ f, df lac; q,p)k(cq2, aql-2n, acq2n+2; q2,p)k q 11.6 Multibasic summation and transformation formulas 329

(acq,acq/d!;q,P)n(acq1-n/d,acq1-n/!;q2,P)n (acq/d,acq/!;q,P)n(acq1-n,acq1-n/d!;q2,P)n x 12 Vu (acq-2n-\ c, d,!, a2c2q/d!, aq-2n-\ ql-2n, q-2n; q2, p) (11.6.17) for n = 0,1, ... ; see Warnaar [2002b, Theorems 4.2 and 4.7]. As in the derivation in Gasper [1989a] of the quadbasic transformation formula in Ex. 3.21, indefinite summation formulas such as in (11.6.5) and (11.6.9) can be extended to transformation formulas by using the identity n n-k n n-k LAkLAj = LAkLAj, (11.6.18) k=O j=O k=O j=O which follows by a change in order of summation. In particular taking Ak to be the kth term in the series in (11.6.5) and Ak to be this term with ak, bk, Ck, dk, and P replaced by Ak, Bk, Ck, Dk, and P, respectively, yields the rather general transformation formula

~ B(ak,ak/bkck,ak/bkdk,ak/ckdk;p) k=O~ B(ak/bkckdk,ak/dk,ak/ck,ak/bk;p) x IT B(bj , Cj, dj , a;/bjcjdj;p) . j=O B(aj /bj , aj /Cj, aj /dj , bjcjdj /aj ,p)

x {I-IT B(Bj,Cj,Dj,A;/BjCjDj;P) } j=O B(Aj/Bj , Aj/Cj , Aj/Dj , BjCjDj/Aj ; P)

= ~ B(Ak' Ak/ BkCk, Ak/ BkDk, Ak/CkDk; P) ~ B(Ak/ BkCkDk, Ak/ Dk, Ak/Ck, Ak/ Bk; P) x IT B(Bj,Cj,Dj,A;/BjCjDj;P). j=O B(Aj / B j , Aj/Cj , Aj/Dj , BjCjDj/Aj , P)

nrr-k B(bj , Cj, dj , a;/bjcjdj;p) } x { 1- . j=O B(aj/bj , aj/cj, aj/dj , bjcjdj/aj;p) (11.6.19) The special case of (11.6.19) corresponding to using (11.6.9) instead of (11.6.5) contains the parameters a, b, c, d, A, B, C, D, and the bases r, s, t, R, S, T, the nomes P and P, and it contains the quadbasic transformation formula in Ex. 3.21 as a special case (see Ex. 11.26). By proceeding as in Gasper and Schlosser [2003] we can use (11.6.11) to derive multibasic extensions of the Fields and Wimp, Verma, and Gasper expansion formulas in (3.7.1)-(3.7.3), (3.7.6)-(3.7.9), and multibasic theta hy­ pergeometric extensions of (3.7.6)-(3.7.8). Set a = 'Y(rst/q)j and b = a(r/q)j in (11.6.11). For j, n = 0,1, ... , let Bn(P) and Cj,n be complex numbers such that Cj,o = 1 and the sequence {Bn (p)} has finite support when P -=I=- O. Then, 330 Elliptic, modular and theta hypergeometric series as in (3.7.5), it follows that Bj(p)xj

n 00 ()(f(rst/q)n, a(r /q)n, ,a- 1 snHtj, ,a- 1 (st)nH, s-kqj-n;p) = {; ~ ()(sj-n-k;p)(q; q,P)n(frtsn+k /q; rt/q,P)n(ars1- n- k /q; rs/q,P)n x (fa-1(stt+lqj-n-\ st/q, p)k-1 (fa-1snHtJ+\ t, P)n-j-l x (arq-j; r, P)n-1 (frstqj-2; rst/q2 ,P)n-1 (q-n; q,p)j

x (_l)n Bn+k(p)Cj,nH_jXnHqn(l+j-n-kHG) (11.6.20) for j = 0,1,00 .. Now multiply both sides of (11.6.20) by Ajwj/(q;q,p)j and sum from n = 0 to 00 to obtain the Gasper and Schlosser multibasic expansion formula

n ()( s -k q j-n ;P ) ( ,rst q j-2 ;rst/ q2) ,p n-1 (arq -j ;r,p) n-1 X L . k j=O ()(sJ-n- ;p)(q; q,p)j

x ()(fa-1snHtj; p)(fa-1(st)n+1qj-n-\ st/q,p)k-1 x (fa-1sn+ktJ+\ t,P)n-j-1AjCj,n+k-jWj qn(j-n-k) , (11.6.21 ) which reduces to (3.7.6) by setting P = 0 and then letting r = P and s = t = q. Setting r = s = t = q in (11.6.21) yields an expansion formula that is equivalent to the following extension of (3.7.7) fAnBn(P) (xw)n = f (a"qn+1/a , a,(3; q,P)n (::f n=O (q; q,P)n n=O (q, ,qn; q,P)n a

00 ()( 2n+2k/. )( n/ -1 n (3 n. ) X '"' ,q a,p ,q a, a ,aq, q ,q,p k Bn+k(P)Xk ~ ()(fq2n /a;p)(q, ,q2n+1; q,p)k

x (q-n"qn;q,p)j k(wq)j ~ (11.6.22) ~ (q, ,qn+1 la, q1-n la, a, (3; q,p)j J , where, as above, {Bn(P)} has finite support when P -=I=- O. Of course, one cannot let a ----+ 00 in (11.6.22) to get an extension of (3.7.3) that holds for any P -=I=- O. Analogous to the q-extension of the Fields and Wimp expansion formula (3.7.1) displayed in (3.7.8), from (11.6.22) one easily obtains the rather general theta hypergeometric expansion formula

f (aR' CT; q,P)n AnBn(p)(xwt n=O (q,bs,dU;q,P)n 11.7 Rosengren's elliptic extension of Milne's fundamental theorem 331

n (-n n f ) x" q ,,,{q ,aR, M;q,P j A.(wq)j (11.6.23) f;:o (q,,,{qn+1/ u,q1-n/u,bs ,eK;q,p)j J , where we used a contracted notation analogous to that used in (3.7.1) and (3.7.8), and to avoid convergence problems it is assumed that {En} has finite support when Pi=- O. Additional formulas are given in the exercises.

11. 7 Rosengren's elliptic extension of Milne's fundamental theorem

Milne's [1985a] fundamental theorem states that

~(zqk) (arzs/zr; qhs = (a1 ... an; q)N IT (11.7.1) ~(z) r,s=1 (qzs/zr;qk (q;q)N'

and the a's and z's are fixed parameters. This is the identity that played a fundamental role in Milne's derivation of the Macdonald identities for the affine Lie algebra A~1), as well as his general approach to hypergeometric series on An or U(n). An important tool for proving (11.7.1) is an easily verifiable identity n n I1(bj - ak) " j=1 b1 ... bn (11. 7.3) ~ ak I1 (aj - ak) = a1 ... an - 1. k=1 #k

To derive an elliptic extension of Milne's identity, Rosengren [2003c] used the following elliptic extension of (11.7.3) n n I1 B(ak/bj;p) " j=1 _ 0 (11. 7.4) k~1 I1 B(ak/aj;p) - , - jf.k where it is assumed that the balancing condition a1 ... an = b1 ... bn holds. Note that (11.7.4) is the same identity as in Ex. 5.23. Slater [1966] gave a proof of it by using special relationships between the parameters in the general transformation formula (5.4.3). Also see Tannery and Molk [1898] for a simple proof via residues. However, since Rosengren [2003c] gave a rather elegant yet 332 Elliptic, modular and theta hypergeometric series elementary proof of (11.7.4) that is similar in spirit to the way Milne proved (11.7.3), we will present his proof. First, it is easy to see that Ex. 2.16(i) is equivalent to the n = 3 case of (11.7.4). Assume that (11.7.4) is true for n = m. Then, by separating the (m + 1)-th term from the series on the left side of (11.7.4), we can rewrite it in the form m-l m-l m-l ()(ak/bm;P) IT ()(ak/bj;p) ()(am/bm;p) IT ()(am/bj;p) 1 L ()(ak/am;P) ji1 ()(ak/aj;p) = - m-l j=1 , (11.7.5) k=1 #k IT ()(am/aj;p) j=1 where the a's and b's are always assumed to satisfy the balancing condition. Considered as a function of am the expression on the left side of (11.7.5) re­ sembles a partial fraction expansion of the product on the right side. When n = m + 1, we write am +l = t, say, and seek an expansion of the form ()(t/bj;p) Ok ()(b1··· bmak/al··· amt;p) IT = f (11. 7.6) j=1 ()(t/aj;p) k=1 ()(ak/t;p) . That such an expansion exists follows by using induction on m and the fact that the m = 2 case is equivalent to (11.4.3). Multiplying both sides of (11.7.6) by ()(t/ak;p) = -()(ak/t;p)t/ak and setting t = ak, we find that m IT ()(ak/bj;p) Ok = _ ...,.-;-::_----:_-:--J_.=_1 __~=-_-:-;-----;-_---:- ()(b1··· bm/al ... am;p) IT#k ()(ak/aj;p)

(11.7.7) ()( am+dbm+1; p) IT ()( ak/ aj; p) #k with al··· am+l = b1··· bm+1. Now substitute (11.7.7) into (11.7.6) and set t = am +l to get m m m ()(ak/bm+1;p) IT ()(ak/bj;p) ()(am+dbm+1;p) IT ()(am+dbj;p) 1 ()(ak/am+l;P) ji1 ()(ak/aj;p) = - fi ( ) k=1L #k () am+lj=~ aj;p j=1 which is the same as (11.7.5) with m replaced by m + 1. This completes the proof of (11.7.4). An elliptic extension of (11.7.1) given in Rosengren [2003c, Theorem 5.1] states that

(b/al, ... , b/an +l; q,p)N (11. 7.8) (q,bz1, ... ,bzn ;q,p)N ' 11.7 Rosengren's elliptic extension of Milne's fundamental theorem 333 where ~(z;p) = ~n(Z;P) = II zrB(zs/zr;P) (11. 7.9) l:S;r

If we set P = 0 and take the limits b ----+ 0, an+1 ----+ 0 such that b/an+1 ----+ a1 ... an Zl ... Zn, then it is easy to see that (11.7.8) approaches Milne's limit (11.7.1). When P = 0, (11.7.8) reduces to Milne's [1988a, Theorem 6.17] An Jackson summation formula, which is a multivariable extension of Jackson's sum (2.6.2). To prove (11.7.8) by induction, observe that when N = 1 we have that k i = Dij for some j, and thus j can be used as the summation index. Then, after a bit of manipulation, 1 n ~(z;p) TI (qzr!zs;q,phr r,s=l and formula (11.7.8) becomes n+1 n+1 n TI B(aszj;p) TI B(b/as;p) L s=l = -=-s=--=,-l ____ (11.7.11) "_ B(bzj;p) B(zj/zs;p) B(b . ) , )-1 sopjTI TIn Zs,p s=l which is the m = n + 1 case of (11. 7.5) with a different set of parameters. So (11.7.8) is true when N = 1. Assume that it is true for a fixed N. We will show that it is also true for N + 1. Denoting the right side of (11. 7.8) by RN, we have that _ (b/a1,"" b/an+1; q,p)N+1 R N+1- (q, bz1, ... ,bzn; q, p)N+1 B(q;p) B(bqN /a1,"" bqN /an+1;p) R (11. 7.12) B(qN+1;p) B(q, bqN Zl, ... ,bqN Zn; p) N· However, by the induction hypothesis we can replace RN by the series on the left side of (11.7.8) and replace the ratio B(bqN /a1,"" bqN /an+1;p) /B(q, bqN Zl, ... , bqN zn;P) by the N = 1 case of the series in (11.7.8) with Zm and b replaced by zmqkTn and bqN, respectively. Thus,

kl,""",kn?O k1+"""+kn=N 334 Elliptic, modular and theta hypergeometric series

(11. 7.13)

In the above summation, replace k by k - j. Since jr E {a, I}, we have (}(bzrqN+kr-jr; p) (}(bzrqN +kr; p) (bzrqN+kr-jr; q,P)jr (bzrqN+kr; q,P)jr' (b) (bzr; q,phr Zr; q,p kr-jr = (b kr-l. ).' zrq ,q,p Jr and n II (Zrq1+kr-ks-jr+js /Zs; q,P)jr = (}(q;p) II(Zrqkr-ks /Zs; q,P)jr· r,s=l Thus

kl, ... ,kn?':O kl +·+kn=N +1 n n (}(bzrqN+kr;p) TI (aszr;q,P)kr x g (}(bzrqN;p)(bzr; q,P) ksr=[e=1 (Zrq/Zs; q,p)kr

(11.7.14) The sum over j can be rewritten in the form n n (}(bzmqk",-\p) TI (}(Zmqk",/Zs;p) ~1 (}(bzmqN+krn;p) rr=~(Zmqkrn-k/Zs;p) m- s#m

r=1 n (11. 7.15) TI (}(bzrqN+kr;p) r=1 by (11.7.11). From (11.7.14) and (11.7.15) it follows that

RN+1 = L kl, ... ,kn?':O kl +·+kn=N+1 11.7 Rosengren's elliptic extension of Milne's fundamental theorem 335 which completes the proof of (11.7.8). Formula (11.7.8) may be regarded as a multivariable extension of Frenkel and Thraev's summation formula (11.4.1) which, as we have seen before, is an elliptic analogue of Jackson's summation formula (2.6.2) for a terminating very-well-poised S

(11.7.17) which resembles the corresponding condition (2.6.1) for the one-dimensional Jackson sum. The transformations of the various terms of the series in (11.7.8) are straightforward, albeit somewhat tedious. First note that b.n+1 (Zn+1qkn+l ;p) qkrB(Zsqks-kr/zr;P) b.n+1 (Zn+1; p) II B(zs/zr;P)

qkrB(Zsqks-kr/zr;P) II B(zs/zr;P)

(11. 7.18)

n+1n+2 n n+2 n+2 II II (aszr ; q,phr = II II (bszr ; q,phr II (a-1bsq-N; q,p)N-lkl, r=ls=l r=ls=l s=l ~1 n II (bzr ; q,p)kr = (q1-N /e; q,p)N-lkl II (aqzr/e; q,p)kr, r=l r=l n+1 n n II (qzr/zs;q,phr = (q;q,p)N-lkl II(azrqN+\q,P)kr II (qzr/zs;q,phr· r,s=l r=l r,s=l

So, n+2 n+1 I1 (aszr ; q,phr ____s_=_l~------II n+1 r=l (bzr ; q, P)kr I1 (qzr/ Zs; q,phr s=l 336 Elliptic, modular and theta hypergeometric series

Using the identities (11.2.49) and (11.2.50), we can simplify the last ex­ pression above and obtain the following well-poised form of (11.7.8) [ ~(zqk;P) n O(azrqlkl+kr;p) S:Q:1 (azs; q,P)lkl kl,.~n20 ~(z;p) 11 O(azr;p) IT (qzr!zs;q,phr Ikl::;N s=l

X n+2III (bsZr; q,p)kr(e, q-N; q,p)lkl qlkl 1 n+2 III (aq/bs; q,p)lkl (aqzr/e, azrqN+1; q,p)kr

= eN s IT (aqzr; q,p)N IT (aq/eb ; q,P)N. (11. 7.19) r=l (aqzr/e; q,p)N s=l (aq/bs; q,p)N For some elliptic multivariable transformation formulas and additional sum­ mation formulas, see the exercises and notes.

Exercises

11.1 Verify the O(a;p) and q,p-shifted factorial identities in (11.2.42)-(11.2.60). 11.2 Verify the q,p-binomial coefficient identities in (11.2.62) and (11.2.64)­ (11.2.66). 11.3 Show that O(aq2n; p) (qa!, -qa!; q,P!)n O(a;p) (a"2,-a-"2;q,p"2)n1 1 1 (qa!, -qa!, q(ap)!, -q(ap)!; q,P)n (a!, -a!, (ap)!, -(ap)!; q,P)n

= (qa!,-qa!,q(a/p)!,-q(ap)!;q,P)n(_ )-n 1 1 1 1 q (a"2, -a"2, (ap) "2 ,-(a/p)"2; q,P)n and convert these identities to the [a; u, Tln notation. 11.4 Prove (11.4.1) via induction by starting with the identity qkO(aq2k, q-n-1, e, aqn+1 /e;p) = O(aqk, qk-n-l, eqk, aqn+k+1 /e;p) - O(aqn+k+l, qk, el-n-l, al/e;p), Exercises 337

which is equivalent to (11.4.3). (Rosengren: June 13, 2002, e-mail message) 11.5 Verify that t ()(aq2k;p) (a, b, e, a/be; q,ph qk = (aq, bq, eq, aq/be; q,P)n ()(a;p) (q,aq/b,aq/e,beq;q,p)k (q,aq/b,aq/e,bcq;q,p)n k=O for n = 0,1, ... , which is an elliptic analogue of Ex. 2.5.

11.6 Derive the following transformation formula for a well-poised 4E3 series

4 E3(a, b, aqn+2/b, q-n; aq/b, bq-n-1, aqn+\ q,p; -1) (aq, q2(ap)~ /b, -q2(a/p)~ /b, -q; q,P)n (q2 /b, q(a/p)~, -q(ap)~, - aq2/b; q,P)n 1 1 1 1 +2 X 12 Vu (-aq/b; (ap) 2, -(alp) 2, qa 2 /b, -qa2 /b, -q, aqn /b, q-n; q,p), where n = 0,1, .... 11.7 Extend the terminating case of the expansion formula in (2.8.2) to f (a, b, e; q,P)n An n=O (q,aq/b, aq/e; q,P)n = f ()(>.. q2n;p)(A, Ab/a, Ae/a, aq/be; q,P)n(a; q,phn (9:)n n=O ()(A; p)(q, aq/b, aq/e, a2q/ Abc; q,P)n(Aq; q,phn A

~ (aq2n,a/A,Abeqn/a;q,p)k A X 6 (q,Aq2n+l,a2qn+1/Abe;q,p)k n+k,

where the sequence {An} has finite support and A is an arbitrary param­ eter. Use this formula to derive (11.5.1). 11.8 Show that t (~,qa~;-qa~,b,c,q-n;q,p)k (_I)k k=O (q, a 2 , -a2 , aq/b, aq/e, aqn+1; q,p)k (aq,aq/be,qb-1(a/p)~,qc-1(a/p)~;q,p)n (aq/b,aq/e,q(a/p)~,ab-1e-1(a/p)~;q,p)n' where be = _ aqn+1, n = 0,1, .... 11.9 (i) Extend Ex. 1.4(i) to the inversion formula

f (a1, ... ,ar ;q,P)n ((_ltqG))1+s-rAnzn (q,b , ... ,bs;q,P)n n=O 1 00 (-1 -1. -1 ) n = '"""' a1 , ... ,ar ,q ,p n A (a1··· arz) 1 1 n=O~ ( q -1 '1b- , ... , b-s ; q -1 ,p) n n q b1··· bs

when the sequence {An} has finite support. 338 Elliptic, modular and theta hypergeometric series

(ii) Extend Ex. 1.4(ii) to the reverse in order of summation formula

when n = 0,1, ... , and {An} is an arbitrary sequence of complex numbers.

11.10 Prove that

~ O(aq2k;p) (a,b,c,d,e,a2q/bcde;q,p)k k O(a;p) (q,aq/b, aq/c, aq/d, aq/e, bcde/a;q,p)k q k=O~ (aq, bcd/a,a2q2/bcde, eq; q,P)n (q,a2q2/bcd,bcde/a, aq/e; q,P)n x 12 Vu (a2q/bcd; aq/bc, aq/bd, aq/cd, e, a2q/bcde, aqn+1, q-n; q,p),

where n = 0,1, ....

11.11 Verify the transformation formulas (11.6.16) and (11.6.17).

11.12 As in (11.2.67), the r( z; q, p) elliptic gamma function is defined by

00 1 _ z-lqj+1 pk+1 r(z; q,p) = II 1 _ z j k ' j,k=O q P

where z, q, p are complex numbers and Iql, Ipl < 1. Show that

(i) r(z; q, 0) = eq(z), (ii) r(zq; q,p) = O(z;p)r(z; q,p), (iii) r(zp; q,p) = O(z; q)r(z; q,p), (iv) r(zqn; q,p) = (z; q,P)nr(z; q,p), (v) r q(z) = (1 - q)l-z(q; q)oor(qZ; q, 0), 0 < q < 1,

and that r(z; q,p) and r(qZ; q,p) are meromorphic functions of z, which are not doubly periodic. (See Jackson [1905d], Ruijsenaars [1997, 2001], Felder and Varchenko [2000], and Spiridonov [2003b].)

11.13 Define the f\z; q,p) elliptic gamma function by

_ (q; q)oo 1-z 00 1 _ qj+1-zpk+1 r(z; q,p) = ( . ) (O(q;p)) II 1 _ j+z k ' p,p 00 j,k=O q P Exercises 339

where z, q, P are complex numbers and Iql, Ipl < 1. Extend the Gauss multiplication formula (1.10.10) and its q-analogue (1.10.11) to

r(nz;q,p)r-- (1;,;r,p ) r - (2;,;r,p ) ···r - (n-n-;r,p - 1 )

()(r;p)]nZ-1_ _( 1 ) _( n-l ) = [()(q;p) r(z; r,p)r z + ;,; r,p ... r z + -n-; r,p

with r = qn, and show that r(z;q,O) = rq(z). (Felder and Varchenko [2003a]) 11.14 (i) Show that if n is a nonnegative integer and k is a positive integer, then

2k+8 TTV2k+7 (b.a ,C, a b/ c, bq, bq 2 , ... , bq k ,aq n ,aqn+1 , ... ,aqn+k-1 ,q,-kn. q k ,p ) (a/c,c/b;q,P)n(qk,abqk;qk,P)n (cqk,abqk/c;qk,P)n(a,l/b;q,P)n· (ii) Show that

and that this formula tends to (lU7) as p ----; O. (iii) Show that

12 Vu(cp; C, -c, -cp, epic, cqp/e, qn+1, q-n; q,p2) (cqp,e/c2;q,p2)n(eq-n;q2,p2)n (qp/c, e; q,p2)n(eq-n /c2; q2,p2)n and that this formula tends to (lU9) as p ----; O. (See Warnaar [2002b] for part (i) and Warnaar [2003f] for parts (ii) and (iii). ) 11.15 Prove that if a, bk, Ck are complex numbers such that Cj i=- Ck and aCjCk i=- 1 for integer j and k, then n L inkgkm = Dn,m k=m with

and 340 Elliptic, modular and theta hypergeometric series

(Warnaar [2002b]) 11.16 Prove that n e(a2q5k;p) (a2; q4,p)k(a, aq, aq2; q3,ph(aqn+1, q-n; q,ph k ~ e(a2;p) (q; q,p)k(a, aq, aq2; q2,ph(aq3-n, a2qn+4; q4,p)k q

_{ (~,q2'l,a:q4;q~'f)n/4 , n:::=0(mod4), - (aq ,aq ,aq ,q/a,q ,P)n/4 0, n =I=- 0 (mod 4). (Warnaar [2002b]) 11.17 Show that if be = a2qn+1 and d = qn+l, then

[n/2]""' e (4k)aq ;P ( b,e;q3) ,p k (a,d;q,p ) k (-n q ;q,p ) 2k k ~ e(a;p)(aq/b, aq/e; q,ph(aq3/d, q3; q3,p)k(aqn+1; q,phk q (aq; q,P)n(aq2-n /b; q3,P)n (aq/b;q,P)n(aq2-n;q3,P)n· (Warnaar [2002b]) 11.18 For be = aq and cd = aqn+1, show that

~ e(aq4k;p)(a, b; q3,ph(d, q-n; q,p)k(e; q,phk k ~ e(a;p)(q, aq/b; q,p)k(aq3/d, aqn+3; q3 ,ph(aq/e; q,phk q

_ { (q,q22,aq3,b2/3a;~3'f)n/3 , n:::= 0 (mod 3), - (bq,bq ,b/a,aq /b,q ,P)n/3 0, n =I=- 0 (mod 3). (Warnaar [2002b])

11.19 Extend the e = bq-n-1 case of (3.8.19) to

[n/2],,",eaq;p ( 4k ) (3b,e;q,Pkd,e;q,Pkq ) ( ) (-n ;q,P2k) k ~ e(a;p) (aq/b, aq/e; q,ph(aq3/d, aq3/e; q3,p)k(aqn+1; q,phk q (aq;q,P)n(aq2-n/b;q3,P)n (aq/b;q,P)n(aq2-n;q3,P)n x 12 Vu (a 2Ide; b, e, aid, ale, q2-n, q1-n, q-n; q3,p), where be = a2qn+1 and de = aqn+1. (Warnaar [2002b]) 11.20 Show that if be = aq and de = aqn+1, then

~ e(aq3k;p) (a,b,e;q2,p)k(d,e,q-n;q,P)k k ~ e(a;p) (q, aq/b, aq/e; q, p)k(aq2/d, aq2/e, aqn+2; q2 ,p)k q (aq2,aq2/be,aq2/bd,aq2/ed;q2,p)n/2 -_ { (aq 2/ b, aq 2/ e, aq 2/ d, aq 2/ bed; q2) ,p n/2 ,n even 0, n odd. Exercises 341

(Warnaar [2002b]) 11.21 Show that if bed = aq, ej = a2q2n+l, and either b = a, or e = a, then ~ ()( aq3k;p) (b,e,d;q,ph(e,j,q-2n;q2,p)k k 6 ()(a;p) (aq2/b, aq2/e, aq2/d; q2,ph(aq/e, aq/ j, aq2n+l; q,P)n q (aq2,a2q2/bee,a2q2/bde,aq2/ed;q2,p)n (a2q2/be,aq2/e,aq2/d,a2q2/bede;q2,p)n· (Warnaar [2002b]) 11.22 Show that if bed = aq, de = a2q3n+l, and either b = a or e = a, then ~ ()(aq4k;p) (b, e; q,ph(d; q,phk 6 ()(a; p) (aq3/b, aq3/e; q3 ,p)k(aq/d; q,phk (e,q-3n;q3,ph k x q (aq/e, aq3n+l; q, p)k (aq3,a2q3/bee,a2q3/bde,aq3/ed;q3,p)n (a2q3/be,aq3/e,aq3/d,a2q3/bede;q3,p)n· (Warnaar [2002b]) 11.23 (i) Verify the quadratic elliptic transformation formula 14 VI3 (a; b, -b, e, -e, dqn, _dqn, q-n, _q-n, qa2/)..2; q,p) (a2q2,d2/)..2,b2d2/a2,e2d2/a2;q2,p2)n (d2/a2,q2)..2,a2q2/b2,a2q2/e2;q2,p2)n x 14 VI3()..2; b2, e2 , d2q2n, a2q2 /)..2, q-2n, _)..2/a, _)..2 q/ a, _)..2/ap, _)..2 qp/ a; q2,p2), where).. = bed/aq and n = 0,1, .... (ii) Prove that the P ----) 0 limit of the above transformation is 12 Wll(a; b, -b, e, -e, dqn, _dqn, q-n, _q-n, qa2/)..2; q, q) (a2q2,d2/)..2,b2d2/a2,e2d2/a2;q2)n (d2/a2,q2)..2,a2q2/b2,a2q2/e2;q2)n

x 10 Wg ()..2; b2, e2 , d2q2n, a2q2 /)..2, q-2n, _)..2/a, _)..2 q/ a; q2, )..2q/a2)

and that the 12 Wll series is VWP-balanced and balanced, while the 10 W g series is VWP-balanced but not balanced. (See (Spiridonov [2002a]) for part (i), and Nassrallah and Rahman [1981] for part (ii). Also see (3.10.15), Andrews and Berkovich [2002] and War­ naar [2003c,e].)

11.24 (i) Verify the quadratic elliptic transformation formula 14VI3(a;a2/)..2,b,bq,e,eq,dqn,dqn+l,q-n,ql-n;q2,p) (aq,d/)..,)..q/b,)..q/e;q,p)n ()..q,d/a,aq/b,aq/e;q,p)n 1 1 x 14 VI3 ()..; a/).., IL, -IL, ILP'l, -ILP-'l, b, e, dqn, dq-n; q,p), 342 Elliptic, modular and theta hypergeometric series

where>.. = bed/aq, J1, = ±>,,(q/a)~, and n = 0,1, .... (ii) Prove that the P ----) 0 limit of the above identity is 12VVll (a; a2/>..2, b, bq, e, eq, dqn, dqn+l, ql-n, q-n;q2,q2) (aq,d/>..,>"q/b,>"q/e;q)n (>"q,d/a,aq/b,aq/e;q)n

x 10 VVg (>..; a/ >.., J1" -J1" b, e, dqn, q-n; q, _q>..2 / a)

and that the 12 VVl1 series is VWP-balanced and balanced, while the 10 VVg series is VWP-balanced but not balanced. (See Warnaar [2002b] for part (i), and Rahman and Verma [1993] for the transformation in part (ii). Also see Andrews and Berkovich [2002] and Warnaar [2003c,e].) 11.25 (i) Extend the quadbasic transformation formula in Ex. 3.21 to t ()(arkqk, brkq-k;p) (a, b; r,p)de, a/be; q,ph k=O ()(a,b;p) (q,aq/b; q,p)k(ar/e, ber;r,p)k (CR-n /A, R-n / BC; R, p)k(Q-n, BQ-n /A; Q, P)k k X (Q-n/C,BCQ-n/A;Q,Ph(R-n/A,R-n/B;R,Ph q (ar, br; r,P)n(eq, aq/be; q,P)n(Q, AQ/ B; Q, P)n(AR/C, BCR; R, P)n (q, aq/b; q,P)n(are, be/r; r,P)n(AR, BR; R, P)n(CQ, AQ/ BC; Q, P)n x t ()(ARkQk,BRkQ-k;p) (A,B;R,Ph(C,A/BC;Q,P)k k=O ()(A, B; P) (Q, AQ/ B; Q, P)dAR/C, BCR; R, P)k x (er-n la, r-n /be; r,p)k(q-n, bq-n fa; q,P)k Qk (q-n /e, beq-n fa; q,p)k(r-n la, r-n /b; r,p)k for n = 0,1, .... (ii) Deduce the following transformation formula for a "split-poised" theta hypergeometric series t ()(aq2k;p) (a,b,e,a/be;q,p)k k=O ()(a;p) (q, aq/b, aq/e, beq; q,p)k (q-n, B/Aqn, C/Aqn, I/BCqn; q,ph k X (I/Aqn, I/Bqn, l/cqn, BC/Aqn; q,P)k q (aq, bq, eq, aq/be, Aq/B, Aq/C, BCq; q,P)n (Aq, Bq, Cq, Aq/ BC, aq/b, aq/e, beq; q,P)n x t ()(Aq2k;p) (A,B,C,A/BC;q,p)k k=O ()(A;p) (q, Aq/ B, Aq/C, BCq; q,ph (q-n,b/aqn,e/aqn,l/beqn;q,p)k k X q (l/aqn,l/bqn, l/eqn, be/aqn; q,p)k for n = 0,1, ... , which is an extension of the transformation formula for a split-poised 1O¢9 series given in Ex. 3.21. Write this formula as a transformation formula for a split-poised 12En series. Exercises 343

(Gasper and Schlosser [2003])

11.26 Extend Ex. l1.25(i) to

t ()(a(rst/q)k, brkq-k, cskq-k, atk /bcqk;p) k=O ()(a, b, e, a/bc;p) (a; rst/q2,p)db; r,p)k(c; s,ph(a/bc; t,P)k x~~~~~~~~~~~~~~~~~~ (q;q,p)k(ast/bq;st/q,p)k(art/cq;rt/q,p)k(bers/q;rs/q,p)k (Q-n; Q, Ph(B(Q/ST)n /A; ST/Q, P)k(C(Q/ RT)n /A; RT/Q, P)k x ((Q2 /RST)n/A; RST/Q2, p)k(R-n / B; R, p)k(s-n /C; S, Ph ((Q/Rs)n/BC;RS/Q,P)k k x (BC/ATn;T,p)k q (arst/q2;rst/q2,P)n(br;r,P)n(cs;s,P)n(at/bc;t,P)n (q;q,P)n(ast/bq;st/q,P)n(art/cq;rt/q,P)n(bers/q;rs/q,p)n (Q; Q,P)n(AST/BQ; ST/Q,P)n X..,....,...'--'--"":""'--:'-""""'------'--::--'-:-"""":--'----'--....,..- (ARST /Q2; RST/Q 2, P)n(BR; R, P)n (ART /CQ; RT/Q, P)n(BCRS/Q; RS/Q, P)n x-'------'-----;-==-=---=-~--;--;-::::-:-=-=:--::::-_=_=_----'-....::...:....--'-- (CS; S,P)n(AT/BC;T, P)n x t ()(A(RST/Q)k, BRkQ-k, CSkQ-k, ATk/BCQk;p) ()(A, B, A/ BC; P) k=O c, (A; RST/Q2, P)k(B; R, P)k x~'--='--=-:-~-===-:-:'::-'-:'---,==-':-::-'--'::-:­ (Q; Q, P)k(AST/BQ; ST/Q, P)k (C; S, P)k(A/BC; T, P)k x (ART/CQ; RT/Q, P)k(BCRS/Q; RS/Q, P)k (q-n; q,p)k(b(q/ st)n fa; st/q,p)k x-'-=------;.-;-:~-"--'-:-=:-----'-----'--;--';::_____'_,-=-:-=---'-'-'­ ((q2 /rst)n fa; rst/q2 ,ph x (c(q/rt)n fa; rt/q,ph((q/rs)n /bc; rs/q,ph Qk (r-n /b; r,p)k(S-n /e; s,p)k(bc/atn; t,p)k

for n = 0,1, .... Use (11.6.9) and (11.6.18) to extend this formula to a transformation formula containing the two additional parameters d and D. (Gasper and Schlosser [2003])

11.27 Show that if 344 Elliptic, modular and theta hypergeometric series

and a~ = bkckdkekfkgk for k = 0, ±1, ±2, ... , then ~ ()(aVbkckdkh, aVbkckdkek, ak, ak/ekfk;P) U f:::n ()(ak/ek, ak/ fk' aVbkCkdkekfk, aVbkCkdk;p) k 1 ()(ak/ckdk,ak/bkdk,ak/bkck,ek,fk,gk;p) ] [ x - ()(ak/bk,ak/ck,ak/dk,a~/bkckdkek,a~/bkckdkfk,a~/bkckdkgk;P)

= Urn - Un+1 for n,m = 0, ±1, ±2, .... (Gasper and Schlosser [2003]) 11.28 Extend the indefinite multibasic theta hypergeometric summation formula in (11.6.9) to

( k a (V ) k a2 ( v2 ) k a2 (V2 ) k a2 ) ~ () av 'J[j Ui 'TiCrJ! qrsl 'vr;cag qrsu 'DCa; P ~ ( a a2 a2 a2 ( v2 ) k. ) k=O () a, J[j' bcd] , bcdg' bcd qr s ,p 3 3 (b; q,p)k(C; r,ph(d; s, p)k(f; t, p)k(g; u, p)k (bcdfg; q;!stu ,p) k (qrstu/v2)k x ------~~~~--~--~---- a v a v a v av v av v bcdfgqrstu qrstu (-b;-,P)k(-;-,P)k(-d;-,P)k(ft;-t,P)k(-;-,P)k( 22 ;-2 ,P)k q c r s gu u a v v ()(cdU!s)k, bd(~s)k, bc(~r)k,ftk,guk, bCdfg(q;!s3tu )k;p) 1 x [ 1------.;------;o;------,;---~---=------;;------2 2 2 2 (a (V)k a (V)k a (V)k a ( V )k a ( v )k fg (tU)k. ) () 7i q 'c r 'd S 'TiCrJ! qr st 'Ocag qr su 'a v ' P ()(a/f,a/g,a2/bcdfg,a2/bcd;p) ()(a 2/bcdg,a2/bcdf,a,a/fg;p) x [1 _ (b; q,P)n+l(C; r,P)n+l(d; S,P)n+l(f; t,P)n+l(g; U,P)n+l (~; ~'P)k+l (%; ¥,P)n+l (a; ~,P)n+l Uj-; ¥,P)n+l (~; *,P)n+l

a3 V 3 x (-oCilfii; 7jFSfU' p) n+ 1 1 ( bcdtg. qrstu) , a ' ----;;r-' P n+l where n = 0,1, .... (Gasper and Schlosser [2003]) 11.29 Define the elliptic beta function BE(t;q,p) via its elliptic beta contour integral representation

BE(t; q,p) = rllE(Z; t; q,p) dz, IJJ' z where l' is the positively oriented unit circle, 4 I1 r(ztk,tk/Z;q,p) ( ) 1 k=O llE z;t;q,p = -2.r( 2 1/ 2 A A/ )' 7rZ Z, z, z, z; q,p Exercises 345

4 t = (to,t1,t2,t3,t4), max(ltol, It11, It21, It31, It41, Iql, Ipl) < 1, A = I1 tk, k=l Iqpl < A, and n r(Zl,"" Zn; q,p) = II r(Zk; q,p), n = 1,2, ... , k=l with f(z; q,p) as defined in Ex. 11.12. Prove that 2 I1 r(tjtk; q,p) O~j

11.30 Let ~E(Z; t) = ~E(Z; t; q,p) and BE(t) = BE(t; q,p), with ~E(Z; t; q,p), BE (t; q, p) and A defined as in the previous exercise, and let

( t3 q q q t3 -m Aqm-1 ) Rm,j (z) = 12 V11 t4; tOt4 ' t1t4 ' t2t4 ,t3Z, ---;' q , -t-4-; q,p

V; (t3 P P P t3 _j Apd-1 ) X1211 t4;tOt4't1t4't2t4,t3z,---;,p ,~;p,q

for j, k, m, n = 0,1, .... Note that the base and nome in the second 12 V11 factors are p and q, respectively. Prove that Rm,j(z) and Tn,k(Z) satisfy the biorthogonality relation

where

hn k = ()(A/qt4;p)(q, qt3/t4, tOt1' tOt2' t1t2, At3; q,P)nq-n , ()(Aq2n /qt4;p)(1/t3t4' tOt3, t1 t3, t2 t3, A/qt3, A/qt4; q,P)n ()(A/pt4; q)(p,pt3/t4, tOt1, tOt2, t1t2, At3;P, q)np-k x ~~~~~~~~-----~~~~~~~ ()(Ap2k /pt4; q)(1/t3t4' tOt3, t1t3, t2t3, A/pt3, A/pt4;p, qh and Cj,k,m,n is a closed positively oriented contour separating the points Z = to,1,2,3pTqs, t4pT-jqs-m, A-1pT+1-k qs+l-n with r, S = 0,1, ... , from their inverses. (Spiridonov [2003b]) 11.31 Show that if 346 Elliptic, modular and theta hypergeometric series

then

kl, ... ,krn?D k1+···+krn=N which is an extension of (11.5.5), (11.7.7), and (11.7.10). (Kajihara and Noumi [2003] and Rosengren [2003c]) 11.32 Let Xl, ... , X n , A 2, ... , An and C be indeterminates. Prove that if, for m = 0, ... , n -1, Pm is a Laurent polynomial of degree less than or equal to m such that Pm(C/X) = Pm(X), then

n II A j X j (l- XdX j)(l- C/XiXj ) II Pi- 1 (1/Ai ), l~i

N where the degree of the Laurent polynomial P(x) = l: aixi, aN of. 0, is i=M defined to be N. (Krattenthaler [1995a]) 11.33 Prove the following elliptic extension of the identity in the above exercise

n II AjX/J(XdXj;p)()(C/XiXj;P) II Pi- 1 (1/Ai ), l~i

(Warnaar [2002b])

11.35 Let Zl, ... , Zn, a, b, c, d, and e be indeterminates and N a nonnegative integer such that a2qN -n+2 = bcde. Show that

B( aZrzsqkrHs; p) II B(azrzs;p)

IT B(az;'q2kr;p) (az;', bzr, cZr, dzr, ezr, q-N; q,p)ks Ikl x r=l B(az;;p) (q, aqzr/b, aqzr/e, aqzr/d, aqzr/e, aqN+l z;; q,pk q = IT (aqz;',aq2-r/bc,aq2-r/bd,aq2-r/cd;q,p)N r=l (aq2-n /bcdzr, aqzr/b, aqzr/c, aqzr/d; q,p)N'

which is an elliptic extension of Schlosser's [2000a] en Jackson sum.

(Warnaar [2002b])

11.36 Prove the following elliptic extension of Gasper's summation formula in Ex. 2.33(i): t B(aq2k;p) (a, b, alb, q-N; q,ph kIT (Cjqm j , aq/Cj; q,ph k=O B(a;p) (q, aq/b, bq, aqN+l; q,p)k q j=l (aql-mj /Cj, Cj; q,ph

= (q, aq; q,p)N IT (Cj/b, cjb/a; q,P)mj (bq, aq/b; q,p)N j=l (Cj, Cj/a; q,P)mj

with ml + ... + mr = N, where ml, ... ,mr are nonnegative integers.

(Rosengren and Schlosser [2003b])

11.37 Prove the following multidimensional extension of Ex. 11.29:

n· I1 rW-ltrts; q,p) n 2 n! II r(t J ; q, p) --'O:s;=:r=<:-s-=:s;=--4...,.....,...,.....,----;;--_---:- (p;p)~(q; q)~ ·-1 r(t; q,p) I1 r(tl-h;l B; q,p) , J- O:S;r:S;4

where B = t2n- 2 I1 tk, max(ltl, Itol, Itll, It21, It31, It41, Ipl,lql) < 1, O:S;k:S;4 Ipql < B, and ,][,n is the n-dimensional unit torus. (van Diejen and Spiridonov [2003, (29)] and Rains [2003b]) 348 Elliptic, modular and theta hypergeometric series

11.38 Prove Rosengren's [2003c] elliptic Dn Jackson summation formula

n-1 I1 (bas, bias; q,p)N = (_qN -1 b) N _----=-s_==-l ----=n:------(q; q,p)N I1 (bzs, b/zs; q,p)N s=l

11.39 From the previous exercise deduce that

x II 1 IT (zsbr, zs/br; q,pk l:<:;r

x IT (azr; q,p)lkl (aq/ Zr; q,p)lkl-kr r=l (aqbr, aq/br; q,P)lkl (q-N,c,a2 qN+1/c;q,p)lkl } x n qlkl I1 (aqzs/c, czs/aqN, azsqN+1; q,phs s=l

= IT (aqzs, aq/zs, aqbs/c, aqbs/c; q,P)N, s=l (aqzs/c,aq/zsc, aqbs, aq/bs; q,p)N

where Ikl = k1 + ... + k n . (Rosengren [2003c])

11.40 Show that m1tmn {~(Zqk;p) qlkl [IT B(azsqks+lkl;p)] kl, ... ,kn=O ~(z;p) s=l B(azs;p) (b, c, d; q,P)lkl IIn (azr; q,P)lkl x (aq/e, aq/ /, aq/g; q,P)lkl r=l (azqmr+1; q,P)lkl x IT (ezs,/zs,gzs;q,pk IT (q-mrzs/zr;q,pk} s=l (aqzs/b, aqzs/c, aqzs/d; q,p)ks r,s=l (qzs/zr; q,p)ks Notes 349

= (~) Iml (Aq/ j, Aq/g; q'P)lml IT (aqzs, Aqzs/d; q,p)Tns A (aq/ j, aq/g; q'P)lml s=l (Aqzs, aqzs/d; q,p)Tns Tn1,···,Tnn" {~( zq k.),p Ikl [nII ()('AZsq ks+lkl.)],p X ~ ~(z;p) q ()(AZs;p) k1, ... ,kn=O s=l (Ab/a, Ac/a, d; q,p)lkl IIn (AZr; q,p)lkl x (aq/e, Aq/ j, Aq/g; q,P)lkl r=l (Aq1+ Tnr zr; q,P)lkl x IT (Aezs/a,jzs,gzs;q,phs IT (q-Tnrzs/zr;q,P)ks}, s=l (aqzs/b, aqzs/c, Aqzs/d; q,p)ks r,s=l (qzs/zr; q,p)ks

where A = a2q/bcd and a2q2+lm l = bcdejg with Ikl = kl + ... + kn and Iml =m1+···+ m n· (Rosengren [2003c])

Notes

§11.2 and 11.3 Totally elliptic multiple hypergeometric series are defined and considered in Spiridonov [2002a, 2003a]. Spiridonov [2003a] showed that the elliptic Milne An, Jackson en and Bhatnagar-Schlosser Dn theta hypergeo­ metric series in the left sides of his equations (21), (18), and (22), respectively, are totally elliptic and modular invariant. He used these results and other observations to motivate his conjecture that, as in the one-variable series case, every totally elliptic multiple hypergeometric series is modular invariant. §11.6 The P = 0, s = q special case of (11.6.21) is equivalent to a cor­ rected version of the generalization of (3.7.6) given in Subbarao and Verma [1999]. §11.7 Rosengren's inductive proof of (11.7.8) is similar to Milne's [1985a, pp. 49-50] inductive proof of (11.7.1) based on the analysis in Milne [1980b, pp. 179-182, and 1985c, pp. 17-19]. For additional material on basic and ellip­ tic summation and transformation formulas, see Bhatnagar [1998, 1999], Bhat­ nagar and Milne [1997], Bhatnagar and Schlosser [1998], van Diejen [1997b], van Diejen and Spiridonov [2000-2003], Gustafson [1987a-1994b], Gustafson and Krattenthaler [1997], Gustafson and Rakha [2000], Ito [2002]' Leininger and Milne [1999a,b], Lilly and Milne [1993], Milne [1980a-2002], Milne and Bhatnagar [1998], Milne and Lilly [1992, 1995], Milne and Schlosser [2002]' Rains [2003a,b], Rosengren [1999-2003f], Rosengren and Schlosser [2003a,b], Schlosser [1997-2003e], Spiridonov [1999-2003b], and Warnaar [1999-2003e]. Ex.11.15 This orthogonality relation is an elliptic analogue of Kratten­ thaler [1996, (1.5)]. Ex.11.17 This summation formula is an elliptic analogue of Gasper [1989a, (5.22) with b = qn+1]. Ex.11.18 This identity is an elliptic extension of W. Chu [1995, (4.6d)]. Ex.11.20 This formula is a generalization of Gessel and Stanton [1983, (6.14)]. 350 Elliptic, modular and theta hypergeometric series

Ex. 11.21 When p = 0 and b = a this formula reduces to Gessel and Stanton [1983, (1.4)]. When p = 0 and e = a it reduces to W. Chu [1995, (5.1d)] and to Rahman [1993, (1.9) with b = q-2n]. Ex.11.22 When p = 0 and b = a this is an elliptic analogue of Gasper [1989a, (5.22) with c = q-3n]. Ex.11.29 Also see the material on elliptic integrals and elliptic gamma functions in Felder, Stevens and Varchenko [2003a,b], Felder and Varchenko [2000-2003b], Narukawa [2003], and Nishizawa [2002]. Ex.11.36 Also see the elliptic summation formula in Rosengren and Schlosser [2003b, Corollary 5.3]. Ex.11.38 The p = 0 case was independently discovered by Bhatnagar [1999] and Schlosser [1997]. Ex.11.39 When p = 0 this reduces to an identity in Schlosser [1997]. Ex.11.40 Equivalent forms of the p = 0 case were independently dis­ covered by Denis and Gustafson [1992, Theorem 3.1], who derived it from a multivariable integral transformations via residues, and by Milne and Newcomb [1996, Theorem 3.1] via series manipulations. Also see Rosengren [2003a]. Appendix I

IDENTITIES INVOLVING q-SHIFTED FACTORIALS, q-GAMMA FUNCTIONS AND q-BINOMIAL COEFFICIENTS

q-Shifted factorials: I, n = 0, (a; q)n = { (1 - a)(l - aq)··· (1 - aqn-l), 1 n = 1,2, ... , (I.1) [(1-aq-l)(1-aq-2) ... (1- aqn)r, n=-1,-2, .... 1 (-q/a)n (n) (a; q)-n = _ = q 2 (1.2) (aq n; q)n (q/a; q)n and (a;q-l)n = (a-\q)n(-a)nq-G), (1.3) where (~) = n(n - 1)/2.

00 (a; q)oo = II (1 - al), (1.4) k=O () (a; q)oo a; q n = ( n ) , (1.5) aq ; q 00 and, for any complex number n,

() (a;q)oo a; q '" = ( '" ) , (1.6) aq ; q 00 where the principal value of q'" is taken and it is assumed that Iql < 1.

(1. 7)

(1.8)

(1.9)

(I.10)

(1.11 )

(I.12)

(I.13)

351 352 Appendix I

(aq-n;q) = (qja;q)n (_~)n-k q(~)-G). (1.14) n-k (qja; q)k q

( -2n. ) = (qja; qhn (_~)n -3G) aq ,q n (qja;q)n q2 q . (1.15)

(aq-kn;q) = (qja;qhn (_atqG)-kn2 • (1.16) n (qja; q)(k-l)n (a; q)n+k = (a; q)n(aqn; q)k. (1.17) n. ) _ (a; qh(aqk; q)n ( aq ,q k - () . (1.18) a;q n kn. ) _ (a; q)(k+l)n ( aq ,q n - ( ) . (1.19) a;q kn k) (a;q)n (aq ; q n-k = -(-) . (1.20) a; q k 2k.) _ (a; q)n(aqn; qh (aq ,q n-k - () . (1.21 ) a; q 2k ( jk.) _ (a; q)n(aqn; q)(j-l)k aq ,qn-k- (a;q)jk . (1.22) (aI, a2,···, ak; q)n = (al; q)n(a2; q)n··· (ak; q)n. (1.23) (aI, a2,···, ak; q)oo = (al; q)oo(a2; q)oo ... (ak; q)oo. (1.24) (a; qhn = (a, aq; q2)n, (1.25) (a; qhn = (a, aq, aq2; q3)n, (1.26) and, in general, (a; q)kn = (a, aq, ... , aqk-\ qk)n. (1.27) (a 2; q2)n = (a, -a; q)n, (1.28) (a3;q3)n = (a,aw,aw2;q)n, W = e27ri/ 3 , (1.29) and, in general, ( a k.,q k) n -_ (a, aWk, ... ,awkk-l. ,q ) n, Wk - _ e 27ri/k . (1.30) (qa~, -qa~; q)n (aq2; q2)n 1 - aq2n (1.31 ) (d,-d;q)n (a;q2)n I-a' (qai, qwai, qw2ai; q)n (aq3; q3) 1 _ aq3n (1.32) (ai,wai,w2ai;q)n (a;q3)n I-a' and, in general, ( qak,qwkak,1. 1. ... qwkk-l ak;q1. ) n (aqk; qk)n (1.33) ( a1.k , Wk ak 1. , ... , wkk-l a1.k ; q) n (a;qk)n 1- a ' where W = e27ri/ 3 and Wk = e27ri /k.

lim (Zqa;q)oo=(I_z)-a, Izll- (z; q)oo Appendix 1 353 q-Gamma function:

0< q < 1, (1.35) q> 1.

(1.36)

(1.37)

rq(nx)rr (~) rr (~) ... rr (n: 1)

= (l+q+ ... +qn-1f x - 1rr(x)rr (x+~) ... rr (x+ n:l), (1.38) with r = qn. q-Binomial coefficient:

(1.39) and, for Iql < 1 and complex a and (3,

(qf3+ 1 , qcx-!3+1; q) 00 (1.40) (q, qcx+l; q)oo rq(a+l) (1.41 )

(1.42)

(1.43)

(1.44)

[a; 1 ] q = [~] q qk + [k ~ 1] q = [~] q + [k ~ 1] q qcx+l-k, (1.45) + ] (qcx+\ q)n [ (1.46) ~ - ~ q (q; q)n_k(qcx+l; q)k'

[~L_l = [~L qk 2 -cxk, (1.47) where n, k are nonnegative integers. For elliptic analogues, see Chapter 11. Appendix II

SELECTED SUMMATION FORMULAS

Sums of basic hypergeometric series:

The two q-exponential functions,

00 zn 1 eq(z) = ~ (q; q)n = (z; q)oo' Izl < 1, (ILl )

~ q(;)zn Eq(z) = ~ -(-. -) = (-z; q)oo. (11.2) n=O q, q n The q-binomial theorem, ( ) (az; q)oo l¢O a;-;q,z = ( ) , Izl < 1, (11.3) z;q 00

Of, when a = q-n, where, as elsewhere in this appendix, n denotes a nonnega­ tive integer, l¢O(q-n;_;q,z) = (zq-n;q)n. (11.4)

The sum of a 1 ¢1 series, ( (cia; q)oo 1¢1 a;c;q,c I)a = ( ) . (11.5) c; q 00 The q-Vandermonde (q-Chu-Vandermonde) sums,

rI.. (-n ) (cla;q)n n 2'1-'1 a, q ; c; q, q = ( ) a (11.6) c;q n and, reversing the order of summation,

rI.. (-n nl) (cia; q)n 2'1-'1 a, q ; c; q, cq a = ( ) . (II. 7) c;q n The q-Gauss sum, ( I) (cia, clb; q)oo 2¢1 a, b; c; q, c ab = ( I b ) . (II.8) c, c a ; q 00 The q-Kummer (Bailey-Daum) sum, 2 rI.. ( b. lb. _ Ib) = (-q; q)oo(aq, aq2 Ib ; q2)00 (11.9) 2'1-'1 a, ,aq ,q, q (_ qIb ,aqlb.) ,qoo . A q-analogue of Bailey's 2F1 ( -1) sum, ( ) (ab, bql a; q2)00 2¢2 a, qI a; -q, b; q, -b = (b.) . (ILlO) ,q 00

354 Appendix II 355

A q-analogue of Gauss' 2F1(-1) sum,

2 2 ! ! ) (a2q, b2q; q2)00 2¢2 ( a ,b ;abq2,-abq2;q,-q = ( 2b2 . 2) . (11.11 ) q,a q,q 00 The q-Saalschiitz (q- Pfaff-Saalschiitz) sum,

A. [ a, b, q-n . ] _ (c/a, c/b; q)n 3'1-'2 1 1 ' q, q - ( / (11.12) c, abc- q -n C, c a b; q ) n . The q-Dixon sum, a,-qa!,b,c qa!] (aq,qb-1a!,qc-1a!,aq/bc;q) 4¢3 [ 1 ; q, - = 1 1 00 , (11.13) -a'2, aq/b, aq/c bc (aq/b, aq/c, qa'2, qa'2 /bc; q)oo or, when c = q-n, a,-qa!,b,q-n. q1+na!]_ (aq,qa!/b;q)n 4¢3 [ 1+ ,q, b - (11.14) -a'2,1 aq/b, aq n (1qa'2, aq/b; q ) n . Jackson's terminating q-analogue of Dixon's sum, ¢ [q-2n, b, c. q2-n] _ (b, c; q)n(q, bc; qhn (11.15) 3 2 q1-2n /b, q1-2n /c' q, bc - (q, bc; q)n(b, c; qhn . A q-analogue of Watson's 3F2 sum, )" q)'!, -q),!, a, b, c, -c, ),q/c2 ),q] 8¢7 [ 1 1 2; q, --b ), '2, -), '2, ),q/a, ),q/b, ),q/ c, -),q/ c, c a

(),q, c2 /),; q)oo (aq, bq, c2q/a, c2q/b; q2)00 (11.16) (),q/a, ),q/b; q)oo(q, abq, c2q, c2q/ab; q2)00' where), = -c(ab/q)!; and Andrews' terminating q-analogue, q-n, a2qn+1, c, -c ] 4¢3 [ 2 ; q, q aq, -aq,c 0, if n is odd, { = cn(q, a2q2 /c2; q2)n/2 (11.17) if n is even. (a2q2, c2q; q2)n/2 ' A q-analogue of Whipple's 3F2 sum,

A. [-c, q(-c)!, -q(-c)!, a, q/a, c, -d, -q/d ] 8'1-'7 1 1 ;q, c (-C)'2, -(-c)'2, -cq/a, -ac, -q, cq/d, cd (-c, -cq; q)oo (acd, acq/d, cdq/a, cq2/ad; q2)00 (11.18) (cd, cq/d, -ac, -cq/a; q)oo

(11.19) 356 Appendix II

The sum of a very-well-poised 6¢5 series,

1 1 e, d '" [a, qa"2, -qa"2, b, aq ] 6'1-'5 1 1 ;q, bed a"2, -a"2, aq/b, aq/e, aq/d (aq,aq/be,aq/bd,aq/ed;q)= (11.20) (aq/b,aq/e,aq/d,aq/bed;q)= or, when d = q-n,

¢ a,qa!2 ,-qa !2 , b ,e,q-n . ~n+l] _ ( aq,aq /b·)e,q n [ (11.21 ) 6 5 a!, -a!, aq/b, aq/e, aqn+l ,q, be - (aq/b, aq/e; q)n·

Jackson's q-analogue of Dougall's 7F6 sum,

1 1 '" [a, qa"2, -qa"2, b, e, d, e, 8'1-'7 1 1 a"2, -a"2, aq/b, aq/e, aq/d, aq/e, (aq,aq/be,aq/bd,aq/ed;q)n (11.22) (aq/b,aq/e,aq/d,aq/bed;q)n' where a2q = bedeq-n.

A nonterminating form of the q-Vandermonde sum,

( ) (q/e, a, b; q)= 2¢1 a, b; e; q, q + (e / q, aq / e, bq / e; q ) = 2 (q/e, abq/e; q)= x 2¢1(aq/e, bq/e; q /e; q, q) = ( / b / ) . (11.23) aq e, q e; q = A nonterminating form of the q-Saalschiitz sum, ¢ [a, b, e] (q/e, a, b, e, qj Ie; q)= 3 2 e,j ;q,q + (e/q,aq/e, bq/e, eq/e, j;q)= '" [aq/e, bq/e, eq/e. ] _ (q/e,j/a,j/b,j/e;q)= (11.24) x 3'1-'2 q 2/ e, qj/ e ' q, q - ( aq / e, bq / e, eq / e, j ; q ) = , where ej = abeq.

Bailey's nonterminating extension of Jackson's 8¢7 sum,

1 1 '" [a, qa"2, -qa"2, b, e, d, e, j ] 8'1-'7 a"2,1 -a"2,1 aq / b, aq / e, aq / d, aq / e, aq/j;q,q b (aq, e, d, e, j, bq/a, bq/e, bq/d, bq/e, bq/ j; q)= a (aq/b, aq/e, aq/d, aq/e, aq/j,be/a, bd/a, be/a, bj/a, b2q/a; q) = b2/a, qba-!, -qba-!, b, be/a, bd/a, be/a, bj /a 1 x 8¢7 [ 1 1 ; q, q ba-"2, -ba-"2, bq/a, bq/e, bq/d, bq/e, bq/ j (aq, b/a, aq/ed, aq/ee, aq/ej,aq/de,aq/dj,aq/ej; q)= (11.25) (aq/e,aq/d,aq/e,aq/j,be/a,bd/a,be/a,bj/a;q)= ' where qa2 = bedej. Appendix II 357 q-Analogues of the Karlsson-Minton sums, a b b qTn1 b qTnr ] A, , ,1 , ... , r . a-I 1-(Tn1 +·+Tnr) r+2'1-'r+l [ b b b ,q, q q, 1,···, r = (q, bq/ a; q)oo (bdb; q)Tn1 ... (br/b; q)Tnr bTn1 +.+Tn r (11.26) (bq, q/a; q)00(b1; q)Tn1 ... (br; q)Tnr and Tn A, a ,1 b q 1, ... , br qTnr . -1 -(Tn 1+···+Tn r) ] - 0 r+ 1 'l-'r [ b b ' q, a q - , (II.27) 1, ... , r where ml, ... ,mr are arbitrary nonnegative integers. Sums of bilateral basic series: Jacobi's triple product,

00 L qk 2 zk = (q2, -qz, -q/z; q2)00. (11.28) k=-oo Ramanujan's sum,

nl, (.b. ) _ (q,b/a,az,q/az;q)oo 1 '1-'1 a, ,q, z - (/ / ) . (11.29) b,q a,z,b az;q 00 The sum of a well-poised 2't/J2 series, 2't/J2(b, e; aq/b, aq/e; q, -aq/be) (aq/be; q)00(aq2/b2, aq2/e2, q2, aq, q/a; q2)00 (II.30) (aq/b, aq/e, q/b, q/e, -aq/be; q)oo Bailey's sum of a well-poised 3't/J3, b, e, d q ] 3't/J3 [ ; q,- q/b, q/e, q/d bed (q,q/be, q/bd,q/ed; q)oo (11.31 ) (q/b, q/e,q/d, q/bed;q)oo· A basic bilateral analogue of Dixon's sum,

nl, [ -qd , b, e, d. qa ~ 1 4'1-'4 1 ,q'bd -a2 , aq/b, aq/e, aq/d e (aq, aq/be, aq/bd, aq/ed, qa! /b, qa! /e, qa! /d, q, q/a; q)oo (11.32) (aq/b, aq/e, aq/d, q/b, q/e, q/d, qa!, qa-!, qa~ /bed; q)oo . The sum of a very-well-poised 6't/J6 series,

nl, qa !2 , -qa !2 , b, e, d, e. qa 2 ] [ 6'1-'6 1 1 ,q,- a 2 , -a2 , aq/b, aq/e, aq/d, aq/e bede (aq, aq/be, aq/bd, aq/be, aq/ed, aq/ee, aq/de, q, q/a;q)oo (11.33) (aq/b,aq/e,aq/d,aq/e,q/b,q/e,q/d, q/e,qa2/bede; q)oo . 358 Appendix II

Bibasic sums: Gosper's indefinite bibasic sum,

~ 1- apkqk (a;ph(c; q)k k (ap;P)n(cq; q)n -n ~ -----:----,----,-----'--:----,----,------'----:--q = c (11.34) k=O 1 - a (q; q)k(ap/c;ph (q; q)n (ap/c; P)n .

An extension of (11.34),

~ (1 - apkqk) (1 - bpkq-k) (a, b; ph (c, a/bc; q)k qk 6 (1 - a)(l - b) (q, aq/b; q)k(ap/c, bCp;p)k (ap,bP;P)n(cq,aq/bc;q)n (11.35) (q,aq/b;q)n(ap/c,bcP;P)n and, more generally,

~ (1 - adpkqk)(l - bpk /dqk) (a, b;P)k(C, ad2/bc; q)k k (1 - ad)(l - bid) (dq, adq/b; qh(adp/c, bcp/d;p)k q k=-m~ (1 - a)(l - b)(l- c)(l- ad2/bc) { (ap, bP;P)n(cq, ad2q/bc; q)n - d(l - ad)(l - b/d)(l - c/d)(l - ad/bc) (dq, adq/b; q)n(adp/c, bcp/d; P)n _ (c/ad, d/bc;P)m+1(1/d, b/ad; q)m+l} (11.36) (l/c, bc/ad2; q)m+l (l/a, l/b;P)m+l ' where m is an integer or +00. An extension of the formula for the n-th q-difference of (apk; q)n-l, k ( 1 _ ~) (1 _ ~) t (apk, bp- ; q)n-l (1 - ap2k /b) (_l)kp(~) = 8n o. q q k=O (p;p)k(P;P)n_k(apk /b;P)n+l ' (11.37) Append ix III

SELECTED TRANSFORMATION FORMULAS

Heine's transformations of 2¢1 series: (b, az; q)oo ( / ) 2¢1 (a, b; C; q, z ) = ( ) 2¢1 C b, Z; az; q, b (IlLl ) c, Z; q 00 (c/b, bz; q)oo (/ / ) = ( ) 2¢1 abz c, b; bz; q, c b (IlL2) c, Z; q 00 (abz/c; q)oo = () 2¢1(c/a, c/b; C; q, abz/c). (IlL3) z;q 00

Jackson's transformations of 2¢1, 2¢2 and 3¢2 series: (az; q)oo (/ ) 2¢1 (a, b; C; q, z ) = ( ) 2¢2 a, c b; c, az; q, bz (IlI.4) z;q 00

_ (abz/c;q)oo A. [a,c/b,o. ] - (bz/c;q)oo 3'1-'2 c,cq/bz,q,q

(a,bz,c/b;q)oo A. [z,abz/c,o. ] /b) 3'1-'2 b b / ,q, q . (IlL5) + (c,z,c z;q 00 z, zq c

Transformations of terminating 2¢1 series: -n ) (c/b;q)n (bZ)n 2¢1 (q ,b;c;q,z = (.) - c,q n q X 3¢2(q-n, q/z, c-1q1-n; bc-1q1-n, 0; q, q) (IlL6) _ (c/b; q)n [q-n, b, bzq-n /c. ] - (.) 3¢2 ,q, q (III. 7) c, q n bq 1-n/C, °

= (c/b;q)nbn A. [q-n,b,q/z. ~] .) 3'1-'1 q, , (IlL8) ( c, q n bq 1-n/C ' C where, as elsewhere in this appendix, n denotes a non-negative integer.

Transformations of 3¢2 series:

A. [a, b, c. de ] 3'1-'2 d, e ,q, abc

_ (e/ a, de/bc; q)oo ¢ [a, d/b, d/ c . 2:] (IlL9) - (e, de/abc; q)oo 3 2 d, de/bc ,q, a _ (b, del ab, de/bc; q)oo [d/b, e/b, de/abc. ] - (d d / b ) 3¢2 / / ,q, b , (IlI.10) ,e, e a C; q 00 de ab, de bc

359 360 Appendix III

q-n, b, e ] [ 3¢2 d, e ; q, q

= (de/be;q)n (be)n A-. [q-n, d/b, die. ] (III. 11) ()e; q n d 3'1'2 d, de /'be q, q

_ (e/e;q)n n A-. [q-n,e,d/b. bq ] ( ) - ( e; q ) n e 3'1'2 d, eq 1 -n /'e q, e , III. 12

A-. [q-n, b, e. deqn ] _ (e/e; q)n A-. [q-n, e, d/b. ] (111.13) 3'1'2 d , e ,q, b e - (.)e, q n 3'1'2 d ,eq I-n/'e q, q .

The Sears-Carlitz transformation of a terminating well-poised 3¢2 series,

a, b, e aqz] [ 3¢2 aq / b, aq / e ;q'-b- e

_ (az;q)oo A-. [d,-a!,(aq)!,-(aq)!,aq/be. ] - ( ) 5'1'4 ,q, q (III.14) z;q 00 aq/b,aq/e,az,q/z provided that a = q-n. See (III.35) for a nonterminating case.

Sears' transformations of terminating balanced 4¢3 series: q-n, a, b, e ] [ 4¢3 d, e, f ; q, q

_ (e/a, f fa; q)n n A-. [ q-n, a, d/b, d/e . ] (III.15) - (f)e, ; q n a 4'1'3 d, aq 1 -n / e, aq 1 -n / f ,q, q _ (a,ef/ab,ef/ae;q)n ¢ [q-n,e/a,f/a,ef/abe. ] (III.16) - (e, f, ef /abe; q)n 4 3 ef lab, eflae, ql-n /a' q, q , where def = abeql-n.

Watson's transformation formulas: a, qa!, -qa!, b, e, d, e, f a2q2 ] [ 8¢7 1 1 ; q, b d f a"2, -a"2, aq/b, aq/e, aq/d, aq/e, aq/ fee (aq, aq/de, aq/df, aq/ef; q)oo ¢ [ aq/be, d, e, f ] = (aq/ d, aq/ e, aq/ f, aq/ def; q)oo 4 3 aq/b, aq/ e, def / a; q, q (III.17) whenever the 8¢7 series converges and the 4¢3 series terminates, and, when f = q-n,

1 1 a, qa"2, -qa"2, b, e, d, e, q-n a2qn+2] 8¢7 [ 1 1 a"2, -a"2, aq/b, aq/e, aq/d, aq/e, aqn+l; q, bede (aq, aq/de; q)n [aq/be, d, e, = 4¢3 (aq/d, aq/e; q)n aq/b, aq/e, de~=: / a ; q, q] (III.18) Appendix III 361 or, equivalently, ¢ [q-n, a, b, c. ] _ (d/b, d/c; q)n 4 3 d,e,j ,q,q - (d,d/bc;q)n

1 1 0", q0"2, -q0"2, j la, e/a, [ b, c, x S¢7 1 1 0"2, -0"2, e, j, ejlab, ejlac, where dej = abcql-n and 0" = ej /aq.

Another transformation of a terminating balanced 4¢3 series to a very-well­ poised S¢7 series, ¢ [q-n, a, b, c . ] _ (abq/ j, acq/ j, bcq/ j, q/ j; q)oo 4 3 d, e, j ,q, q - (aq/ j, bq/ j, cq/ j, abcq/ j; q)oo b, c, eqn. de] p,q/b, JLq/c, d ,q, abc ' (III.20) where dej = abcql-n and JL = abc/ f. Singh's quadratic transformation: a2,b2,c,d ] [ 4¢3 a bq2, 1 -ab q2,-C 1 d; q, q a2, b2, c2, d2 2 2] 4¢3 [ 2b2 d d; q ,q , (II1.21 ) = a q, -c ,-c q provided the series terminate. A q-analogue of Clausen's formula:

a,b,abz,ab/z . { [ ]}2 4¢3 a bq2, 1 -ab q2,-a 1 b,q,q _ [a2, b2, ab, abz, ab/z . ] - 5¢4 b 1 b 1 b 2b2 ' q, q (III.22) a q 2 ,-a q 2 ,-a ,a provided both series terminate. A non-terminating q-analogue of Clausen's formula is given in (8.8.17).

Transformations of very-well-poised S¢7 series:

1 1 a, qa2 , -qa2 , b, c, d, e, j a2 q2 ] S¢7 [ 1 1 a2 , -a2 , aq/b, aq/c, aq/d, aq/e, aq/ j; q, bcdej (aq,aq/ej,Aq/e, Aq/j;q)oo (aq/e,aq/j,Aq, Aq/ej;q)oo A, qA!, -qA!, e, Ab/a, Ac/a, Ad/a, aq ] x S¢7 [ j \ 1 _ \! Aq/j;q, ej /\2, /\, aq/b, aq/c, aq/d, Aq/e, (III.23) 362 Appendix III

(aq, b, bcf..t/a, bdf..t/a, bef..t/a, bff..t/a; q)oo (aq/c,aq/d,aq/e,aq/f,f..tq,bf..t/a;q)oo 1 1 f..t, qf..t2, -qf..t2, aq/bc, aq/bd, aq/be, aq/bf, [ bf..t/a 1 x 8¢7 1 1 aq/b; q, b , f..t2, -f..t2, bcf..t/a, bdf..t/a, bef..t/a, bff..t/a, (III.24)

Transformations of a nearly-poised 5¢4 series:

a, b, c, d, 5¢4 [ aq/b, aq/c, aq/d, 1 1 _ (Aq/a,A2q/a;q)n rI-. [A, qA2, -qA2, bA/ a, CA/ a, dA/ a, - 12'i"11 \ .1 (Aq, A2q/a2; q)n 1\ 2 , -A! , aq/b, aq/c, aq/d,

1 a 2 , -a!, (aq)!, -(aq)!, A2qn+1/a, Aq/d, -Aq/d, A(q/a)!, -A(q/a)!, aq-n /A,

5 r1-. 4 [q-n, b, c, d, e ] 'i" q1-n/b, q1-n/c, q1-n/d, eq-2n/f..t2;q,q (f..t2 qn+1, J.tq/ e; q)n (f..t2qn+1/e,f..tq;q)n f..t' qf..t2,1 -qf..t2,1 f..tbqn, f..tcqn, f..tdqn, x 12¢11 [ .1 1 f..t2, -f..t2, q1-n /b, q1-n /c, q1-n /d, q-n/2, _ q-n/2, q(1-n)/2, _q(1-n)/2, e, f..t2 qn+1/e ] f..tq(n+2)/2, _f..tq(n+2)/2, f..tq(n+1)/2, _f..tq(n+1)/2, f..tq/e, eq-n / f..t; q, q , (III.26) where A = qa2/bcd and f..t = q1-2n /bcd.

Transformation of a nearly-poised 7¢6 series:

1 1 rI-. [a, qa2 , -qa2 , b, c, d, q-n ] 7'i"6 1 1 2 2 2 ; q, q a2, -a2, aq/b, aq/c, aq/d, a q -n /A (Ajaq, A2/aq; q)n (1 - A2q2n-1/a) [A, qA!, -qA!, bAja, cA/a, = 12¢11 (Aq,A2/a2q;q)n (1-A2/aq) A!, -A!, aq/b, aq/c,

dA/a, (aq)2,1 -(aq)2,11 qa2, -qa2,121 A qn- la, q-n 1 1 1 1 1 ; q, q , aq/d, A(q/a)2, -A(q/a)2, Aja2, -A/a2, aq2-n /A, Aqn+1 (III.27) where A = qa2/bcd. Appendix III 363

Bailey's 1O¢9 transformation formula:

a, qa!! 2 , -qa 2, b ,C, d ,e, j ,/\aq\ n+l/ e,j q -n 1 1O¢9 [ 1 1 ;q,q a2,-a2,aq/b,aq/c,aq/d,aq/e,aq/j,ejq-n/A,aqn+l

_ (aq,aq/ej,Aq/e,Aq/j;q)n ¢ [A, qA!, -qA!, Ab/a, Ac/a, - (aq/e, aq/ j, Aq/ej, Aq; q)n 10 9 A!, -A!, aq/b, aq/c,

Ad/a, e, j, Aaqn+l/ej, q-n, ] (III.28) aq/d, Aq/e, Aq/ j, ejq-n la, Aqn+l ; q, q , where A = qa2/bcd.

Transformations of r+2¢r+1 series:

a b b qml b qmr ] do [ ' ,1 , ... , r . a-I m+l-(ml +···+mr ) r+2'!-'r+l b l+m b b' q, q q ,1,···, r

= (q, bq/a; q)oo (bq; q)m(bI/b; q)ml ... (br/b; q)mr bml+.+mr-m (bq, q/a; q)oo (q; q)m(b1 ; q)ml ... (br ; q)mr

q-m, b, bq/b1 , ... , bq/br ] [ (III.29) X r+2¢r+l bq / a, bq I-m1/b 1,···, bq I-mr/b r ; q, q and a b b qml b qmr ] do [ ' ,1 , ... , r . a-I 1-(ml+···+mr ) r+2'!-'r+ 1 b b b ,q, q cq, 1,···, r

(III.30) where m, ml, ... ,mr are arbitrary nonnegative integers.

Three-term transformation formulas:

(abz/c, q/c; q)oo ( / / / / ) 2¢1 ( a, b; c; q, z ) = ( / / ) 2¢1 C a, cq abz; cq az; q, bq c az c, q a; q 00 (b, q/c, cia, az/q, q2 /az; q)oo (/ / 2/ ) ( ) - (/ b / / / / ) 2¢1 aq c,bq c;q c;q,z. III.31 c q, q c, q a, az c, cq az; q 00

(b,c/a,az,q/az;q)oo ( / / /) 2¢1 ( a, b; c; q, z ) = (b/ /) 2¢1 a, aq c; aq b; q, cq abz c, a, z, q z; q 00 (a, c/b, bz, q/bz; q)oo ( / / /) + ( /b / ) 2¢1 b, bq c; bq a; q, cq abz . (III.32) c, a , z, q z; q 00 364 Appendix III

a, b, e de] [ 3¢2 d,e ;q, abc

_ (e/b, e/e, eq/ a, q/ d; q)oo ¢ [e, d/ a, eq/ e . bq] - (e, eq/d, q/a, e/be; q)oo 3 2 eq/a, beq/e' q, d (q/ d, eq/ d, b, e, d/ a, de/beq, beq2 / de; q)oo ¢ [aq/ d, bq/ d, eq/ d . de ] - (d/q, e, bq/d, eq/d, q/a, e/be, beq/e; q)oo 3 2 q2/d, eq/d ,q, abc . (III.33)

a, b, e. ~] _ (e/b, e/e; q)oo [d/a, b, e. ] 3¢2 [ ,q, b - ( /b ) 3¢2 / ,q, q d, e ace, e e; q 00 d, beq e (d/ a, b, e, de/be; q)oo [e/b, e/ e, del abc ] (III. 34) + (d, e, be/e, de/abc; q)oo 3¢2 de/be, eq/be ; q, q .

¢ [a, b, e aqx] 3 2 aq/b, aq/e; q, bc

_ (ax;q)oo r/, [a!, -a!, (aq)!, -(aq)!, aq/bc. ] - (x; q)oo 5'1'4 aq / b, aq / e, ax, q/x ,q,q (a,aq/be,aqx/b,aqx/e;q)oo + (aq/b, aq/e, aqx/be, X-I; q)oo

r/, [xa!,-xd,x(aq)!,-x(aq)!,aqx/be ] x 5'1'4 ; q, q . (III.35) aqx/b,aqx/e,xq,ax2

¢ [a, qa!, -qa!, b, e, d, e, f. a2q2 ] 8 7 a!, -a!, aq/b, aq/e, aq/d, aq/e, aq/ f' q, bedef _ (aq, aq/de, aq/df, aq/ef; q)oo ¢ [ aq/be, d, e, f . ] - (aq/ d, aq/ e, aq/ f, aq/ def; q)oo 4 3 aq/b, aq/ e, def / a' q, q (aq,aq/be,d,e,f,a2q2/bdef,a2q2/edef;q)00 +~~~~--~~~~-7~~--~~~-- (aq/b, aq/e, aq/d, aq/e,aq/f, a2q2/bedef,def/aq; q)oo aq/de,aq/df,aq/ef,a2q2/bedef ] x 4¢3 [ 2 2 2 2 2; q, q . (III.36) a q /bdef,a q /edef,aq /def

a, qa!, -qa!, b, e, d, e, f a2q2 ] [ 8¢7 a!, -a!, aq/b, aq/e, aq/d, aq/e, aq/f;q, bedef (aq,aq/de,aq/df,aq/ef,eq/e, fq/e, b/a, bef/a;q)oo (aq/d, aq/e, aq/f,aq/def,q/e, efq/e, be/a, bf/a;q)oo

ef/e,q(ef/e)! ,-q(ef/e)! ,aq/be,aq/ed,ef/a,e,J bd] x 8¢7 [ 1 1 ; q, --;;: (ef /ep ,- (ef /ep ,befla, def la, aq/e, fq/e, eq/e Appendix III 365

b (aq, bq/a, bq/c, bq/d, bq/e, bq/ j, d, e, j, aq/bc; q)oo +-~~~~~~~~~~~~~~~ a (aq/b,aq/c,aq/d,aq/e,aq/j,bd/a,be/a;q)oo (bdej /a2, a2q/bdej; q)oo x-:-::--:--;--.:..,,-::-'-:-'-----'---:-:-=-'-::---;:..-.c-~=:------:-- (bj/a,dej/a,aq/dej,q/c,b2q/a;q)oo ¢ [b2/a, bqa-!, -bqa-!, b, bc/a, bd/a, be/a, bj /a. a2q2 ] x 8 7 ba-!,-ba-!,bq/a,bq/c,bq/d,bq/e,bq/j ,q, bcdej . (III.37)

Transformation of an 8'¢8 series: (aq/b, aq/c, aq/d, aq/e,q/ab, q/ac, q/ad, q/ae; q)oo (ja,ga, j/a,g/a,qa2, q/a2;q)oo qa, -qa ba, ca, da, ea, ja, x 8'¢8 [ a, -a, aq/b, aq/c, aq/d, aq/e, aq/ j, (q,q/bj,q/cj,q/dj,q/ej,qj/b,qj/c,qj/d,qj/e;q)oo (ja,q/ja,aq/j,j/a,g/j,jg,qj2;q)oo P, qj, -qj, jb, jc, jd, je, jg q2 x 8¢7 [ 1 j, - j, jq/b, jq/c, jq/d, jq/e, jq/g; q, bcdejg + idem (1; g). (III.38)

Bailey's four-term 1O¢9 transformation: 1 1 a,qa'l,-qa'l,b,c,d,e,j,g,h ] 1O¢9 [ 1 1 ;q,q a'l,-a'l,aq/b,aq/c,aq/d,aq/e,aq/j,aq/g,aq/h (aq,b/a,c,d,e,j,g,h,bq/c;q)oo + (b2q/a,a/b,aq/c,aq/d,aq/e,aq/j,aq/g,aq/h,bc/a;q)oo (bq/ d, bq/ e, bq/ j, bq/ g, bq/ h; q)oo x-c--.:..,------'-,---:'--c--'-7------'---:---':-- (bd/a, be/a, bj la, bg/a, bh/a; q)oo b2/a, qba-!, -qba-!, b, bc/a, bd/a, be/a, bj la, bg/a, bh/a 1 x 1O¢9 [ 1 1 ; q, q ba-'l, -ba-'l, bq/a, bq/c, bq/d, bq/e, bq/ j, bq/g, bq/h (aq,b/a, Aq/j,Aq/g, Aq/h,bj/A,bg/A,bh/A; q)oo (Aq,b/A, aq/j,aq/g, aq/h, bj/a, bg/a, bh/a; q)oo A, qA!, -qA!, b, Ac/a, Ad/a, Ae/a, j, g, h 1 X 1O¢9 [ 1 1 ; q, q A'l, -A'l, Aq/b, aq/c, aq/d, aq/e, Aq/ j, Aq/g, Aq/h (aq,b/a,j,g,h,bq/j,bq/g,bq/h,Ac/a,Ad/a,Ae/a,abq/Ac;q)oo + (b2q/A,A/b,aq/c,aq/d,aq/e,aq/j,aq/g,aq/h,bc/a,bd/a,be/a,bj/a;q)oo (abq/Ad, abq/Ae; q)oo [ b2/A, qbA-!, -qbA-!, b, bc/a, bd/a, x (bg/a,bh/a;q)oo 1O¢9 bA-!,-bA-!,bq/A,abq/cA,abq/dA,

be/a, bj /A, bg/A, bh/A ] . q q (III.39) abq/eA,bq/j,bq/g,bq/h" , 366 Appendix III where a3q2 = bcdefgh and A = qa2/cde.

Transformation of a 1O'l)i1O series: (aq/b,aq/c,aq/d,aq/e,aq/f,q/ab,q/ac,q/ad,q/ae,q/af;q)oo (ag,ah,ak,g/a,h/a,k/a,qa2,q/a2;q)00 qa, -qa, ba, ca, da, ea, fa, ga, ha, ka q3 ] [ X 1O'l)i1O a, -a, aq/b, aq/ c, aq/ d, aq/ e, aq/ f, aq/ g, aq/ h, aq/ k ; q, bcdefghk (q,q/bg,q/cg,q/dg,q/eg,q/fg,qg/b,qg/c,qg/d,qg/e,qg/f;q)00 (gh,gk,h/g,k/g,ag,q/ag,g/a,aq/g, qg2;q)00

g2, qg, _qg, gb, ge, gd, ge, gf, gh, gk q3] [ x 1O

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(a)n 2, 5 Cj (;3lq) 187 (a; q)n 3, 6 cn(x; a; q) 202 (a; q)l/ 208 c~(x; k) 204 (al' a2, ... ,am; q)n 6 C 312 (a; q)oo 6 C N (Contour) 126 (al' a2, ... ,am; q)oo 6 Cj,k,m,n (Contour) 345 (a; q,P)n, (al' a2,···, am; q,P)n 304 C(¢) 265 (a;p, q)r,s 112 Cosq(x) 28 ,al, laJ 205 Cq(x;w) 212 [a]q 7 Cn(x; ;3lq) 31 [n]q! 7 C~(x; ;3lq) 255 [a]q,n 7 C~(x) 2 [al' a2,·· . ,am]q,n 7 C.T. 171 'Dq 27 [a; 0"] 7 Dq 197 [n; O"]! 8 212 [a;O"]n 8 Dn(x; ;3lq) 312 [a;O",T] 17 r+ler res 316, 319 [n;O",T]! 312 eq(z) 11 [a; 0", T]n, [al' a2,··· ,am; 0", T]n 312 eXPq(z) 12 [al' a2,·· . ,am; 0", T]n 312 Eq(z) 11 an(a, ;3lq) 192 E± 197 A(z) 185 q bk,j 195 r+1Er 304 309 b(k, n;;3) 185 rEs Eq(x; a) 12 bn(a,;3; q) 192 Eq(x; a, b) 12 B(z) 189 Eq(x, y; a) 111 B((}) 265 F(a, b; c; z) 2 B(x,y) 23 3 Bq(x, y) 23 rFs Fl ,F2,F3 283 B~(x; k) 205 F4 219, 283 BE(t; q,p) 344 F(x, t) 259 cosq(x) 28 F(x, ylq) 265 Ck,n 195 Fl (x, ylq), F2(x, ylq) 266

415 416 Symbol index

FA:B;C D:E;F 283 Lt(x, y; a, b, e, a", M, N; q) 231 r9s 316,319 L~(x) 4 rGr 309 L~(x; q) 210 rGs 310 Mn(x; a, e; q) 202 9o(a,b,e,d,f) 159 M(x, y), M(x, y; a, b, e, e'; q) 291 G(x, t) 260 p(n) 239 269 Gl(x, ylq) PN (n), Pe (n), Pdist (n), Po( n) 240 G2(x, ylq) 270 Peven (n), Podd (n) 241 Gt(x; a, b, e, dlq) 262 Pn(x) 175 Gt(x; a, elq) 264 Pn(x; a, (3) 195 h(x; a), h(x; a; q) 154 Pn(x; a, b; q) 32, 181 h(x; aI, a2,··· ,am) 154 Pn(eos(B + ¢); a, blq) 194 h(x; aI, a2,··· ,am; q) 154 Pn(x; a, b, e, dlq) 59, 189 hn(q),hn(a,b,e,N;q) 180 Pv(qX; a, b; q) 253 ht(a) 261 P(z) 125 hn(a, b, e; q) 182 Pn(x) 2 hn(a, b, e, dlq) 190 Pn(x; a, b, e; q) 182 hn(a, b, e, d, 1) 256 Pn(xlq) 255 hn((3lq) 186 Pn(x; a, b, e, d, a2, a3, . .. , aslq) 255 hn(x; q) 209 p~a,(3)(x) 2 Hn(x) 4 p~a,(3) (x; q) 191 Hn(xlq) 31 Hn(x;q) 209 p~a,(3) (xlq) 191 Ht(x), Ht(x; a, b, e, dlq) 263 Pz(x, y) 229 H(x,y, t) 259 Pz(x,y;a,b,e,a",](,M,N;q) 229 idem (b; e) 69 Qn(x), Qn(x; a, b, N; q) 180 idem (al;a2, ... ,ar+1) 121 Qn(x; a, blq) 273 I(a,b,e,d) 156 Qn(z; a, b, e, dlq) 189 1m 125 r2k(n) 243 Ja(x) 4 rn(x;a,b,e,dlq),rn(x) 224, 265 JP) (x; q), JS2) (X; q), JS3) (x; q) 30 r~(x), r~(x; a, b, e, dlq) 254 J(a,b,e,d,j,9) 157 RN 333 kn 265 Rn(~(x)), Rn(~(x); b, e, N; q) 181 ]( (Contour) 125 Rm(x), Rm(x; a, b, e, d, 1) 257 ]((x, y, z) 225 Rm,j(z) 345 ]((x, y, Z; (3lq) 223 sn(x) 216 ](n(x; a, N; q) 201 sinq(x) 28 ](n(x; a, Nlq) 201 Sinq(x) 28 ](t(x, y) 227, 261 SL(2, 2':) 315 ](t(x, y; (3lq) 227 S(a,b,e,d,j,;A,~) 294 ](:ff (x; a, N; q) 202 S(A,~,V,p) 61 Lm(a; b, e, d) 164 Sn(x;p, q) 214 Lt(x, y; (3lq) 227 Sn(x),Sn(x;a,b,e,d,j) 257 Symbol index 417

Sq(x; w) 212 312 rts 7 [~Lp '][' 344 a(x) 175 ,][,n 347 r(x) 21 Tn (X) 2 rq(X) 20, 29 Tn,k(Z) 345 r(z; q,p) 311,338 Un (X) 2 r(Zl'oo 0 ,zn;q,p) 345 U~a)(x; q) 209 t(z; q,p) 338 Vn 176 Om,n 42 r+lVr 312 Oq 11, 197 v(x; a, b, c, d, f) 159 /).J(z), '\l J(z) 11 V(eiO ) 198 /).Uk 80 r+1 v;. 306 /).bJ(Z) 32 Wj 175, 179 /).(Z) , /).(zqk) w(x) 175 331 /).(z;p), /).n(z;p) w(x; a, b, c, d) 157 333 /).E(Z; t; q,p) 344 w(x; a, b, c, dlq) 190 w( 0; q) 192 /).E(Z; t) 345 0 0 0, 303 wk(a, b, c, dlq) 191 O(x;p), O(Xl' xm;P) 4 16 W(O) 194 171 (x, q), 172 (X, q), 173 (X, q), 17 (x, q) K(a, b, c, dlq) 190 W(eiO ), W(eiO,a,b,c,dlq) 198 Aj 180 W,a(xlq) 185 An(q) 156 Wn(x; b,q) 214 An(a, b, c, d, a2, a3, 0 0 0, aslq) 156 Wn(x;q), Wn(x;a,b,c,fV;q) 59, 180 JL(x) 181 r+lWr 39 Vn 269 x+ 114 1rn 270 z,zqk 331 p(x;q),p(x;a,b,c,fV;q) 180 Z 312 194 2Z 320 Pn(a, blq) Z+rZ 312 p(xlq) 156 J J(t) dqt 23 p(x; a, b, c, d, a2, a3, 0 0 0, aslq) 156 J(n) "-' g(n) 185 ¢(a,b;c;q,z) 3 L:~=m ak 81 2¢1 (a, b; c; q, z) 3 r¢s 4 rr~=m ak 325 [0 0 oj 95 24 (l) , (2) ,(3), (4) 283 [~L' [~L 1(a; {3, (3'; 'I; q; x, y) 294 25 D(a; bl , 0 0 0, br; c; q; Xl, 0 0 0, xr) 300 [kl' 0 ~ ,kmL A:B;C D:E;F 283 [~Lp' [~Lp 311 r1/Js 137 r-lW r-2 315 Author index

Adams, C. R., 36 Bateman, H., 221, 222 Adiga, C., 35, 67, 152 Baxter, R. J., 35, 67, 302, 307 Agarwal, A. K., 111, 257 Beckmann, P., 257 Agarwal, N., 112 Beerends, R. J., 301 Agarwal, R. P., 34-36, 68, 111, 112, 119, Bellman, R., 35 122, 124, 136, 301 Bender, E. A., 34 Aigner, M., 35 Berg, C., 214 Alder, H. L., 257 Berkovich, A., 35, 67, 341, 342 Alexanderson, G. L., 35 Berman, G., 35 Alladi, K., 35, 67 Berndt, B. C., 35, 67, 152, 257 Allaway, Wm. R., 204, 205, 214, 216 Bhargava, S., 35, 67, 152 AI-Salam, W. A., 35, 36, 52, 83, 86, 204, Bhatnagar, G., 257, 258, 325, 349, 350 205, 209, 214-216, 273 Biedenharn, L. C., 213 Andrews, G. E., 9, 15, 21, 26, 29, 33-37, Bohr, H., 29 52, 53, 59, 65, 67, 68, 111, 112, 138, 141, Baing, H., 34 147, 149, 152, 174, 181, 182, 184, 213, Borwein, J. M., 67 215, 216, 241-243, 245, 257, 296, 297, Borwein, P. B., 67 300, 341, 342 Bowman, D., 36 Aomoto, K., 174 de Branges, L., xv, 84, 232, 237, 257 Appell, P., xv, 282 Bressoud, D. M., 35, 36, 67, 68, 83, 111, Artin, E., 36 112, 213, 226, 257, 301 Askey, R., 2, 9, 17, 21, 26, 29, 31, 33-35, Bromwich, T. J. I'A., 5 52, 53, 59, 67, 84, 99, 113, 125, 129, 136, Brown, B. M., 214, 216 138, 141, 149, 152, 154, 165, 170-172, Burchnall, J. L., 111, 112, 291 174, 177, 180-182, 185, 188, 191, 193, Burge, W. H., 67 195, 197, 198, 200, 204-206, 214, 215, Bustoz, J., 36, 213, 273, 280, 281 232, 236, 242, 257, 273, 274, 281 Atakishiyev, M. N., 215, 257 Atakishiyev, N. M., 12, 35, 195, 213-215, Carlitz, L., 28, 31, 35, 36, 58, 64, 111, 152, 257 209, 210, 215, 216, 275, 281 Atkin, A. O. L., 152 Carlson, B. C., 34 Atkinson, F. V., 176 Carmichael, R. D., 36 Carnovale, G., 35 Bailey, W. N., xxiii, 5, 9, 18, 41, 46, 47, Cauchy, A.-L., 9, 112, 132 50, 53, 54, 57, 58, 60, 61, 64, 67, 73, 96- Cayley, A., 112 100, 112, 140, 148-150, 152, 219, 236, Charris, J., 215 261, 287-290, 293, 298, 299 Chaundy, T. W., 111, 291 Baker, M., 35 Cheema, M. S., 35 Baker, T. H., 68 Chen, Y., 216 Bannai, E., 213 Cherednik, 1., 68 Barnes, E. W., 113, 117 Chihara, L., 213-215 418 Author index 419

Chihara, T. S., 34, 176, 209, 213-216, 259, Floris, P. G. A., 258 273 Foata, D., 36, 112 Chu Shih-Chieh, 2 Foda, 0., 67 Chu, VV., 111, 151, 152, 257, 325, 349, 350 Forrester, P. J., 67, 68, 257 Chudnovsky, D. V., xv, 232 Fox, C., 19 Chudnovsky, G. V., xv, 232 Frenkel, I. B., 7, 17, 258, 302, 307, 315 Chung, VV. S., 258 Freud, G., 176 Ciccoli, N., 215 Fryer, K. D., 35 Cigler, J., 33, 35, 112, 216 Fiirlinger, J., 112 Clausen, T., xiv, 103, 232 Cohen, H., 258 Gangolli, R., 195 Comtet, L., 258 Garoufalidis, S., 34 Coon, D., 35 Garrett, K., 67, 148 Cooper, S., 68, 258 Garsia, A. M., 67, 112 Crippa, D., 34 Gartley, M. G., 12 Garvan, F. G., 34, 35, 67, 174 Date, E., 302, 307 Gasper, G., 19, 32-34, 36, 58, 61, 65-67, Daum, J. A., 18 74,78,80,81,83-85,88,91,93,94,105- Dehesa, J. S., 216 107, 109-111, 130, 135, 136, 172, 174, Delsarte, P., 181, 202, 213, 215 186, 195, 207, 208, 213, 215, 217, 223, Denis, R Y., 301, 350 226, 227, 229, 232, 233, 235, 236, 238, Desarmenien, J., 112, 216 239, 248, 250, 251, 256-258, 279, 300, Dickson, 1. E., 243 302, 326, 327, 329, 343, 344, 349, 350 van Diejen, J.F., 258, 302, 317, 347, 349 Gauss, C. F., xiv, 1 Di Vizio, L., 36 Gegenbauer, 1., 225, 249 Dixon, A. C., 38 Geronimo, J. S., 258 Dobbie, J. M., 67 Gessel, I., 83, 93, 112, 258, 349, 350 Dougall, J., 38 Goethals, J. M., 181, 215 Dowling, T. A., 35 Goldman, J., 35, 258 Dunkl, C. F., 35, 181, 202, 259 Gonnet, G. H., 34, 174 Duren, P. L., 257 Gordon, B., 152 Dyson, F. J., 67, 68, 242, 257 Gosper, R. VVm., 21, 34, 36, 81, 93, 111, 148 Edwards, D., 112 Goulden, I. P., 37, 68, 257 Eichler, M., 307 Greiner, P. C., 19 Erdelyi, A., 34, 66, 111, 174, 236, 249, 265, Grosswald, E., 243 284, 287, 302 Gupta, D. P., 215 Euler, 1., 3, 62, 239, 241 Gustafson, R A., 35, 152, 173, 258, 349, Evans, R J., 68, 174 350 Evans, VV. D., 214 Ewell, J. A., 35 Habsieger, L., 174 Exton, H., 36, 283 Hahn, VV., 28, 30, 36, 138, 180, 181 Hall, N. A., 72 Faddeev, L. D., 36 Handa, B. R., 35 Fairlie, D. B., 34 Hardy, G. H., 44, 45, 138, 240-242 Favard, J., 176 Heine, E., xviii, 3, 9, 13, 14, 21, 26-28 Feinsilver, P., 33 Henrici, P., 32, 34 Felder, G., 312, 338, 339, 350 Hickerson, D., 242, 257 Feldheim, E., 281 Hilbert, D., 243 Fields, J. L., 84, 86 Hirschhorn, M. D., 152, 245, 258 Fine, N. J., 67, 112, 242 Hodges, J., 215 Floreanini, R, 215, 301 Hofbauer, J., 112 420 Author index

Hou, Q.-H., 216 Lakin, A., 141 Hua, L. K., 243 Lam, H. Y., 258 Lapointe, L., 301 Ihrig, E., 35 Lascoux, A., 216 Ismail, M. E. H., 12, 22, 30, 31, 35-37, 61, Lassalle, M., 174, 201 65, 67, 68, 86, 111, 138, 141, 148, 152, Le, T. T., 34 155, 172-174, 177, 184, 185, 204, 209, Leininger, V., 258, 349 212-216, 251, 254, 257-259, 261, 272- Leonard, D. A., 213 277, 280, 281, 301 Lepowsky, J., 35, 68 Ito, M., 174,349 LeTourneux, J., 215 Ito, T., 213 Lewis, R. P., 35 Li, X., 215 Jackson, D. M., 37 Libis, C. A., 215 Jackson, F. H., 14, 18, 21-23 28 30 32 Lilly, G. M., 258, 349 35, 36, 43, 58, 87, 103, 1'12, '136: 11~, Littlewood, J. E., 118 138, 232, 282-284, 311, 338 Liu, Zhi-Guo, 258 Jackson, M., 144, 146, 152 Lorch, L., 22 Jacobi, C. G. J., 15, 34 Louck, J. D., 213 Jain, V. K., 36, 60, 61, 63, 78, 97, 98, 101, Lubinsky, D. S., 216 111, 112, 152, 232, 242, 301 Luke, Y. L., 34, 88 Jimbo, M., 35, 302, 307 Joichi, J. T., 34, 35 Macdonald, 1. G., 35, 68, 174 Joshi, C. M., 68 MacMahon, P. A., 35 Kac, V. G., 35, 152 Masson, D. R., 36, 173, 213, 215, 275, 276, Kadell, K W. J., 36, 68, 152, 174, 258 280, 281 Kairies, H.-H., 36 Masuda, T., 37, 302 Kajihara, Y., 302, 346 McCoy, B. M., 67, 68 Kajiwara, K, 302 Meijer, H. G., 36 Kalnins, E. G., 135, 214, 257, 258 Menon, P. K., 35 Kampe de Feriet, J., xv, 282 Merkes, E., 36 van Kampen, N. G., 35 Milin, 1. M., 257 Kaneko, J., 68, 174 Miller, W., 34, 36, 135, 214, 258 Karlsson, Per W., 18 Milne, S. C., 35, 67, 152, 174, 213, 257, Kashaev, R. M., 36 258, 303, 325, 331, 333, 335, 349, 350 Kendall, M. G., 35 Mimachi, K., 36 Kirillov, A. N., 258 Minton, B. M., 18, 19 Klein, F., xv Misra, K. C., 68 Miwa, T., 302, 307 Klimyk, A. U., 215, 257 Knopfmacher, A., 34 Moak, D. S., 21, 29, 211, 216 Mohanty, S. G., 35 Knuth, D., 35, 36 Koekoek, R., 259, 273 Molk, J., 151, 331 Mollerop, J., 29 Koelink, H. T., 213-215, 252, 256, 258, 274, 279, 280, 302 Monsalve, S., 215 Koepf, W., 34 Mordell, L. J., 258 Koornwinder, T. H., 21, 33-35, 101, 172, Morris, W., 68, 174 202, 210, 213-216, 257, 258, 349 Mu, Y.-P., 216 Krattenthaler, C., 34, 83, 111, 112, 258, Muldoon, M. E., 22, 36 301, 346, 349 Mulla, F. S., 257 Kummer, E. E., xv Mullin, R., 258 Kuniba, A., 302, 307 Muttalib, K A., 216

Laine, T. P., 216 Nakagami, Y., 37 Author index 421

Narukawa, A., 350 Rogers, L. J., xviii, 31, 68, 184, 226 Nassrallah, B., 96, 104, 105, 112, 158, 163, Rogov, V.-B. K., 35 232, 295, 301, 341 Rosengren, H., 258, 281, 301-303, 331, 332, Needham, J., 2 335, 337, 346-350 Nelson, C. A., 12 Rota, G.-C., 35, 258 Nevai, P. G., 186, 190, 215, 259 Rothe, H. A., 9 Newcomb, J. W., 258, 350 Roy, R, 9, 113, 125, 129, 172 Nikiforov, A. F., 34, 135, 136, 213 Ruedemann, R. W., 213 Nishizawa, M., 350 Ruijsenaars, S. N. M., 311, 338 van Norden, Y., 302 Noumi, M., 258, 302, 346 Saalschiitz, L., 17, 113 Saff, E. B., 216 O'Hara, K. M., 35 Sahai, V., 301 Ohta, Y., 302 Sauloy, J., 36 Okado, M., 302, 307 Schempp, W., 257 Onofri, E., 257 Schilling, A., 67, 68 Opdam, E. M., 68 Schlosser, M., 152, 174, 257, 258, 302, 326, Orr, W. McF., 112 327, 329, 343, 344, 347, 349, 350 Schur, I. J., 68 Pastro, P. I., 216 Schiitzenberger, M.-P., 33 Paule, P., 34, 67, 68, 112 Schwartz, A. L., 257 Pearce, P. A., 35, 67 Schweins, Ferd. F., 9 Perline, R, 37 Sears, D. B., 15, 49, 51, 61, 64, 70, 71, 74, Perron, 0., 176 130, 131, 136, 152 Petkovsek, M., 34 Selberg, A., 174 Pfaff, J. F., xiv, 17 Shohat, J., 214 Phragmen, E., 302 Sills, A. V., 34, 68 Pitman, J., 301 Simon, K., 34 Pitre, S. N., 301 Singh, V. N., 99, 105, 232 P6lya, G., 35 Slater, L. J., 5, 9, 32, 36, 68, 125, 126, 128, Potter, H. S. A., 33 130, 141-144, 151, 152, 242, 331 Spiridonov, V. P., 215, 302-306, 309, 310, Qazi, Tariq M., 214, 254, 255, 257 313, 316, 317, 320, 338, 341, 345, 347, Quano, Y.-H., 67 349 Srinivasa Rao, K., 301 Rademacher, H., 241 Srivastava, B., 301 Rahman, M., 34, 36, 58, 61, 65, 74, 78, 81, Srivastava, H. M., 84, 107, 232, 283 91-94, 96, 104, 108-112, 136, 152, 153, Stanton, D., 34-36, 65, 67, 68, 83, 93, 111, 155, 158, 159, 161, 163, 169, 171-174, 112, 148, 155, 174, 181, 201, 213, 215, 185, 191, 207, 208, 211-217, 223, 225- 216, 258, 276, 277, 280, 349, 350 229, 232, 235, 248, 249, 251, 253-258, Starcher, G. W., 36 269, 274-279, 281, 295, 341, 342, 350 Stembridge, J. R., 68 Rains, E., 174, 302, 347, 349 Stevens, L., 350 Rainville, E. D., 34 Stokman, J. V., 174, 215, 253, 258, 279 Rakha, M. A., 258, 349 Stone, M. H., 176 Ramanujan, S., 68, 138, 152, 172, 232, Stuart, A., 35 236, 240 Styer, D., 36 Ramanathan, K. G., 67 Subbarao, M. V., 152, 349 Rankin, R A., 67 Sudler, C., 35 Remmel, J., 112 Suslov, S. K., 12, 34-36, 111, 135, 136, Reshetikhin, N. Yu, 258 152, 153, 195, 213, 214, 216, 257, 258, Riese, A., 34 274-276, 280, 281 422 Author index

Swarttouw, R. F., 36, 258, 259, 273 Vilenkin, N. Ja., 34 Swinnerton-Dyer, P., 152 Vinet, L., 215, 301 Sylvester, J. J., 35 Volkov, A. Yu., 36 Szego, G., 31, 34, 35, 176, 177, 213, 216, 249, 259, 260, 265 Wallisser, R., 36 Warnaar, S. 0., 34, 67, 68, 302-304, 306, Takacs, L., 2 323, 325, 329, 339-342, 346, 347, 349 Tamarkin, T., 214 Watson, G. N., xv, 16, 17, 21, 34, 35, 42, Tannery, J., 151, 331 67, 68, 112, 114, 115, 117, 119, 124, 151, Tarasov, V., 174 152, 220, 317 Temme, N., 259 Whipple, F. J. W., 49, 68, 112 Terwilliger, P., 213 Whittaker, E. T., 16, 17, 21, 34, 151 Thomae, J., xviii, 21, 23, 24, 69, 70 Wilf, H. S., 34 Toeplitz, 0., 35 Wilson, J. A., 59, 99, 132, 152, 154, 165, Touhami, N., 258 177, 180, 188, 191, 193, 195, 197, 198, Tratnik, M. V., 213, 215, 258 200, 206, 214, 215, 259, 261, 272, 274, Trebels, W., 257 317 Trjitzinsky, W. J., 36 Wilson, R. L., 68 Trutt, D., 257 Wimp, J., 37, 84 Turaev, V. G., 7, 17, 258, 302, 307, 315 Wintner, A., 176 Wright, E. M., 35, 241, 242 Wu, M.-Y., 34 Ueno, K., 37 Upadhyay, M., 301 Uvarov, V. B., 34, 136, 213 Xu, Y., 259

Valent, G., 213 Yamada, Y., 302 Vandermonde, A. T., 2 Yang, K.-W., 33 Van Assche, W., 258 Yoon, G. J., 213 Van der Jeugt, J., 252, 258, 280, 301 Varchenko, A., 174, 312, 339, 350 Zagier, D., 307 Verma, A., 36, 52, 61, 63, 65, 68, 78, 83, Zaslavsky, T., 35 86, 97, 98, 111, 112, 185, 207, 214, 216, Zeilberger, D., 34, 35, 68, 174 225, 228, 242, 249, 257, 342, 349 Zhang, R., 12, 204, 251 Vidyasagar, M., 152 Zhedanov, A. S., 215, 302 Viennot, G., 67, 216 Zimmermann, B., 34 Subject index

Addition formula for Bailey's 3'I/J3 and 6'I/J6 summation formulas, continuous q-ultraspherical polynomials, 149, 150, 357 249 Balanced series (and k-balanced), 5 Eq functions, 111 Barnes' beta integral, 114 little q-Jacobi polynomials, 210 Barnes' contour integral, 113 little q-Legendre functions, 254 Barnes' first and second lemmas, 113 Affine q-Krawtchouk polynomials, 202 q-analogues of, 119 Al-Salam-Chihara polynomials, 273 Base modular parameter, 304 Almost-poised series, 111 Basic contour integrals, 113 Analytic continuation of Basic hypergeometric functions, 5 21 series, 117 Basic hypergeometric series, 1-4 r+" cPr series, 120 Basic integrals, 23 Andrews' q-Dyson conjecture, 68 Basic number, 4 Appell functions and series, 282, 283 Bateman's product formula, 221, 222 Askey-Gasper inequality, 232 Bessel function, 4 q-analogues of, 236-238 Beta function, 22 Askey-Wilson polynomials, 59, 188 Beta function integral, 23 multivariable extension of, 255 Bibasic Askey-Wilson q-beta integral, 154, expansion formulas, 84-87 163-168 series, 80-87 Associated Askey-Wilson polynomials, 254 summation formulas, 80-83, 328, 358 Associated q-ultraspherical polynomials, 255 transformation formulas, 105-107 Big q-Jacobi polynomials, 181, 182 Bailey-Daum summation formula, 18, 354 Bilateral basic hypergeometric series, 137 Bailey's four-term transformation formu­ Bilateral bibasic series, 82 las for balanced very-well-poised lOcP9 se­ Bilateral q-integral, 23 ries, 55-58, 64, 365 Bilateral theta hypergeometric series, 309, Bailey's identity, 61 316 Bailey's lemma, 41 Bilinear generating functions, 227, 259, 281 Bailey's product formulas, 219, 236 Binomial theorem, 8 Bailey's sum of a very-well-poised 6'I/J6 se- q-analogue of, 8, 354 ries, 140, 357 Biorthogonal rational functions, 35, 173, Bailey's summation formula, 54, 124, 356 213, 257, 258, 345 Bailey's three-term transformation formula for a very-well-poised 8cP7 series, 53, 364 Cauchy's beta integral, 132 Bailey's transformation formulas for Christoffel-Darboux formula, 200 2'I/J2 series, 150 Chu-Vandermonde formula, 3 terminating 5cP4 and 7cP6 series, 45-47, q-analogue of, 14, 354 363 Clausen's formula, 103, 232 terminating lOcP9 series, 47, 263 q-extensions of, 232, 235, 236, 251, 261 limiting cases of, 48-53, 360-362 Connection coefficients, 33, 195-197 423 424 Subject index

Continuous q-Hahn polynomials, 193 Expansion formulas (also see Addition for- in base q-l, 279 mula, Connection coefficients, Lineariza­ Continuous q-Hermite polynomials, 31 tion formula, Nonnegativity, Product for­ Continuous q-Jacobi polynomials, 191 mulas, and Transformation formulas), Continuous q-ultraspherical polynomials, basic series, 19, 40, 41, 62--67, 84-87, 31, 184 107,143 Contour integral representations of elliptic and theta hypergeometric series, 2Fl series, 113 329-331, 337 2<1>1 series, 115 multi basic series, 95-97 very-well-poised series, 121-124 Convergence of Fields-Wimp expansion, 84 basic hypergeometric series, 5 q-extensions of, 84-87 bilateral basic series, 137 q,p-extensions of, 329-331 hypergeometric series, 5 Cubic summation and transformation for­ Gamma function, 2, 21, 22 mulas, 93, 108-110 q-analogues of, 20, 29, 353 Gauss' hypergeometric series, 1, 2 Gauss' multiplication formula, 22 Darboux's method, 259 Gauss' summation formula, 3 Discrete q-Hermite polynomials, 209 Gegenbauer's addition formula, 249 Dixon's formula, 38 q-extension of, 249 q-analogue of, 44, 355 Gegenbauer polynomials, 2 Double product theta function, 303 Gegenbauer's product formula, 225 Dougall's summation formulas, 38, 39 General basic contour integral formulas, q-analogues of, 43, 44, 356 126 Dual orthogonality, 176 General transformations for r'l/Jr series, Dual q-Hahn polynomials, 181 138-145 Dyson's conjecture, 68 Generalized hypergeometric series, 3 Generalized Stieltjes-Wigert polynomials, 214 Generating functions for Elliptic analogue of Bailey's transforma­ orthogonal polynomials, 31, 184, 203, tion formula for a terminating 1O9 se­ 259-281 ries, 307, 323 partitions, 239, 240 Elliptic analogue of Jackson's 87 summa­ q-Bessel functions, 31 tion formula, 307, 321 Gosper's sums, 81, 93, 358 Elliptic balancing conditions (E-balanced), extensions, 81, 82, 93, 358 305, 309, 313, 314 Elliptically balanced series, see Elliptic bal- Hall's formula, 72 ancing conditions Heine's series, 3 Elliptic beta function, 344 Heine's summation formula, 14 Elliptic en Jackson sum, 347 Heine's transformation formulas, 13, 14, Elliptic Dn Jackson sum, 348 359 Elliptic gamma function, 311, 338 Hermite polynomials, 4 Elliptic hypergeometric series, 302, 305 q-analogues of, 31 Elliptic integrals, 344, 345, 347 Hypergeometric functions, 5 Elliptic numbers, 17 Elliptic shifted factorials, 304, 312 Identities involving q-shifted factorials, 6, Erdelyi's formula, 174 24, 25, 351-353 Euler's identity, 62 Indefinite summation formulas, 80-83, 322, Euler's integral representation, 24 324-329, 342-344 Euler's partition identity, 240 Infinite products, 6, 20-23, 352, 353 Euler's transformation formulas, 13, 86 sums of, 61, 150-152, 304-310, 312-349 Subject index 425

Integral representations of Multibasic hypergeometric series, 95-97, 2Fl series, 24 105-107, 112 21 series, 24 Multibasic summation and transformation associated Askey-Wilson polynomials, 255 formulas for theta hypergeometric series, 325-331 very-well-poised 81>7 series, 157, 158 Multinomial coefficients, 68 very-well-poised 101>9 series, 159-161 q-analogue of, 25, 68 Integral representations for q-Appell se­ Multivariable sums, 331-336, 345-350 ries, 284, 286, 288, 289, 294 Inversion in the base of a basic series, 25 Nearly poised series, 38, 39 theta hypergeometric series, 337 of the first and second kinds, 38, 39 Nome modular parameter, 304 Jackson's product formula, 93 N onnegativity of Jackson's summation formulas, 43, 355, 3F2 series, 232, 237 356 4F3 series, 239 Jacobi polynomials, 2 51>4 series, 237 Jacobi's triple product identity, 15, 357 61>5 series, 238, 239 Jain's transformation formula, 60 71>6 series, 238 Jain and Verma's transformation formu­ connection coefficients, 197, 215 las, 63, 78 linearization coefficients, 226, 227, 257 Jump function, 175 Poisson and other kernels, 229-232, 257

Kampe de Feriet series, 284 Karlsson-Minton summation formulas, 18, Orthogonality relations for 19 affine q-Krawtchouk polynomials, 201 q-extensions of, 19, 20 Al-Salam-Carlitz polynomials, 209 Kummer's formula, 18 Askey-Wilson polynomials, 190, 191, q-analogue of, 18, 354 206-208 big q-Jacobi polynomials, 182 Laguerre polynomials, 4 continuous q-Hermite polynomials, 31, Legendre polynomials, 2 204 Level basic series, 111 continuous q-Jacobi polynomials, 191, 192 Linearization formula for Cn(x;,Blq), 226 continuous q-ultraspherical polynomials, inverse of, 249 31, 184-187 Linearization formula for discrete q-Hermite polynomials, 209 continuous q-Jacobi polynomials, 253 dual q-Hahn polynomials, 181 associated q-ultraspherical polynomials, little q-Jacobi polynomials, 182 255 q-Charlier polynomials, 202 Little q-Jacobi polynomials, 32, 181, 182, q-Hahn polynomials, 180 245 q-Krawtchouk polynomials, 201, 203 Little q-Legendre function, 253 q-Laguerre polynomials, 210 q-Meixner polynomials, 202 Milne's fundamental theorem, 331 q-Racah polynomials, 180 Rosengren's elliptic extension of, 331, 332 sieved orthogonal polynomials, 204, 205 Modular balancing conditions (M-balanced), Orthogonal system of polynomials, 175 315-317 in several variables, 213, 255, 258 Modular group, 315 Modular hypergeometric function, 315 Modular invariant, 316 Partial sums, 58, 63, 80-83, 106, 358 Modular parameters, 304, 312 Partitions, 239-242 Modular series, 316 Pentagonal numbers, 241 Modular symmetry relations, 317 Pfaff-Saalschiitz summation formula, 17 Moments, 175 q-analogue of, 17, 355 426 Subject index

Poisson kernels for q-difference equations for Askey-Wilson continuous q-ultraspherical poly­ polynomials, 199 nomials, 227-229 q-difference operators, 32 q-Racah polynomials, 229-232 forward and backward, 11 Product formulas for symmetric, 11 2Fl series, 103, 219-221, 232, 236 q-differential equations, 27 Askey-Wilson polynomials, 222 q-Dixon sums, 44, 58, 355 balanced 43 polynomials, 218-223 q-Dougall sum, 44, 356 basic hypergeometric series, 103-105, q-exponential functions, 11, 12,33,34, 111, 232-236, 251, 361 354 big q-Jacobi polynomials, 245 q-extension, 7 continuous q-Jacobi polynomials, 253 q-gamma functions, 20-22, 29, 353 continuous q-ultraspherical polynomials, q-Gauss sum, 14, 354 223, 227, 236, 249 q-generalization, 7 Gegenbauer polynomials, 225 q-Hahn polynomials, 180, 181 hypergeometric series, 103,219-221,232, q-integral representations of 236 2(1)1 series, 24 Jacobi polynomials, 220, 222 Askey-Wilson polynomials, 207 little q-Jacobi polynomials, 210, 245 continuous q-ultraspherical polynomials, q-Hahn polynomials, 221, 246 185 q-Racah polynomials, 221, 246, 247 very-well-poised 87 series, 52 Projection formulas, 195 q-integrals, 23, 24, 52-55, 57, 58, 65, 76, 149, 156-158, 162, 163, 166, 171, 172, 183, 185, 203, 207 q-Kampe de Feriet series, 284 q-analogue of Fl (a;b,b' ;c;x,y), 283-285, q-Karlsson-Minton sums, 19, 20, 357 294-296 q-Krawtchouk polynomials, 201, 202 q-analogue of F4 (a,b;c,c';X(1-y),Y(1-X)), 219, 220, 283, 290-294 q-Kummer sum, 18,354 q-Appell functions, 282, 283 q-Lagrange inversion theorem, 107 q-Bessel functions, 30, 104 q-Laguerre polynomials, 210 q-beta function, 22, 23 q-Lauricella function, 300, 301 q-beta integrals of q-Leibniz formula, 27 Andrews and Askey, 53 q-Mehler's formula, 275 Askey, 170, 171 q-Meixner polynomials, 202 Askey and Roy, 129 q-multinomial coefficients, 25, 68 Askey and Wilson, 154, 155 q-multinomial theorem, 25 Gasper, 130 q-number,7 Nassrallah and Rahman, 158,295 q-number factorial, 7 Ramanujan, 172 q-number shifted factorial, 7 Wilson, 132 q-quadratic lattice, 12, 294 q-binomial theorem, 8, 354 q-Racah polynomials, 59, 197-180 q-binomial coefficients, 24, 353 q-Saalschiitz sum, 17, 355 q-Cayley-Orr type formulas, 105, 112 q-series, 4, 8, 282 q-Charlier polynomials, 202 q-shifted factorial, 3, 351 q-Clausen formulas, 232-235, 361 q-sine functions, 28 q-contiguous relations for q-trigonometric functions, 28, 212 q-ultraspherical function of the second kind, 43 polynomials, 200 Heine's series, 27 211 q-Vandermonde sum, 14,354 q-Cosine functions, 28 q-Watson sum, 61, 355 q-deformation, 7 q-Whipple sum, 61, 355 q-derivative operators, 27, 197, 208, 251 q,p-binomial coefficient, 311 q-difference equations, 32, 199 Subject index 427

q,p -shifted factorial, 304 Split-poised series, 106 Quadratic elliptic transformation formula, Squares of

341 2 F1 series, 232 Quadratic summation and transformation 2 3 transformation formula, 49, 360 quartic summation formulas, 94, 109 Sears' nonterminating extension of the Ramanujan's 1 'l/J1 sum, 138, 357 q-Saalschiitz sum, 51, 356 Sears' 3

Transformation formulas (selected): very-well-poised 2r'