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Crm Proceedings & Lecture Notes Volume 22 C CRM R PROCEEDINGS & M LECTURE NOTES Centre de Recherches Mathématiques Montréal Algebraic Methods and q-Special Functions Jan Felipe van Diejen Luc Vinet Editors American Mathematical Society Algebraic Methods and q-Special Functions https://doi.org/10.1090/crmp/022 Volume 22 C CRM R PROCEEDINGS & M LECTURE NOTES Centre de Recherches Mathématiques Montréal Algebraic Methods and q-Special Functions Jan Felipe van Diejen Luc Vinet Editors The Centre de Recherches Mathématiques (CRM) of the Université de Montréal was created in 1968 to promote research in pure and applied mathematics and related disciplines. Among its activities are special theme years, summer schools, workshops, postdoctoral programs, and publishing. The CRM is supported by the Université de Montréal, the Province of Québec (FQRNT), and the Natural Sciences and Engineering Research Council of Canada. It is affiliated with the Institut des Sciences Mathématiques (ISM) of Montréal, whose constituent members are Concordia University, McGill University, the Université de Montréal, the Université de Québec à Montréal, and the Ecole Polytechnique. The CRM may be reached on the Web at www.crm.umontreal.ca. American Mathematical Society Providence, Rhode Island USA The production of this volume was supported in part by the Fonds pour la Formation de Chercheurs et l’Aide `alaRecherche(FondsFCAR)andtheNaturalSciencesand Engineering Research Council of Canada (NSERC). 1991 Mathematics Subject Classification.Primary33D45;Secondary05E05,33C50, 33C80, 43A85, 43A90, 81R05, 81R10. Library of Congress Cataloging-in-Publication Data Algebraic methods and q-special functions / Jan Felipe van Diejen, Luc Vinet, editors. p. cm. — (CRM proceedings & lecture notes, ISSN 1065-8580 ; v. 22) Includes bibliographical references. ISBN 0-8218-2026-5 (alk. paper) 1. Orthogonal polynomials Congresses. 2. Functions, Special Congresses. I. Van Diejen, Jan Felipe, 1965– . II. Vinet, Luc. III. Series. QA404.5.A56 1999 515′.55—dc21 99-34734 CIP Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 1999 by the American Mathematical Society. All rights reserved. ⃝ The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines ∞ ⃝ established to ensure permanence and durability. This volume was typeset using -TEX, AMS the American Mathematical Society’s TEXmacrosystem, and submitted to the American Mathematical Society in camera ready form by the Centre de Recherches Math´ematiques. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 04 03 02 01 00 99 Contents WaleedAl-Salam,1926–1996 ix Preface xiii Science Fiction and Macdonald’s Polynomials 1 F. Bergeron and A. M. Garsia On the Expansion of Elliptic Functions and Applications 53 Raouf Chouikha Generalized Hypergeometric Functions—Classification of Identities and Explicit Rational Approximations 59 D. V. Chudnovsky and G. V. Chudnovsky Tensor Products of q-Superalgebras and q-Series Identities. I 93 Won Sang Chung, E. G. Kalnins, and W. Miller, Jr. q-Racah Polynomials for BC Type Root Systems 109 J. F. van Diejen and J. V. Stokman Intertwining Operators of Type BN 119 Charles F. Dunkl Symmetries and Continuous q-Orthogonal Polynomials 135 Roberto Floreanini, Jean LeTourneux, and Luc Vinet Addition Theorems for Spherical Polynomials on a Family of Quantum Spheres 145 Paul G. A. Floris On a q-Analogue of the String Equation and a Generalization of the Classical Orthogonal Polynomials 171 F. Alberto Gr¨unbaum and Luc Haine The q-Bessel Function on a q-Quadratic Grid 183 Mourad E. H. Ismail, David R. Masson, and Sergei K. Suslov Three Statistics on Lattice Paths 201 D. Kim and D. Stanton Quantum Grothendieck Polynomials 215 Anatol N. Kirillov vii viii CONTENTS q-Difference Raising Operators for Macdonald Polynomials and the Integrality of Transition Coefficients 227 A. N. Kirillov and M. Noumi Great Powers of q-Calculus 245 Boris A. Kupershmidt q-Special Functions: Differential-Difference Equations, Roots of Unity, and All That 257 Vyacheslav Spiridonov On Algebras of Creation and Annihilation Operators 267 Aleksander Strasburger Waleed Al-Salam, 1926–1996 Mourad E. H. Ismail Waleed (Abdul) Al-Salam passed away at home on April 14, 1996 in Edmonton, Alberta with many of his family at his side. He died from complications of cancer (multiple myeloma.). He is survived by his wife, Nadhla (n´ee Abdul-Halim), six children: Nabeel, Leila, Ramsey, Selma, Haithem, and Yaseen, and his brother, Khalid. His math- ematical family includes his wife and frequent collaborator, Nadhla, n´ee Abdul- Halim), four mathematical sons Iftikhar Ahmad (M.Sc. 1969), William Allaway (Ph.D. 1972), M. N. Bajaj (M.Sc. 1969) and Mourad Ismail (Ph.D. 1974), and seven mathematical grandchildren. He was born July 15, 1926 in Baghdad, Iraq. His undergraduate education was at the University of California, Berkeley. His graduate education was at Duke University in mathematics, where he wrote a dissertation under Leonard Carlitz. He held positions at the University of Baghdad, Texas Technological University, and the University of Alberta. He visited Cambridge University, the American University of Beirut, and Arizona State University. He spent most of his professional life, 30 years, in the Mathematics Department of the University of Alberta, Edmonton. Waleed’s research work encompassed the broad area of special functions, or- thogonal polynomials and their q-analogues. Two families of orthogonal polynomi- als have been named after Waleed Al-Salam. Waleed’s 1965 paper with Carlitz [4] introduced a one parameter generaliza- tions of the discrete Hermite polynomials, found their orthogonality relation and de- (a) rived many of their structural properties. The polynomials are denoted by Un (x) (a) and Vn (x) and are called the Al-Salam-Carlitz q-polynomials, after Chihara [11]. From a combinatorial perspective they are q-analogues of the Charlier polynomials. (a) The moment problem associated with Vn (x)canbeindeterminateandboththe Hamburger and Stieltjes moment problems were completely solved in 1994 by Berg and Valent [9]. Waleed worked extensively on characterization theorems for classical orthogonal polynomials. His survey paper [1]onthesubjectisveryinformativeandhasbeen influential. His last paper [2]gaveacharacterizationoftheq-Hermite polynomials. A noteworthy paper in this group is Al-Salam’s joint paper with Ted Chihara [5] where they characterized pairs of orthogonal polynomials, rn(x) ,and sn(x) n for which rk(y)sn k(x)areorthogonalinx for infinitely many y’s. This k=0 − ! " ! " class contains the Hermite and Laguerre polynomials and contained one new and # ix xMOURADE.H.ISMAIL interesting set of orthogonal polynomials which became known as the Al-Salam- Chihara polynomials [7]. Their weight function was found shortly after by Askey and Ismail [7]. Askey and Wilson [8]gaveanotherproof.ChiharaandI[12] continued the investigation in [7] of the corresponding moment problem in the case q>1, where the solution may not be unique. As time went by it was realized that the Al-Salam-Chihara polynomials are part of a larger scheme of q-orthogonal polynomials, the Askey scheme, where the Al-Salam-Chihara polynomials play the role of the Laguerre polynomials in the q =1scheme. Waleed worked on q-orthogonal polynomials during the time when the subject was not fashionable. The major developments in the 1970’s and 80’s in the subject of q-orthogonal polynomials in one variable and the current intense activity in q-orthogonal polynomials in several variables owe a very deep debt to the early pioneers of whom Waleed was an important figure. Some of Waleed’s contributions to the subject are included in the extensive treatise on q-series [13], by Gasper and Rahman. Al-Salam, Allaway and Askey [3] introduced the sieved ultraspherical polyno- mials in 1984. This work is now a classic and is nicknamed the AAA paper. The weight function of these polynomials vanishes at isolated points in the interval of orthogonality and later authors managed to use polynomials of a similar type to explicitly get polynomials orthogonal on several intervals. The sieved ultraspherical polynomials were also used to construct polynomials whose spectrum has imbedded discrete masses [14]. A sample of the structural properties of these polynomials is in [10]. Waleed also worked on q-fractional integrals and derivatives, and his early work was used, for example, to construct reproducing kernels for the little q-Jacobi polynomials. He also worked on q-beta integrals and biorthogonal polynomials and functions. In particular our joint paper [6]containsanevaluationofaq-beta integral and introduces a system of rational functions biorthogonal on the unit circle. These functions are unit circle analogues of the Askey-Wilson polynomials, and some refer to them as the Al-Salam-Ismail functions. Waleed also worked on a variety of other topics including Bessel polynomials, Turan type inequalities, generating functions for various polynomials and functions, and other technical aspects of special functions.
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