Springer Monographs in Mathematics

Hypergeometric Orthogonal Polynomials and Their q-Analogues

Bearbeitet von Roelof Koekoek, Peter A Lesky, René F Swarttouw, Tom H Koornwinder

1. Auflage 2010. Buch. xix, 578 S. Hardcover ISBN 978 3 642 05013 8 Format (B x L): 15,5 x 23,5 cm Gewicht: 2230 g

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Foreword ...... v

Preface ...... xi

1 Definitions and Miscellaneous Formulas ...... 1 1.1 Orthogonal Polynomials ...... 1 1.2 The Gamma and Beta Function ...... 3 1.3 The Shifted Factorial and Binomial Coefficients ...... 4 1.4 Hypergeometric Functions ...... 5 1.5 The Binomial Theorem and Other Summation Formulas ...... 7 1.6 SomeIntegrals...... 8 1.7 TransformationFormulas...... 10 1.8 The q-ShiftedFactorial...... 11 1.9 The q-Gamma Function and q-Binomial Coefficients ...... 13 1.10 Basic Hypergeometric Functions ...... 15 1.11 The q-Binomial Theorem and Other Summation Formulas ...... 16 1.12 MoreIntegrals...... 18 1.13 TransformationFormulas...... 19 1.14 Some q-Analogues of Special Functions ...... 22 1.15 The q-Derivative and q-Integral...... 24 1.16 Shift Operators and Rodrigues-Type Formulas ...... 26

2 Polynomial Solutions of Eigenvalue Problems ...... 29 2.1 Hahn’s q-Operator...... 29 2.2 EigenvalueProblems...... 30 2.3 The Regularity Condition ...... 33 2.4 Determination of the Polynomial Solutions ...... 35 2.4.1 First Approach ...... 35 2.4.2 Second Approach ...... 37 2.5 Existence of a Three-Term Recurrence Relation ...... 40 2.6 Explicit Form of the Three-Term Recurrence Relation ...... 45

xv xvi Contents

3 Orthogonality of the Polynomial Solutions ...... 53 3.1 Favard’s Theorem ...... 53 3.2 Orthogonality and the Self-Adjoint Operator Equation...... 55 3.3 The Jackson-Thomae q-Integral ...... 59 3.4 Rodrigues Formulas ...... 62 3.5 Duality ...... 71

Part I Classical Orthogonal Polynomials

4 Orthogonal Polynomial Solutions of Differential Equations ...... 79 Continuous Classical Orthogonal Polynomials ...... 79 4.1 Polynomial Solutions of Differential Equations ...... 79 4.2 Classification of the Positive-Definite Orthogonal Polynomial Solutions ...... 80 4.3 Properties of the Positive-Definite Orthogonal Polynomial Solutions 83

5 Orthogonal Polynomial Solutions of Real Difference Equations ...... 95 Discrete Classical Orthogonal Polynomials I...... 95 5.1 Polynomial Solutions of Real Difference Equations ...... 95 5.2 Classification of the Positive-Definite Orthogonal Polynomial Solutions ...... 97 5.3 Properties of the Positive-Definite Orthogonal Polynomial Solutions 101

6 Orthogonal Polynomial Solutions of Complex Difference Equations ... 123 Discrete Classical Orthogonal Polynomials II ...... 123 6.1 Real Polynomial Solutions of Complex Difference Equations ...... 123 6.2 Classification of the Real Positive-Definite Orthogonal Polynomial Solutions ...... 130 6.3 Properties of the Positive-Definite Orthogonal Polynomial Solutions 131

7 Orthogonal Polynomial Solutions in x(x + u) of Real Difference Equations ...... 141 Discrete Classical Orthogonal Polynomials III ...... 141 7.1 Motivation for Polynomials in x(x + u) Through Duality ...... 141 7.2 Difference Equations Having Real Polynomial Solutions with Argument x(x + u)...... 142 7.3 The Hypergeometric Representation ...... 144 7.4 The Three-Term Recurrence Relation ...... 148 7.5 Classification of the Positive-Definite Orthogonal Polynomial Solutions ...... 150 7.6 The Self-Adjoint Difference Equation ...... 156 7.7 Orthogonality Relations for Dual Hahn Polynomials ...... 158 7.8 Orthogonality Relations for Racah Polynomials ...... 162 Contents xvii

8 Orthogonal Polynomial Solutions in z(z + u) of Complex Difference Equations ...... 171 Discrete Classical Orthogonal Polynomials IV ...... 171 8.1 Real Polynomial Solutions of Complex Difference Equations ...... 171 8.2 Orthogonality Relations for Continuous Dual Hahn Polynomials . . . 173 8.3 Orthogonality Relations for Wilson Polynomials ...... 177

Askey Scheme of Hypergeometric Orthogonal Polynomials ...... 183

9 Hypergeometric Orthogonal Polynomials ...... 185 9.1 Wilson ...... 185 9.2 Racah ...... 190 9.3 Continuous Dual Hahn ...... 196 9.4 Continuous Hahn ...... 200 9.5 Hahn...... 204 9.6 DualHahn...... 208 9.7 Meixner-Pollaczek ...... 213 9.8 Jacobi ...... 216 9.8.1 Gegenbauer / Ultraspherical ...... 222 9.8.2 Chebyshev ...... 225 9.8.3 Legendre / Spherical ...... 229 9.9 Pseudo Jacobi ...... 231 9.10 Meixner ...... 234 9.11 Krawtchouk ...... 237 9.12 Laguerre ...... 241 9.13 Bessel...... 244 9.14 Charlier...... 247 9.15 Hermite...... 250

Part II Classical q-Orthogonal Polynomials

10 Orthogonal Polynomial Solutions of q-Difference Equations ...... 257 Classical q-Orthogonal Polynomials I ...... 257 10.1 Polynomial Solutions of q-Difference Equations ...... 257 10.2 The Basic Hypergeometric Representation ...... 258 10.3 The Three-Term Recurrence Relation ...... 266 10.4 Classification of the Positive-Definite Orthogonal Polynomial Solutions ...... 267 10.5 Solutions of the q-Pearson Equation ...... 293 10.6 Orthogonality Relations ...... 307

11 Orthogonal Polynomial Solutions in q−x of q-Difference Equations .... 323 Classical q-Orthogonal Polynomials II ...... 323 11.1 Polynomial Solutions in q−x of q-Difference Equations ...... 323 11.2 The Basic Hypergeometric Representation ...... 324 11.3 The Three-Term Recurrence Relation ...... 328 xviii Contents

11.4 Orthogonality and the Self-Adjoint Operator Equation...... 329 11.5 Rodrigues Formulas ...... 333 11.6 Classification of the Positive-Definite Orthogonal Polynomial Solutions ...... 334 11.7 Solutions of the q−1-Pearson Equation ...... 344 11.8 Orthogonality Relations ...... 354

12 Orthogonal Polynomial Solutions in q−x + uqx of Real q-Difference Equations ...... 369 Classical q-Orthogonal Polynomials III ...... 369 12.1 Motivation for Polynomials in q−x + uqx Through Duality ...... 369 12.2 Difference Equations Having Real Polynomial Solutions with Argument q−x + uqx ...... 370 12.3 The Basic Hypergeometric Representation ...... 373 12.4 The Three-Term Recurrence Relation ...... 377 12.5 Classification of the Positive-Definite Orthogonal Polynomial Solutions ...... 379 12.6 Solutions of the q-Pearson Equation ...... 383 12.7 Orthogonality Relations ...... 389

a uz 13 Orthogonal Polynomial Solutions in z + a of Complex q-Difference Equations ...... 395 Classical q-Orthogonal Polynomials IV ...... 395 a uz ∈ R \{ } 13.1 Real Polynomial Solutions in z + a with u 0 and a,z ∈ C \{0} ...... 395 13.2 Classification of the Positive-Definite Orthogonal Polynomial Solutions ...... 398 13.3 Solutions of the q-Pearson Equation ...... 401 13.4 Orthogonality Relations ...... 407

Scheme of Basic Hypergeometric Orthogonal Polynomials ...... 413

14 Basic Hypergeometric Orthogonal Polynomials ...... 415 14.1 Askey-Wilson...... 415 14.2 q-Racah ...... 422 14.3 Continuous Dual q-Hahn...... 429 14.4 Continuous q-Hahn...... 433 14.5 Big q-Jacobi ...... 438 14.5.1 Big q-Legendre ...... 443 14.6 q-Hahn...... 445 14.7 Dual q-Hahn ...... 450 14.8 Al-Salam-Chihara...... 455 14.9 q-Meixner-Pollaczek ...... 460 14.10 Continuous q-Jacobi ...... 463 14.10.1 Continuous q-Ultraspherical / Rogers ...... 469 14.10.2 Continuous q-Legendre ...... 475 Contents xix

14.11 Big q-Laguerre ...... 478 14.12 Little q-Jacobi ...... 482 14.12.1 Little q-Legendre ...... 486 14.13 q-Meixner...... 488 14.14 Quantum q-Krawtchouk ...... 493 14.15 q-Krawtchouk ...... 496 14.16 Affine q-Krawtchouk ...... 501 14.17 Dual q-Krawtchouk ...... 505 14.18 Continuous Big q-Hermite...... 509 14.19 Continuous q-Laguerre ...... 514 14.20 Little q-Laguerre / Wall ...... 518 14.21 q-Laguerre ...... 522 14.22 q-Bessel...... 526 14.23 q-Charlier...... 530 14.24 Al-Salam-Carlitz I ...... 534 14.25 Al-Salam-Carlitz II ...... 537 14.26 Continuous q-Hermite ...... 540 14.27 Stieltjes-Wigert ...... 544 14.28 Discrete q-HermiteI...... 547 14.29 Discrete q-HermiteII...... 550

Bibliography ...... 553

Index ...... 575