Hypergeometric Orthogonal Polynomials and Their q-Analogues (eBook)

Foreword 6
Preface 12
Contents 16
Definitions and Miscellaneous Formulas 21
Orthogonal Polynomials 21
The Gamma and Beta Function 23
The Shifted Factorial and Binomial Coefficients 24
Hypergeometric Functions 25
The Binomial Theorem and Other Summation Formulas 27
Some Integrals 28
Transformation Formulas 30
The q-Shifted Factorial 31
The q-Gamma Function and q-Binomial Coefficients 33
Basic Hypergeometric Functions 35
The q-Binomial Theorem and Other Summation Formulas 36
More Integrals 38
Transformation Formulas 39
Some q-Analogues of Special Functions 42
The q-Derivative and q-Integral 44
Shift Operators and Rodrigues-Type Formulas 46
Polynomial Solutions of Eigenvalue Problems 48
Hahn's q-Operator 48
Eigenvalue Problems 49
The Regularity Condition 52
Determination of the Polynomial Solutions 54
First Approach 54
Second Approach 56
Existence of a Three-Term Recurrence Relation 59
Explicit Form of the Three-Term Recurrence Relation 64
Orthogonality of the Polynomial Solutions 71
Favard's Theorem 71
Orthogonality and the Self-Adjoint Operator Equation 73
The Jackson-Thomae q-Integral 77
Rodrigues Formulas 80
Duality 89
Part I Classical Orthogonal Polynomials 94
Orthogonal Polynomial Solutions of Differential Equations 95
Continuous Classical Orthogonal Polynomials 95
Polynomial Solutions of Differential Equations 95
Classification of the Positive-Definite Orthogonal Polynomial Solutions 96
Properties of the Positive-Definite Orthogonal Polynomial Solutions 99
Orthogonal Polynomial Solutions of Real Difference Equations 110
Discrete Classical Orthogonal Polynomials I 110
Polynomial Solutions of Real Difference Equations 110
Classification of the Positive-Definite Orthogonal Polynomial Solutions 112
Properties of the Positive-Definite Orthogonal Polynomial Solutions 116
Orthogonal Polynomial Solutions of Complex Difference Equations 137
Discrete Classical Orthogonal Polynomials II 137
Real Polynomial Solutions of Complex Difference Equations 137
Classification of the Real Positive-Definite Orthogonal Polynomial Solutions 144
Properties of the Positive-Definite Orthogonal Polynomial Solutions 145
Orthogonal Polynomial Solutions in x(x+u) of Real Difference Equations 154
Discrete Classical Orthogonal Polynomials III 154
Motivation for Polynomials in x(x+u) Through Duality 154
Difference Equations Having Real Polynomial Solutions with Argument x(x+u) 155
The Hypergeometric Representation 157
The Three-Term Recurrence Relation 161
Classification of the Positive-Definite Orthogonal Polynomial Solutions 163
The Self-Adjoint Difference Equation 169
Orthogonality Relations for Dual Hahn Polynomials 171
Orthogonality Relations for Racah Polynomials 175
Orthogonal Polynomial Solutions in z(z+u) of Complex Difference Equations 184
Discrete Classical Orthogonal Polynomials IV 184
Real Polynomial Solutions of Complex Difference Equations 184
Orthogonality Relations for Continuous Dual Hahn Polynomials 186
Orthogonality Relations for Wilson Polynomials 190
Askey Scheme of Hypergeometric Orthogonal Polynomials 195
Hypergeometric Orthogonal Polynomials 196
Wilson 196
Racah 201
Continuous Dual Hahn 207
Continuous Hahn 211
Hahn 215
Dual Hahn 219
Meixner-Pollaczek 224
Jacobi 227
Gegenbauer / Ultraspherical 233
Chebyshev 236
Legendre / Spherical 240
Pseudo Jacobi 242
Meixner 245
Krawtchouk 248
Laguerre 252
Bessel 255
Charlier 258
Hermite 261
Part II Classical q-Orthogonal Polynomials 265
Orthogonal Polynomial Solutions of q-Difference Equations 266
Classical q-Orthogonal Polynomials I 266
Polynomial Solutions of q-Difference Equations 266
The Basic Hypergeometric Representation 267
The Three-Term Recurrence Relation 275
Classification of the Positive-Definite Orthogonal Polynomial Solutions 276
Solutions of the q-Pearson Equation 302
Orthogonality Relations 316
Orthogonal Polynomial Solutions in q-x of q-Difference Equations 332
Classical q-Orthogonal Polynomials II 332
Polynomial Solutions in q-x of q-Difference Equations 332
The Basic Hypergeometric Representation 333
The Three-Term Recurrence Relation 337
Orthogonality and the Self-Adjoint Operator Equation 338
Rodrigues Formulas 342
Classification of the Positive-Definite Orthogonal Polynomial Solutions 343
Solutions of the q-1-Pearson Equation 353
Orthogonality Relations 363
Orthogonal Polynomial Solutions in q-x+uqx of Real q-Difference Equations 377
Classical q-Orthogonal Polynomials III 377
Motivation for Polynomials in q-x+uqx Through Duality 377
Difference Equations Having Real Polynomial Solutions with Argument q-x+uqx 378
The Basic Hypergeometric Representation 381
The Three-Term Recurrence Relation 385
Classification of the Positive-Definite Orthogonal Polynomial Solutions 387
Solutions of the q-Pearson Equation 391
Orthogonality Relations 397
Orthogonal Polynomial Solutions in az+uza of Complex q-Difference Equations 403
Classical q-Orthogonal Polynomials IV 403
Real Polynomial Solutions in az+uza with uR{0} and a,zC{0} 403
Classification of the Positive-Definite Orthogonal Polynomial Solutions 406
Solutions of the q-Pearson Equation 409
Orthogonality Relations 415
Scheme of Basic Hypergeometric Orthogonal Polynomials 420
Basic Hypergeometric Orthogonal Polynomials 421
Askey-Wilson 421
q-Racah 428
Continuous Dual q-Hahn 435
Continuous q-Hahn 439
Big q-Jacobi 444
Big q-Legendre 449
q-Hahn 451
Dual q-Hahn 456
Al-Salam-Chihara 461
q-Meixner-Pollaczek 466
Continuous q-Jacobi 469
Continuous q-Ultraspherical / Rogers 475
Continuous q-Legendre 481
Big q-Laguerre 484
Little q-Jacobi 488
Little q-Legendre 492
q-Meixner 494
Quantum q-Krawtchouk 499
q-Krawtchouk 502
Affine q-Krawtchouk 507
Dual q-Krawtchouk 511
Continuous Big q-Hermite 515
Continuous q-Laguerre 520
Little q-Laguerre / Wall 524
q-Laguerre 528
q-Bessel 532
q-Charlier 536
Al-Salam-Carlitz I 540
Al-Salam-Carlitz II 543
Continuous q-Hermite 546
Stieltjes-Wigert 550
Discrete q-Hermite I 553
Discrete q-Hermite II 556
Bibliography 559
Index 581