Classical and Quantum Orthogonal Polynomials in One Variable

Foreword page xi
Preface xvi
1 Preliminaries 1
1.1 Hermitian Matrices and Quadratic Forms 1
1.2 Some Real and Complex Analysis 3
1.3 Some Special Functions 8
1.4 Summation Theorems and Transformations 12
2 Orthogonal Polynomials 16
2.1 Construction of Orthogonal Polynomials 16
2.2 Recurrence Relations 22
2.3 Numerator Polynomials 26
2.4 Quadrature Formulas 28
2.5 The Spectral Theorem 30
2.6 Continued Fractions 35
2.7 Modifications of Measures: Christoffel and Uvarov 37
2.8 Modifications of Measures: Toda 41
2.9 Modification by Adding Finite Discrete Parts 43
2.10 Modifications of Recursion Coefficients 45
2.11 Dual Systems 47
3 Differential Equations, Discriminants and Electrostatics 52
3.1 Preliminaries 52
3.2 Differential Equations 53
3.3 Applications 63
3.4 Discriminants 67
3.5 An Electrostatic Equilibrium Problem 70
3.6 Functions of the Second Kind 73
3.7 Lie Algebras 76
4 Jacobi Polynomials 80
4.1 Orthogonality 80
4.2 Differential and Recursion Formulas 82
4.3 Generating Functions 88
4.4 Functions of the Second Kind 93
4.5 Ultraspherical Polynomials 94
4.6 Laguerre and Hermite Polynomials 98
4.7 Multilinear Generating Functions 106
4.8 Asymptotics and Expansions 114
4.9 Relative Extrema of Classical Polynomials 120
4.10 The Bessel Polynomials 123
5 Some Inverse Problems 133
5.1 Ultraspherical Polynomials 133
5.2 Birth and Death Processes 136
5.3 The Hadamard Integral 141
5.4 Pollaczek Polynomials 147
5.5 A Generalization 151
5.6 Associated Laguerre and Hermite Polynomials 158
5.7 Associated Jacobi Polynomials 162
5.8 The J-Matrix Method 168
5.9 The Meixner–Pollaczek Polynomials 171
6 Discrete Orthogonal Polynomials 174
6.1 Meixner Polynomials 174
6.2 Hahn, Dual Hahn, and Krawtchouk Polynomials 177
6.3 Difference Equations 186
6.4 Discrete Discriminants 190
6.5 Lommel Polynomials 194
6.6 An Inverse Operator 199
7 Zeros and Inequalities 203
7.1 A Theorem of Markov 203
7.2 Chain Sequences 205
7.3 The Hellmann–Feynman Theorem 211
7.4 Extreme Zeros of Orthogonal Polynomials 219
7.5 Concluding Remarks 221
8 Polynomials Orthogonal on the Unit Circle 222
8.1 Elementary Properties 222
8.2 Recurrence Relations 225
8.3 Differential Equations 231
8.4 Functional Equations and Zeros 240
8.5 Limit Theorems 245
8.6 Modifications of Measures 247
9 Linearization, Connections and Integral Representations 254
9.1 Connection Coefficients 256
9.2 The Ultraspherical Polynomials and Watson’s Theorem 262
9.3 Linearization and Power Series Coefficients 264
9.4 Linearization of Products and Enumeration 269
9.5 Representations for Jacobi Polynomials 274
9.6 Addition and Product Formulas 277
9.7 The Askey–Gasper Inequality 281
10 The Sheffer Classification 283
10.1 Preliminaries 283
10.2 Delta Operators 286
10.3 Algebraic Theory 288
11 q-Series Preliminaries 294
11.1 Introduction 294
11.2 Orthogonal Polynomials 294
11.3 The Bootstrap Method 295
11.4 q-Differences 297
12 q-Summation Theorems 300
12.1 Basic Definitions 300
12.2 Expansion Theorems 303
12.3 Bilateral Series 308
12.4 Transformations 311
12.5 Additional Transformations 314
12.6 Theta Functions 316
13 Some q-Orthogonal Polynomials 319
13.1 q-Hermite Polynomials 320
13.2 q-Ultraspherical Polynomials 327
13.3 Linearization and Connection Coefficients 331
13.4 Asymptotics 335
13.5 Application: The Rogers–Ramanujan Identities 336
13.6 Related Orthogonal Polynomials 341
13.7 Three Systems of q-Orthogonal Polynomials 345
14 Exponential and q-Bessel Functions 352
14.1 Definitions 352
14.2 Generating Functions 357
14.3 Addition Formulas 359
14.4 q-Analogues of Lommel and Bessel Polynomials 360
14.5 A Class of Orthogonal Functions 364
14.6 An Operator Calculus 366
14.7 Polynomials of q-Binomial Type 372
14.8 Another q-Umbral Calculus 376
15 The Askey–Wilson Polynomials 378
15.1 The Al-Salam–Chihara Polynomials 378
15.2 The Askey–Wilson Polynomials 382
15.3 Remarks 387
15.4 Asymptotics 389
15.5 Continuous q-Jacobi Polynomials and Discriminants 391
15.6 q-Racah Polynomials 396
15.7 q-Integral Representations 400
15.8 Linear and Multilinear Generating Functions 405
15.9 Associated q-Ultraspherical Polynomials 411
15.10 Two Systems of Orthogonal Polynomials 416
16 The Askey–Wilson Operators 426
16.1 Basic Results 426
16.2 A q-Sturm–Liouville Operator 433
16.3 The Askey–Wilson Polynomials 437
16.4 Connection Coefficients 443
16.5 Bethe Ansatz Equations of XXZ Model 446
17 q-Hermite Polynomials on the Unit Circle 455
17.1 The Rogers–Szeg˝ o Polynomials 455
17.2 Generalizations 460
17.3 q-Difference Equations 464
18 Discrete q-Orthogonal Polynomials 469
18.1 Discrete Sturm–Liouville Problems 469
18.2 The Al-Salam–Carlitz Polynomials 470
18.3 The Al-Salam–Carlitz Moment Problem 476
18.4 q-Jacobi Polynomials 477
18.5 q-Hahn Polynomials 484
18.6 q-Differences and Quantized Discriminants 486
18.7 A Family of Biorthogonal Rational Functions 488
19 Fractional and q-Fractional Calculus 491
19.1 The Riemann–Liouville Operators 491
19.2 Bilinear Formulas 495
19.3 Examples 496
19.4 q-Fractional Calculus 501
19.5 Some Integral Operators 504
20 Polynomial Solutions to Functional Equations 509
20.1 Bochner’s Theorem 509
20.2 Difference and q-Difference Equations 514
20.3 Equations in the Askey–Wilson Operators 516
20.4 Leonard Pairs and the q-Racah Polynomials 518
20.5 Characterization Theorems 525
21 Some Indeterminate Moment Problems 530
21.1 The Hamburger Moment Problem 530
21.2 A System of Orthogonal Polynomials 534
21.3 Generating Functions 537
21.4 The Nevanlinna Matrix 542
21.5 Some Orthogonality Measures 544
21.6 Ladder Operators 547
21.7 Zeros 550
21.8 The q-Laguerre Moment Problem 553
21.9 Other Indeterminate Moment Problems 563
21.10 Some Biorthogonal Rational Functions 572
22 The Riemann-Hilbert Problem for Orthogonal Polynomials 578
22.1 The Cauchy Transform 578
22.2 The Fokas–Its–Kitaev Boundary Value Problem 581
22.2.1 The three-term recurrence relation 584
22.3 Hermite Polynomials 586
22.3.1 A Differential Equation 586
22.4 Laguerre Polynomials 589
22.4.1 Three-term recurrence relation 591
22.4.2 A differential equation 592
22.5 Jacobi Polynomials 596
22.5.1 Differential equation 597
22.6 Asymptotic Behavior 601
22.7 Discrete Orthogonal Polynomials 603
22.8 Exponential Weights 604
23 Multiple Orthogonal Polynomials 607
23.1 Type I and II Multiple Orthogonal Polynomials 608
23.1.1 Angelesco systems 610
23.1.2 AT systems 611
23.1.3 Biorthogonality 613
23.1.4 Recurrence relations 614
23.2 Hermite–Padé Approximation 621
23.3 Multiple Jacobi Polynomials 622
23.3.1 Jacobi–Angelesco polynomials 622
23.3.2 Jacobi–Pi˜ neiro polynomials 626
23.4 Multiple Laguerre Polynomials 628
23.4.1 Multiple Laguerre polynomials of the first kind 628
23.4.2 Multiple Laguerre polynomials of the second kind 629
23.5 Multiple Hermite Polynomials 630
23.5.1 Random matrices with external source 631
23.6 Discrete Multiple Orthogonal Polynomials 632
23.6.1 Multiple Charlier polynomials 632
23.6.2 Multiple Meixner polynomials 632
23.6.3 Multiple Krawtchouk polynomials 634
23.6.4 Multiple Hahn polynomials 634
23.6.5 Multiple little q-Jacobi polynomials 635
23.7 Modified Bessel Function Weights 636
23.7.1 Modified Bessel functions 637
23.8 The Riemann–Hilbert Problem for Multiple Orthogonal Poly-
nomials 639
23.8.1 Recurrence relation 644
23.8.2 Differential equation for multiple Hermite polynomials 645
24 Research Problems 648
24.1 Multiple Orthogonal Polynomials 648
24.2 A Class of Orthogonal Functions 649
24.3 Positivity 649
24.4 Asymptotics and Moment Problems 650
24.5 Functional Equations and Lie Algebras 652
24.6 Rogers–Ramanujan Identities 653
24.7 Characterization Theorems 654
24.8 Special Systems of Orthogonal Polynomials 658
24.9 Zeros of Orthogonal Polynomials 661
Bibliography 663
Index 699
Author index 705