Lectures on Orthogonal Polynomials and Special Functions

Contributors page x
Preface xi
1 Exceptional Orthogonal Polynomials viaKrall Discrete Poly-
nomials Antonio J. Durán 1
1.1 Background on classical and classical discrete
polynomials 4
1.1.1 Weights on the real line 4
1.1.2 The three-term recurrence relation 5
1.1.3 The classical orthogonal polynomial families 6
1.1.4 Second-order differential operator 10
1.1.5 Characterizations of the classical families of
orthogonal polynomials 12
1.1.6 The classical families and the basic quantum
models 13
1.1.7 The classical discrete families 15
1.2 The Askey tableau. Krall and exceptional polynomi-
als. Darboux Transforms 18
1.2.1 The Askey tableau 18
1.2.2 Krall and exceptional polynomials 21
1.2.3 Krall polynomials 23
1.2.4 Darboux transforms 25
1.3 D-operators 30
1.3.1 D-operators 30
1.3.2 D-operators on the stage 32
1.3.3 D-operators of type 2 37
1.4 Constructing Krall polynomials by using D-operators 38
1.4.1 Back to the orthogonality 39
1.4.2 Krall–Laguerre polynomials 40
1.4.3 Krall discrete polynomials 42
1.5 First expansion of the Askey tableau. Exceptional
polynomials: discrete case 48
1.5.1 Comparing the Krall continuous and discrete
cases (roughly speaking): Darboux transform 48
1.5.2 First expansion of the Askey tableau 50
1.5.3 Exceptional polynomials 53
1.5.4 Constructing exceptional discrete polynomi-
als by using duality 56
1.6 Exceptional polynomials: continuous case. Second
expansion of the Askey tableau 60
1.6.1 Exceptional Charlier polynomials: admissi-
bility 60
1.6.2 Exceptional Hermite polynomials by passing
to the limit 62
1.6.3 Exceptional Meixner and Laguerre polyno-
mials 64
1.6.4 Second expansion of the Askey tableau 68
1.7 Appendix: Symmetries for Wronskian type deter-
minants whose entries are classical and classical
discrete orthogonal polynomials 68
References 70
2 A Brief Review of q-Series Mourad E.H. Ismail 76
2.1 Introduction 76
2.2 Notation and q-operators 77
2.3 q-Taylor series 81
2.4 Summation theorems 85
2.5 Transformations 90
2.6 q-Hermite polynomials 94
2.7 The Askey–Wilson polynomials 101
2.8 Ladder operators and Rodrigues formulas 106
2.9 Identities and summation theorems 113
2.10 Expansions 115
2.11 Askey–Wilson expansions 119
2.12 A q-exponential function 125
References 128
3 ApplicationsofSpectral TheorytoSpecialFunctions Erik
Koelink 131
3.1 Introduction 133
3.2 Three-term recurrences in ? 2 (Z) 139
3.2.1 Exercises 147
3.3 Three-term recurrence relations and orthogonal
polynomials 149
3.3.1 Orthogonal polynomials 149
3.3.2 Jacobi operators 152
3.3.3 Moment problems 154
3.3.4 Exercises 154
3.4 Matrix-valued orthogonal polynomials 155
3.4.1 Matrix-valued measures and related polyno-
mials 156
3.4.2 The corresponding Jacobi operator 163
3.4.3 The resolvent operator 167
3.4.4 The spectral measure 171
3.4.5 Exercises 173
3.5 More on matrix weights, matrix-valued orthogonal
polynomials and Jacobi operators 174
3.5.1 Matrix weights 174
3.5.2 Matrix-valued orthogonal polynomials 176
3.5.3 Link to case of ? 2 (Z) 178
3.5.4 Reducibility 179
3.5.5 Exercises 180
3.6 The J-matrix method 180
3.6.1 Schrödinger equation with the Morse potential 182
3.6.2 A tridiagonal differential operator 186
3.6.3 J-matrix method with matrix-valued orthog-
onal polynomials 190
3.6.4 Exercises 198
3.7 Appendix: The spectral theorem 199
3.7.1 Hilbert spaces and operators 199
3.7.2 Hilbert C ∗ -modules 201
3.7.3 Unbounded operators 202
3.7.4 The spectral theorem for bounded self-
adjoint operators 202
3.7.5 Unbounded self-adjoint operators 204
3.7.6 The spectral theorem for unbounded self-
adjoint operators 205
3.8 Hints and answers for selected exercises 206
References 207
4 Elliptic Hypergeometric Functions Hjalmar Rosengren 213
4.1 Elliptic functions 216
4.1.1 Definitions 216
4.1.2 Theta functions 217
4.1.3 Factorization of elliptic functions 220
4.1.4 The three-term identity 222
4.1.5 Even elliptic functions 223
4.1.6 Interpolation and partial fractions 226
4.1.7 Modularity and elliptic curves 229
4.1.8 Comparison with classical notation 233
4.2 Elliptic hypergeometric functions 235
4.2.1 Three levels of hypergeometry 235
4.2.2 Elliptic hypergeometric sums 237
4.2.3 The Frenkel–Turaev sum 239
4.2.4 Well-poised and very well-poised sums 243
4.2.5 The sum 12 V 11 245
4.2.6 Biorthogonal rational functions 248
4.2.7 A quadratic summation 250
4.2.8 An elliptic Minton summation 253
4.2.9 The elliptic gamma function 255
4.2.10 Elliptic hypergeometric integrals 256
4.2.11 Spiridonov’s elliptic beta integral 258
4.3 Solvable lattice models 262
4.3.1 Solid-on-solid models 262
4.3.2 The Yang–Baxter equation 265
4.3.3 The R-operator 266
4.3.4 The elliptic SOS model 268
4.3.5 Fusion and elliptic hypergeometry 270
References 276
5 Combinatorics of Orthogonal Polynomials and their Mo-
ments Jiang Zeng 280
5.1 Introduction 280
5.2 General and combinatorial theories of formal OPS 283
5.2.1 Formal theory of orthogonal polynomials 283
5.2.2 The Flajolet–Viennot combinatorial approach 291
5.3 Combinatorics of generating functions 294
5.3.1 Exponential formula and Foata’s approach 294
5.3.2 Models of orthogonal Sheffer polynomials 297
5.3.3 MacMahon’s Master Theorem and a Mehler-
type formula 299
5.4 Moments of orthogonal Sheffer polynomials 304
5.4.1 Combinatorics of the moments 304
5.4.2 Linearization coefficients of Sheffer polyno-
mials 309
5.5 Combinatorics of some q-polynomials 314
5.5.1 Al-Salam–Chihara polynomials 314
5.5.2 Moments of continuous q-Hermite,
q-Charlier and q-Laguerre polynomials 315
5.5.3 Linearization coefficients of continuous q-
Hermite, q-Charlier and
q-Laguerre polynomials 318
5.5.4 A curious q-analogue of Hermite polynomials 323
5.5.5 Combinatorics of continued fractions and
γ -positivity 328
5.6 Some open problems 330
References 331