Table of Contents

Preface vii

Chapter I Elementary Theory of Orthogonal Polynomials 1

1 Introduction 1

2 The moment functional and orthogonality 6

3 Existence of OPS 11

4 The fundamental recurrence formula 18

5 Zeros 26

6 Gauss quadrature 31

7 Kernel polynomials 35

8 Symmetric moment functionals 40

9 Certain related recurrence relations 45

Chapter II The Representation Theorem and Distribution Functions 51

1 Introduction 51

2 Some preliminary theorems 52

3 The representation theorem 56

4 Spectral points and zeros of orthogonal polynomials 59

5 Determinacy of L in the bounded case 63

6 The classical moment problems 71

Chapter III Continued Fractions and Chain Sequences 77

1 Basic concepts 77

2 The fundamental recurrence formulas 80

3 A convergence theorem 82

4 Jacobi fractions and orthogonal polynomials 85

5 Chain sequences 91

6 Additional results on chain sequences 100

Chapter IV The Recurrence Formula and Properties of Orthogonal Polynomials 107

1 Introduction 107

2 Chain sequences and orthogonal polynomials 108

3 Some spectral analysis 113

4 OPS whose zeros are dense in intervals 120

5 Preliminaries to Krein's theorem 128

6 Krein's theorem 133

Chapter V Special Functions 142

1 General remarks 142

2 The classical orthogonal polynomials 142

3 The Hahn class of orthogonal polynomials 159

4 The Meixner class of orthogonal polynomials 163

5 Other classes of orthogonal polynomials 166

Chapter VI Some Specific Systems of Orthogonal Polynomials 170

1 The Charlier polynomials 170

2 The Stieltjes-Wigert polynomials 172

3 The Meixner polynomials 175

4 The Bessel polynomials 181

5 The Pollaczek polynomials 184

6 Modified Lommel polynomials 187

7 Tricomi-Carlitz polynomials 190

8 OPS related to Bernoulli numbers 191

9 OPS related to Jacobi elliptic functions 193

10 The q-polynomials of Al-Salam and Carlitz 195

11 Wall polynomials 198

12 Associated Legendre polynomials 201

13 Miscellaneous OPS 203

Notes 209

Appendix: Table of Recurrence Formulas 215

List of Frequently Used Symbols 223

Bibliography 225

Index 243