Book:T.S. Chihara/An Introduction to Orthogonal Polynomials

T.S. Chihara: An Introduction to Orthogonal Polynomials

Published $1978$, Gordon and Breach.

Contents

PREFACE

CHAPTER I. ELEMENTARY THEORY OF ORTHOGONAL POLYNOMIALS

1 Introduction

$(1.1)$

$(1.2)$

$(1.3)$

$(1.4)$

2 The moment functional and orthogonality

3 Existence of OPS

4 The fundamental recurrence formula

5 Zeros

6 Gauss quadrature

7 Kernel polynomials

8 Symmetric moment functionals

9 Certain related recurrence relations

CHAPTER II. THE REPRESENTATION THEOREM AND DISTRIBUTION FUNCTIONS

1 Introduction

2 Some preliminary theorems

3 The representation theorem

4 Spectral points and zeros of orthogonal polynomials

5 Determinacy of $\mathscr{L}$ in the bounded case

6 The classical moment problems

CHAPTER III. CONTINUED FRACTIONS AND CHAIN SEQUENCES

1 Basic concepts

2 The fundamental recurrence formulas

3 A convergence theorem

4 Jacobi fractions and orthogonal polynomials

5 Chain sequences

6 Additional results on chain sequences

CHAPTER IV. THE RECURRENCE FORMULA AND PROPERTIES OF ORTHOGONAL POLYNOMIALS

1 Introduction

2 Chain sequence and orthogonal polynomials

3 Some spectral analysis

4 OPS whose zeros are dense in intervals

5 Preliminaries to Krein's theorem

6 Krein's theorem

CHAPTER V. SPECIAL FUNCTIONS

1 General remarks

2 The classical orthogonal polynomials

3 The Hahn class of orthogonal polynomials

4 The Meixner class of orthogonal polynomials

5 Other classes of orthogonal polynomials

CHAPTER VI. SOME SPECIFIC SYSTEMS OF ORTHOGONAL POLYNOMIALS

1 The Charlier polynomials

2 The Stieltjes-Wigert polynomials

3 The Meixner polynomials

4 The Bessel polynomials

5 The Pollaczek polynomials

6 Modified Lommel polynomials

7 Tricomi-Carlitz polynomials

8 OPS related to Bernoulli numbers

9 OPS related to Jacobi elliptic functions

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

11 Wall polynomials

12 Associated Legendre polynomials

13 Miscellaneous OPS

NOTES

APPENDIX TABLE OF RECURRENCE FORMULAS

LIST OF FREQUENTLY USED SYMBOLS

BIBLIOGRAPHY

INDEX