Book:Roelof Koekoek/Hypergeometric Orthogonal Polynomials and Their q-Analogues

Roelof Koekoek, Peter A. Lesky and René F. Swarttouw: Hypergeometric Orthogonal Polynomials and Their q-Analogues

Published $2010$, Springer Verlag.

Contents

Foreward

Preface

1. Definitions and Miscellaneous Formulas

1.1 Orthogonal Polynomials

1.2 The Gamma and Beta Function

1.3 The Shifted Factorial and Binomial Coefficients

1.4 Hypergeometric Functions

1.5 The Binomial Theorem and Other Summation Formulas

1.6 Some Integrals

1.7 Transformation Formulas

1.8 The $q$-Shifted Factorial

1.9 The $q$-Gamma Function and $q$-Binomial Coefficients

1.10 Basic Hypergeometric Functions

1.11 The $q$-Binomial Theorem and Other Summation Formulas

1.12 More Integrals

1.13 Transformation Formulas

1.14 Some $q$-Analogues of Special Functions

1.15 The $q$-Derivative and $q$-Integral

1.16 Shift Operators and Rodrigues-Type Formulas

2. Polynomial Solutions of Eigenvalue Problems

2.1 Hahn's $q$-Operator

2.2 Eigenvalue Problems

2.3 The Regularity Condition

2.4 Determination of the Polynomial Solutions

2.4.1 First Approach

2.4.2 Second Approach

2.5 Existence of a Three-Term Recurrence Relation

2.6 Explicit Form of the Three-Term Recurrence Relation

3. Orthogonality of the Polynomial Solutions

3.1 Favard's Theorem

3.2 Orthogonality and the Self-Adjoint Operator Equation

3.3 The Jackson-Thomae $q$-Integral

3.4 Rodrigues Formulas

3.5 Duality

4. Orthogonal Polynomial Solutions of Differential Equations: Continuous Classical Orthogonal Polynomials

4.1 Polynomial Solutions of Differential Equations

4.2 Classification of the Positive-Definite Orthogonal Polynomial Solutions

4.3 Properties of the Positive-Definite Orthogonal Polynomial Solutions

5. Orthogonal Polynomial Solutions of Real Difference Equations: Discrete Classical Orthogonal Polynomials I

5.1 Polynomial Solution of Real Difference Equations

5.2 Classification of the Positive-Definite Orthogonal Polynomial Solutions

5.3 Properties of the Positive-Definite Orthogonal Polynomial Solutions

6. Orthogonal Polynomial Solutions of Complex Difference Equations: Discrete Classical Orthogonal Polynomials II

6.1 Real Polynomial Solutions of Complex Difference Equations

6.2 Classification of the Real Positive-Definite Orthogonal Polynomial Solutions

6.3 Properties of the Positive-Definite Orthogonal Polynomial Solutions

7. Orthogonal Polynomial Solutions in $x(x+u)$ of Real Difference Equations: Discrete Classical Orthogonal Polynomials III

7.1 Motivation for Polynomials in $x(x+u)$ Through Duality

7.2 Difference Equations Having Real Polynomial Solutions with Argument $x(x+u)$

7.3 The Hypergeometric Representation

7.4 The Three-Term Recurrence Relation

7.5 Classification of the Positive-Definite Orthogonal Polynomial Solutions

7.6 The Self-Adjoint Difference Equation

7.7 Orthogonality Relations for Dual Hahn Polynomials

7.8 Orthogonality Relations for Racah Polynomials

8. Orthogonal Polynomial Solutions in $z(z+u)$ of Complex Difference Equations: Discrete Classical Orthogonal Polynomials IV

8.1 Real Polynomial Solutions of Complex Difference Equations

8.2 Orthogonality Relations for Continuous Dual Hahn Polynomials

8.3 Orthogonality Relations for Wilson Polynomials

Askey Scheme of Hypergeometric Orthogonal Polynomials

9 Hypergeometric Orthogonal Polynomials

9.1 Wilson

9.2 Racah

9.3 Continuous Dual Hahn

9.4 Continuous Hahn

9.5 Hahn

9.6 Dual Hahn

9.7 Meixner-Pollaczek

9.8 Jacobi

9.8.1 Gegenbauer/Ultraspherical

9.8.2 Chebyshev

9.8.3 Legendre/Spherical

9.9 Pseudo Jacobi

9.10 Meixner

9.11 Krawtchouk

9.12 Laguerre

9.13 Bessel

9.14 Charlier

9.15 Hermite

10 Orthogonal Polynomial Solutions of $q$-Difference Equations: Classical $q$-Orthogonal Polynomials I

10.1 Polynomial Solutions of $q$-Difference Equations

10.2 The Basic Hypergeometric Representation

10.3 The Three-Term Recurrence Relation

10.4 Classification of the Positive-Definite Orthogonal Polynomial Solutions

10.5 Solutions of the $q$-Pearson Equation

10.6 Orthogonality Relations

11. Orthogonal Polynomials Solutions in $q^{-x}$ of $q$-Difference Equations: Classical $q$-Orthogonal Polynomials II

11.1 Polynomial Solutions in $q^{-x}$ of $q$-Difference Equations

11.2 The Basic Hypergeometric Representation

11.3 The Three-Term Recurrence Relation

11.4 Orthogonality and the Self-Adjoint Operator Equation

11.5 Rodrigues Formulas

11.6 Classification of the Positive-Definite Orthogonal Polynomial Solutions

11.7 Solutions of the $q^{-1}$-Pearson Equation

11.8 Orthogonality Relations

12 Orthogonal Polynomial Solutions in $q^{-x}+uq^x$ of Real $q$-Difference Equations: Classical $q$-Orthogonal Polynomials III

12.1 Motivation for Polynomials in $q^{-x}+uq^x$ Through Duality

12.2 Difference Equations Having Real Polynomial Solutions with Argument $q^{-x}+uq^x$

12.3 The Basic Hypergeometric Representation

12.4 The Three-Term Recurrence Relation

12.5 Classification of the Positive-Definite Orthogonal Polynomial Solutions

12.6 Solutions of the $q$-Pearson Equation

12.7 Orthogonality Relations

13. Orthogonal Polynomial Solutions in $\dfrac{a}{z} + \dfrac{uz}{a}$ of Complex $q$-Difference Equations: Classical $q$-Orthogonal Polynomials IV

13.1 Real Polynomial Solutions in $\dfrac{a}{z} + \dfrac{uz}{a}$ with $u \in \mathbb{R} \setminus \{0\}$ and $a,z \in \mathbb{C} \setminus \{0\}$

13.2 Classification of the Positive-Definite Orthogonal Polynomial Solutions

13.3 Solutions of the $q$-Pearson Equation

13.4 Orthogonality Relations

14. Basic Hypergeometric Orthogonal Polynomials

14.1 Askey-Wilson

14.2 $q$-Racah

14.3 Continuous Dual $q$-Hahn

14.4 Continuous $q$-Hahn

14.5 Big $q$-Jacobi

14.5.1 Big $q$-Legendre

14.6 $q$-Hahn

14.7 Dual $q$-Hahn

14.8 Al-Salam-Chihara

14.9 $q$-Meixner-Pollaczek

14.10 Continuous $q$-Jacobi

14.10.1 Continuous $q$-Ultraspherical/Rogers

14.10.2 Continuous $q$-Legendre

14.11 Big $q$-Laguerre

14.12 Little $q$-Jacobi

14.12.1 Little $q$-Legendre

14.13 $q$-Meixner

14.14 Quantum $q$-Krawtchouk

14.15 $q$-Krawtchouk

14.16 Affine $q$-Krawtchouk

14.17 Dual $q$-Krawtchouk

14.18 Continuous Big $q$-Hermite

14.19 Continuous $q$-Laguerre

14.20 Little $q$-Laguerre/Wall

14.21 $q$-Laguerre

14.22 $q$-Bessel

14.23 $q$-Charlier

14.24 Al-Salam-Carlitz I

14.25 Al-Salam-Carlitz II

14.26 Continuous $q$-Hermite

14.27 Stieltjes-Wigert

14.28 Discrete $q$-Hermite I

14.29 Discrete $q$-Hermite II

Bibliography

Index