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Roelof KoekoekPeter 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