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

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{{Book|Hypergeometric Orthogonal Polynomials and Their q-Analogues|2010|Springer Verlag||Roelof Koekoek|author2=Peter A. Lesky|author3=René F. Swarttouw}}

===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