# Book:George E. Andrews/Special Functions

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## George E. Andrews, Richard Askey and Ranjan Roy: *Special Functions*

Published $1999$, **Cambridge University Press**.

### Contents

- Preface
- 1 The Gamma and Beta Functions
- 1.1 The Gamma and Beta Integrals and Functions
- 1.2 The Euler Reflection Formula
- 1.3 The Hurwitz and Riemann Zeta Functions
- 1.4 Stirling's Asymptotic Formula
- 1.5 Gauss's Multiplication Formula for $\Gamma(mx)$
- 1.6 Integral Representations for $\mathrm{Log} \Gamma(x)$ and $\psi(x)$
- 1.7 Kummer's Fourier Expansion of $\mathrm{Log} \Gamma(x)$
- 1.8 Integrals of Dirichlet and Volumes of Ellipsoids
- 1.9 The Bohr-Mollerup Theorem
- 1.10 Gauss and Jacobi Sums
- 1.11 A Probabilistic Evaluation of the Beta Function
- 1.12 The $p$-adic Gamma Function

- 2 The Hypergeometric Functions
- 2.1 The Hypergeometric Series
- 2.2 Euler's Integral Representation
- 2.3 The Hypergeometric Equation
- 2.4 The Barnes Integral for the Hypergeometric Function
- 2.5 Contiguous Relations
- 2.6 Dilogarithms
- 2.7 Binomial Sums
- 2.8 Dougall's Bilateral Sum
- 2.9 Fractional Integration by Parts and Hypergeometric Integrals

- 3 Hypergeometric Transformations and Identities
- 3.1 Quadratic Transformations
- 3.2 The Arithmetic-Geometric Mean and Elliptic Integrals
- 3.3 Tranformations of Balnaced Series
- 3.4 Whipple's Transformation
- 3.5 Dougall's Formula and Hypergeometric Identities
- 3.6 Integral Analogs of Hypergeometric Sums
- 3.7 Contiguous Relations
- 3.8 The Wilson Polynomials
- 3.9 Quadratic Transformations -- Riemann's View
- 3.10 Indefinite Hypergeometric Summations
- 3.11 The W-Z Method
- 3.12 Contiguous Relations and Summation Methods

- 4 Bessel Functions and Confluent Hypergeometric Functions
- 4.1 The Confluent Hypergeometric Equation
- 4.2 Barnes's Integral for ${}_1F_1$
- 4.3 Whittaker Functions
- 4.4 Examples of ${}_1F_1$ and Whittaker Functions
- 4.5 Bessel's Equation and Bessel Functions
- 4.6 Recurrence Relations
- 4.7 Integral Representations of Bessel Functions
- 4.8 Asymptotic Expansions
- 4.9 Fourier Transforms and Bessel Functions
- 4.10 Addition Theorems
- 4.11 Integrals of Bessel Functions
- 4.12 The Modified Bessel Functions
- 4.13 Nicholson's Integral
- 4.14 Zeros of Bessel Functions
- 4.15 Monotonicity Properties of Bessel Functions
- 4.16 Zero-Free Regions for ${}_1F_1$ Functions

- 5 Orthogonal Polynomials
- 5.1 Chebyshev Polynomials
- 5.2 Recurrence
- 5.3 Gauss Quadrature
- 5.4 Zeros of Orthogonal Polynomials
- 5.5 Continued Fractions
- 5.6 Kernel Polynomials
- 5.7 Parseval's Formula
- 5.8 The Moment-Generating Function

- 6 Special Orthogonal Polynomials
- 6.1 Hermite Polynomials
- 6.2 Laguerre Polynomials
- 6.3 Jacobi Polynomials and Gram Determinants
- 6.4 Generating Functions for Jacobi Polynomials
- 6.5 Completeness of Orthogonal Polynomials
- 6.6 Asymptotic Behavior of $P_n^{(\alpha,\beta)}(x)$ for Large $n$
- 6.7 Integral Represetnations of Jacobi Polynomials
- 6.8 Linearization of Products of Orthogonal Polynomials
- 6.9 Matching Polynomials
- 6.10 The Hypergeometric Orthogonal Polynomials
- 6.11 An Extension of the Ultraspherical Polynomials

- 7 Topics in Orthogonal Polynomials
- 7.1 Connection Coefficients
- 7.2 Rational Functions with Positive Power Series Coefficients
- 7.3 Positive Polynomial Sums from Quadrature and Vietoris's Inequality
- 7.4 Positive Polynomial Sums and the Bieberback Conjecture
- 7.5 A Theorem of Turan
- 7.6 Positive Summability of Ultraspherical Polynomials
- 7.7 The Irrationality of $\zeta(3)$

- 8 The Selberg Integral and Its Applications
- 8.1 Selberg's and Aomoto's Integrals
- 8.2 Aomoto's Proof of Selberg's Formula
- 8.3 Extensions of Aomoto's Integral Formula
- 8.4 Anderson's Proof of Selberg's Formula
- 8.5 A Problem of Stieltjes and the Discriminant of a Jacobi Polynoial
- 8.6 Siegel's Inequality
- 8.7 The Stieltjes Problem on the Unit Circle
- 8.8 Contant-Term Identities
- 8.9 Nearly Poised ${}_3F_2$ Identities
- 8.10 The Hasse-Davenport Relation
- 8.11 A Finite-Field Analog of Selberg's Integral

- 9 Spherical Harmonics
- 9.1 Harmonic Polynomials
- 9.2 The Laplace Equation in Three Dimensions
- 9.3 Dimension of the Space of Harmonic Polynomials of Degree $k$
- 9.4 Orthogonality of Harmonic Polynomials
- 9.5 Action of an Orthogonal Matrix
- 9.6 The Addition Theorem
- 9.7 The Funk-Hecke Formula
- 9.8 The Addition Theorem for Ultraspherical Polynomials
- 9.9 The Poisson Kernel and Dirichlet Problem
- 9.10 Fourier Transforms
- 9.11 Finite-Dimensional Representations of Compact Groups
- 9.12 The Group $SU(2)$
- 9.13 Representations of $SU(2)$
- 9.14 Jacobi Polynomials as Matrix Entries
- 9.15 An Addition Theorem
- 9.16 Relation of $SU(2)$ to the Rotation Group $SO(3)$

- 10 Introduction to $q$-series
- 10.1 The $q$-integral
- 10.2 The $q$-Binomial Theorem
- 10.3 The $q$-Gamma function
- 10.4 The Triple Product Identity
- 10.5 Ramanujan's Summation Formula
- 10.6 Representations of Numbers as Sums of Squares
- 10.7 Elliptic and Theta Functions
- 10.8 $q$-Beta Integrals
- 10.9 Basic Hypergeometric Series
- 10.10 Basic Hypergeometric Identities
- 10.11 $q$-Ultraspherical Polynomials
- 10.12 Mellin Transforms

- 11 Partitions
- 11.1 Background on Partitions
- 11.2 Partition Analysis
- 11.3 A Library for Partition Analysis Algorithm
- 11.4 Generating Functions
- 11.5 Some Results on Partitions
- 11.6 Graphical Methods
- 11.7 Congruence Properties of Partitions

- 12 Baily Chains
- 12.1 Roger's Second Proof of the Rogers-Ramanujan Identities
- 12.2 Baily's Lemma
- 12.3 Waton's Transformation Formula
- 12.4 Other Applications

- A Infinite Products
- A.1 Infinite Products

- B Summability and Fractional Integration
- B.1 Abel and Cesaro Means
- B.2 The Cesaro Means $(C,\alpha)$
- B.3 Fractional Integrals
- B.4 Historical Remarks

- C Asymptotic Expansions
- C.1 Asymptotic Expansion
- C.2 Properties of Asymptotic Expansions
- C.3 Watson's Lemma
- C.4 The Ratio of Two Gamma Functions

- D Euler-Maclaurin Summation Formula
- D.1 Introduction
- D.2 The Euler-Maclaurin Formula
- D.3 Applications
- D.4 The Poisson Summation Formula

- E Lagrange Inversion Formula
- E.1 Reversion of Series
- E.2 A Basic Lemma
- E.3 Lambert's Identity
- E.4 Whipple's Transformation

- F Series Solutions of Differential Equations
- F.1 Ordinary Points
- F.2 Singular Points
- F.3 Regular Singular Points

- Bibliography
- Index
- Subject Index
- Symbol Index