Book:George E. Andrews/Special Functions

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

$q$-number

$q$-factorial

$(10.2.1)$

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