Book:George E. Andrews/Special Functions

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

Published $\text {1999}$, Cambridge University Press

ISBN 0-521-62321-9

Contents

Preface

CHAPTER 1: The Gamma and Beta Functions

$\S 1$. The Gamma and Beta Integrals and Functions

$\S 2$. The Euler Reflection Formula

$\S 3$. The Hurwitz and Riemann Zeta Functions

$\S 4$. Stirling's Asymptotic Formula

$\S 5$. Gauss's Multiplication Formula for $\map \Gamma {mx}$

$\S 6$. Integral Representation for $\map \Log {\map \Gamma x}$ and $\map \psi x$

$\S 7$. Kummer's Fourier Expansion of $\map \Log {\map \Gamma x}$

$\S 8$. Integrals of Dirichlet and Volumes of Ellipsoids

$\S 9$. The Bohr-Mollerup Theorem

$\S 10$. Gauss and Jacobi Sums

$\S 11$. A Probabilistic Evaluation of the Beta Function

$\S 12$. The p-adic Gamma Function

Exercises

CHAPTER 2: The Hypergeometric Functions

$\S 1$. The Hypergeometric Series

$\S 2$. Euler's Integral Representation

$\S 3$. The Hypergeometric Equation

$\S 4.$ The Barnes Integral for the Hypergeometric Function

$\S 5$. Contiguous Relations

$\S 6$. Dilogarithms

$\S 7$. Binomial Sums

$\S 8$. Dougall's Bilateral Sum

$\S 9$. Fractional Integration by Parts and Hypergeometric Integrals

Exercises

CHAPTER 3: Hypergeometric Transformations and Identities

$\S 1$. Quadratic Transformations

$\S 2$. The Arithmetic-Geometric Mean and Elliptic Integrals

$\S 3$. Transformations of Balanced Series

$\S 4.$ Whipple's Transformation

$\S 5$. Dougall's Formula and Hypergeometric Identities

$\S 6$. Integral Analogs of Hypergeometric Sums

$\S 7$. Contiguous Relations

$\S 8$. The Wilson Polynomials

$\S 9$. Quadratic Transformations - Riemann's View

$\S 10$. Indefinite Hypergeometric Summation

$\S 11$. The W-Z Method

$\S 12$. Contiguous Relations and Summation Methods

Exercises

More to complete later...(10+ additional chapters)

Bibliography

Index

Subject Index

Symbol Index