View source for Book:George E. Andrews/Special Functions

{{Book|Special Functions|1999|Cambridge University Press||George E. Andrews|author2=Richard Askey|author3=Ranjan Roy}}

===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|$q$-number]]
:::[[q-factorial|$q$-factorial]]
:::[[q-shifted 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

[[Category:Book]]