ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS. Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS

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1 ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS

2 Preface page xiii 1 The Gamma and Beta Functions The Gamma and Beta Integrals and Functions The Euler Reflection Formula The Hurwitz and Riemann Zeta Functions Stirling's Asymptotic Formula Gauss's Multiplication Formula for T(mx) Integral Representations for Log T(x) and xjr (x) Kummer's Fourier Expansion of Log F (x) Integrals of Dirichlet and Volumes of Ellipsoids The Bohr-Mollerup Theorem Gauss and Jacobi Sums A Probabilistic Evaluation of the Beta Function The /? : adic Gamma Function The Hypergeometric Functions \ The Hypergeometric Series Euler's Integral Representation The Hypergeometric Equation The Barnes Integral for the Hypergeometric Function Contiguous Relations Dilogarithms Binomial Sums Dougall's Bilateral Sum Fractional Integration by Parts and Hypergeometric Integrals

3 viii 3 Hypergeometric Transformations and Identities Quadratic Transformations The Arithmetic-Geometric Mean and Elliptic Integrals Transformations of Balanced Series Whipple's Transformation Dougall's Formula and Hypergeometric Identities Integral Analogs of Hypergeometric Sums Contiguous Relations The Wilson Polynomials Quadratic Transformations - Riemann's View Indefinite Hypergeometric Summation The W-Z Method Contiguous Relations and Summation Methods Bessel Functions and Confluent Hypergeometric Functions The Confluent Hypergeometric Equation Barnes's Integral for i F] Whittaker Functions Examples of i Fi and Whittaker Functions Bessel's Equation and Bessel Functions Recurrence Relations Integral Representations of Bessel Functions Asymptotic Expansions Fourier Transforms and Bessel Functions Addition Theorems Integrals of Bessel Functions The Modified Bessel Functions Nicholson's Integral Zeros of Bessel Functions Monotonicity Properties of Bessel Functions Zero-Free Regions for ifi Functions Orthogonal Polynomials Chebyshev Polynomials Recurrence Gauss Quadrature Zeros of Orthogonal Polynomials Continued Fractions 256

4 5.6 Kernel Polynomials Parseval's Formula The Moment-Generating Function Special Orthogonal Polynomials Hermite Polynomials Laguerre Polynomials Jacobi Polynomials and Gram Determinants Generating Functions for Jacobi Polynomials Completeness of Orthogonal Polynomials Asymptotic Behavior of P^a- P) {x) for Large n Integral Representations of Jacobi Polynomials Linearization of Products of Orthogonal Polynomials Matching Polynomials The Hypergeometric Orthogonal Polynomials An Extension of the Ultraspherical Polynomials Topics in Orthogonal Polynomials Connection Coefficients Rational Functions with Positive Power Series Coefficients Positive Polynomial Sums from Quadrature and Vietoris's Inequality Positive Polynomial Sums and the Bieberback Conjecture A Theorem of Turan Positive Summability of Ultraspherical Polynomials The Irrationality of? (3) The Selberg Integral and Its Applications Selberg's and Aomoto's Integrals Aomoto's Proof of Selberg's Formula Extensions of Aomoto's Integral Formula Anderson's Proof of Selberg's Formula A Problem of Stieltjes and the Discriminant of a Jacobi Polynomial Siegel's Inequality The Stieltjes Problem on the Unit Circle Constant-Term Identities Nearly Poised 3F 2 Identities The Hasse-Davenport Relation 430 ix

5 x 8.11 A Finite-Field Analog of Selberg's Integral Spherical Harmonics Harmonic Polynomials The Laplace Equation in Three Dimensions Dimension of the Space of Harmonic Polynomials of Degree k Orthogonality of Harmonic Polynomials Action of an Orthogonal Matrix The Addition Theorem The Funk-Hecke Formula The Addition Theorem for Ultraspherical Polynomials The Poisson Kernel and Dirichlet Problem Fourier Transforms Finite-Dimensional Representations of Compact Groups The Group SU(2) Representations of SU (2) Jacobi Polynomials as Matrix Entries An Addition Theorem Relation of SI/ (2) to the Rotation Group S O (3) Introduction to ^-Series The ^-Integral The ^-Binomial Theorem The q -Gamma Function * The Triple Product Identity Ramanujan's Summation Formula Representations of Numbers as Sums of Squares Elliptic and Theta Functions q -Beta Integrals Basic Hypergeometric Series Basic Hypergeometric Identities g-ultraspherical Polynomials, Mellin Transforms Partitions Background on Partitions Partition Analysis A Library for the Partition Analysis Algorithm 557

6 12 B D E 11.4 Generating Functions 11.5 Some Results on Partitions 11.6 Graphical Methods 11.7 Congruence Properties of Partitions Bailey Chains 12.1 Rogers's Second Proof of the Rogers-Ramanujan Identities 12.2 Bailey's Lemma 12.3 Watson's Transformation Formula 12.4 Other Applications Infinite Products A. 1 Infinite Products Summability and Fractional Integration B.I Abel and Cesaro Means B.2 The Cesaro Means (C, a) B.3 Fractional Integrals B.4 Historical Remarks Asymptotic Expansions C.I Asymptotic Expansion C.2 Properties of Asymptotic Expansions C.3 Watson's Lemma C.4 The Ratio of Two Gamma Functions Euler-Maclaurin Summation Formula D.I Introduction D.2 The Euler-Maclaurin Formula D.3 Applications D.4 The Poisson Summation Formula Lagrange Inversion Formula E. 1 Reversion of Series E.2 A Basic Lemma E.3 Lambert's Identity E.4 Whipple's Transformation

7 xii F Series Solutions of Differential Equations 637 F.I Ordinary Points 637 F.2 Singular Points 638 F.3 Regular Singular Points 639 Bibliography 641 Index 655 Subject Index 659 Symbol Index 661

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