特殊函数 (2010 年度畅销榜NO.42404 )

Special functions, natural generalizations of the elementary functions, have been studied for centuries. The greatest mathematicians, among them Euler, Gauss, Legendre, Eisenstein, Riemann, and Ramanujan, have laid the foundations for this beautiful and useful area of mathematics. For instance, Euler found the gamma function, which extends the factorial. The Bessel fufictions and Legendrepolynomials play a role in three dimensions similar to the role of sine and cosine in two dimensions.This treatise presents an overview of special functions, focusing primarily on hypergeometric functions and the associated hypergeometric series, including Bessel functions and classical orthogonal polynomials. The basic building block of the functions studied in this book is the gamma function. In addition to relatively new work on gamma and beta functions, such as Selberg's multidimensional integrals, a number of important but relatively unknown nineteenth century results are included.The authors discuss Wilson's beta integral and the associated orthogonal polynomials. S omeqextensions of beta integrals and of hypergeometric series are presented with Bailey chains employed to derive some results. An introduction to spherical harmonics and applications of special functions to combinatorial problems are included. The book also deals with finite field versions of some beta integrals.
The authors provide organizing ideas, motivation, and historical background for the study and application of some important special functions. This clearly expressed and readable work can serve as a learning tool and lasting reference for students and researchers in special functions, mathematicalphysics, differential equations, mathematical computing, number theory, and combinatorics.

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 F(rnx)
1.6 Integral Representations'for Log F (x) and lp (x)
1.7 Kummer's Fourier Expansion of Log F (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
Exercises
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 Hyperge0metric Integrals
Exercises
3 Hypergeometric Transformations and Identities
3.1 Quadratic Transformations
3.2 The Arithmetic-Geometric Mean and Elliptic Integrals
3.3 Transformations of Balanced 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 Summation
3.11 The W-Z Method
3.12 Contiguous Relations and Summation Methods
Exercises
4 Bessei Functions and Confluent Hypergeometric Functions
4.1 The Confluent Hypergeometric Equation
4.2 Barnes's Integral for j Fx
4.3 Whittaker Functions
4.4 Examples of t Fl 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 ~ Fl Functions
Exercises
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
Exercises
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?'~)(x) for Large n
6.7 Integral Representations 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
Exercises.
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 Turfin
7.6 Positive Summability of Ultraspherical Polynomials
7.7 The Irrationality of ~ (3)
Exercises
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 Polynomial
8.6 Siegel's Inequality
8.7 The Stieltjes Problem on the Unit Circle
8.8 Constant-Term Identities
8.9 Nearly Poised 3 F2 Identities
8.10 The Hasse-Davenport Relation
8.11 A Finite-Field Analog of Selberg's Integral
Exercises
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 $U (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)
Exercises
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 Them 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
Exercises
11 Partitions
11.1 Background on Partitions
11.2 Partition Analysis
11.3 A Library for the Partition Analysis Algorithm
Generating Functions
Some Results on Partitions
Graphical Methods
Congruence Properties of Partitions
Exercises
12 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
Exercises
A Infinite Products
A. 1 Infinite Products
Exercises
B Summability and Fractional Integration
B. 1 Abel and Ceiaro Means
B.2 The Cesaro Means (C. or}
B.3 Fractional Integrals
B.4 Historical Remarks
Exercises
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
Exercises
D Euler-Maclaurin Summation Formula
D. 1 Introduction
D.2 The Euler-Maclaurin Formula
D.3 Applications
D.4 The Poisson Summation Formula
Exercises
E Lagrange Inversion Formula
E. 1 Reversion of Series
E.2 A Basic Lemma
E.3 Lambert's Identity
E.4 Whipple's Transformation
Exercises
Series Solutions of Differential Equations
F. 1 Ordinary Points
F.2 Singular Points
F.3 Regular Singular Points
Bibliography
Index
Subject Index
Symbol Index