A Handbook of Generalized Special Functions

for Statistical and Physical Sciences

A. M. Mathai Department of Mathematics and Statistics, McGill University

CLARENDON PRESS • OXFORD

1993

Contents

Mathematical preliminaries 1.1 The gamma function 1

1.1.1 Some elementary properties 1 1.1.2 Evaluation of r ( l / 2 ) 3 1.1.3 Other equivalent definitions for a gamma function 4 1.1.4 Continuation 4 1.1.5 Multiplication formula for the gamma function 5

1.2 Bernoulli polynomials 6 1.2.1 Elementary properties of Bernoulli polynomials 6 1.2.2 The first six generalized Bernoulli polynomials 7 1.2.3 The first six Bernoulli polynomials 8 1.2.4 The first nine Bernoulli numbers 8 1.2.5 A special case of the generalized Bernoulli polynomial 8

1.3 Asymptotic expansions of gamma functions 9 1.4 The psi function 11 1.5 The generalized zeta functions 12 1.6 The beta function 15

1.6.1 Basic properties of the beta function 16 1.7 Calculation of residues for gamma functions 16

1.7.1 Residue when one gamma function is involved 17 1.7.2 Residues when a product of two gamma

functions is involved 17 1.7.3 Residues when products of several gammas

are involved 19 1.7.4 A differential Operator 21 1.7.5 The residue theorem 22

1.8 The Meilin transform 23 1.9 Density functions 24

1.9.1 Some commonly used probability modeis for scalar random variables 26

1.9.2 Moments 38 1.9.3 Joint, marginal and conditional densities 42 1.9.4 Transformation of variables 44 1.9.5 Statistical independeiice 45

1.10 Methods of deriving distributions 49 Exercises 52

The G-function 2.0 Introduction 58 2.1 The G-function 60

Vlll Contents

2.1.1 Different types of contours 61 2.1.2 Existence conditions for the G-function 63 2.1.3 Verification of the existence conditions 63

2.2 Some basic properties of the G-function 69 2.3 The Mellin transform of a G-function 78

2.3.1 Conditions of validity for the Mellin transform of a G-function 79

2.3.2 Verification of the conditions of validity of the Mellin transform 80

2.3.3 Products of independent real gamma random variables 82

2.3.4 Products of independent real type-1 beta random variables 83

2.3.5 Products of independent real type-2 beta or F-random variables 84

2.4 Properties connected with the derivatives of a G-function 94

2.5 Series representations for a G-function 96 2.5.1 The hypergeometric series 96 2.5.2 Computable representations of a G-function:

simple poles 98 2.5.3 Computable representations of a G-function:

multiple poles 99 2.6 G-functions as multiple integrals or as

solutions of integral equations 106 2.6.1 Type A integral equation 106 2.6.2 Type B integral equation 106 2.6.3 Type C integral equation 108

2.7 Differential equation for a G-function 111 2.8 Asymptotic expansions for a G-function 112

Exercises 113

Elementary special functions and the G-functon 3.0 Introduction 117 3.1 Gamma and related functions:

notations and definitions 117 3.1.1 Gamma function 117 3.1.2 Beta function 117 3.1.3 Psi function 117 3.1.4 Zeta function 117 3.1.5 Generalized Riemann zeta function 118 3.1.6 Euler's dilogarithm 118

IX

3.2 Hypergeometric functions: notations and special cases 118 3.2.1 Generalized hypergeometric series 118 3.2.2 Exponential series 118 3.2.3 Binomial series 118 3.2.4 Gauss' hypergeometric function 119 3.2.5 Incomplete beta funtion 119

3.3 Confluent hypergeometric function and related functions 119 3.3.1 Whittaker functions 119 3.3.2 Parabolic cylinder function 120 3.3.3 Bateman's function 120 3.3.4 Incomplete gamma functions 120 3.3.5 Coulomb wave functions 121 3.3.6 Error functions and related functions 121

3.4 Exponential integral and related functions 121 3.5 Bessel functions and associated functions 121

3.5.1 Kelvin's functions 122 3.6 Other special functions 122

3.6.1 Lommel's functions 122 3.6.2 Elliptic functions 123 3.6.3 Struve's functions 123 3.6.4 Anger-Weber functions 123 3.6.5 Neumann polynomials 123 3.6.6 Theta functions 123

3.7 Orthogonal polynomials 124 3.7.1 Jacobi polynomials 124 3.7.2 Shifted Jacobi polynomials 124 3.7.3 Legendre polynomials 124 3.7.4 Legendre polynomials: modified 125 3.7.5 Gegenbauer or ultraspherical polynomials 125 3.7.6 Chebyshev polynomials 125 3.7.7 Chebyshev polynomials:shifted 125 3.7.8 Chebyshev polynomials:second kind 125 3.7.9 Chebyshev polynomials:second kind, shifted 126 3.7.10 Laguerre polynomials 126 3.7.11 Hermite polynomials 126

3.8 Elementary special functions expressed in terms of G-functions 127

3.9 G-functions expressed in terms of elementary special functions 129

3.10 Some integrals involving G-functions 132 3.10.1 The Meilin transform of a G-function 132

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3.10.2 Hankel transform of a G-function 132 3.10.3 The K-transform of a G-function 132 3.10.4 The Y-transform of a G-function 132 3.10.5 The H-transform of a G-function 133 3.10.6 Stieltjes transform of a G-function 133 3.10.7 Whittaker transform of a G-function 133 3.10.8 Gauss' hypergeometric transform of a G-function 133 3.10.9 Laguerre transform of a G-function 134 3.10.10 Laplace transform of a G-function 134 3.10.11 Integral involving product of two G-functions 134 3.10.12 Some examples from mathematical statistics 136 3.10.13 Some examples from communication theory 138 3.10.14 Some examples from astrophysics 139

3.11 The H-function 140 3.11.1 Conditions for the existence of an H-function 141 3.11.2 Some basic properties of an H-function 142 3.11.3 Some examples from statistics 142 3.11.4 Some examples from astrophysics 144

3.12 Computational aspects of G-and H-functions 144 3.13 Orders of the special functions for small

and large values of the argument 145 Exercises 148

4. Generalizations to matrix variables 4.0 Introduction 152 4.1 Scalar functions of a Symmetrie positive definite matrix 152

4.1.1 Some Jacobians of transformations 153 4.2 Scalar functions of matrix arguments 158

4.2.1 Matrix variate gamma density 158 4.3 Laplace transform 160

4.3.1 Functions of matrix arguments through Laplace transform 162

4.3.2 Matrix variate beta density 163 4.4 Hypergeometric functions of matrix arguments 171 4.5 Generalized matrix transform or M-transform 177

4.5.1 Hypergeometric functions of matrix arguments through M-transforms 178

4.5.2 Canonical correlation matrix 183 4.5.3 G- and H-functions of matrix arguments:

M-transforms 189

Contents XI

4.6 Zonal polynomial 194 4.6.1 Some basic properties of zonal polynomials 194 4.6.2 Hypergeometric function of matrix arguments

through zonal polynomials 195 4.7 Matrix variate Dirichlet distribution 197 4.8 Hypergeometric functions of many scalar variables 205

4.8.1 Lauricella functions of scalar variables 205 4.8.2 Integral representations for Lauricella functions 206 4.8.3 Some cases of reducibility for Lauricella functions 209 4.8.4 Lauricella functions of matrix arguments 210 4.8.5 Definitions through M-transforms 212

4.9 Hypergeometric functions of many matrix arguments 215 4.9.1 An example from Statistical distribution theory 215

4.10 G- and H-functions of two variables 217 Exercises 218

Bibliography 227 Glossary of symbols 231 Author index 233 Subject index 234