The H-Function (eBook)

Preface 7
Contents 9
1 On the H-Function With Applications 13
1.1 A Brief Historical Background 13
1.2 The H-Function 14
1.3 Illustrative Examples 19
1.4 Some Identities of the H-Function 23
1.4.1 Derivatives of the H-Function 25
1.5 Recurrence Relations for the H-Function 28
1.6 Expansion Formulae for the H-Function 29
1.7 Asymptotic Expansions 31
1.8 Some Special Cases of the H-Function 33
1.8.1 Some Commonly Used Special Cases of the H-Function 38
1.9 Generalized Wright Functions 41
1.9.1 Existence Conditions 42
1.9.2 Representation of Generalized Wright Function 43
2 H-Function in Science and Engineering 56
2.1 Integrals Involving H-Functions 56
2.2 Integral Transforms of the H-Function 56
2.2.1 Mellin Transform 56
2.2.2 Illustrative Examples 57
2.2.3 Mellin Transform of the H-Function 58
2.2.4 Mellin Transform of the G-Function 59
2.2.5 Mellin Transform of the Wright Function 59
2.2.6 Laplace Transform 59
2.2.7 Illustrative Examples 60
2.2.8 Laplace Transform of the H-Function 61
2.2.9 Inverse Laplace Transform of the H-Function 62
2.2.10 Laplace Transform of the G-Function 63
2.2.11 K-Transform 64
2.2.12 K-Transform of the H-Function 65
2.2.13 Varma Transform 66
2.2.14 Varma Transform of the H-Function 66
2.2.15 Hankel Transform 67
2.2.16 Hankel Transform of the H-Function 68
2.2.17 Euler Transform of the H-Function 69
2.3 Mellin Transform of the Product of Two H-Functions 71
2.3.1 Eulerian Integrals for the H-Function 71
2.3.2 Fractional Integration of a H-Function 73
2.4 H-Function and Exponential Functions 78
2.5 Legendre Function and the H-Function 80
2.6 Generalized Laguerre Polynomials 82
3 Fractional Calculus 86
3.1 Introduction 86
3.2 A Brief Historical Background 87
3.3 Fractional Integrals 88
3.3.1 Riemann–Liouville Fractional Integrals 90
3.3.2 Basic Properties of Fractional Integrals 90
3.3.3 Illustrative Examples 92
3.4 Riemann–Liouville Fractional Derivatives 94
3.4.1 Illustrative Examples 99
3.5 The Weyl Integral 102
3.5.1 Basic Properties of Weyl Integrals 102
3.5.2 Illustrative Examples 103
3.6 Laplace Transform 105
3.6.1 Laplace Transform of Fractional Integrals 105
3.6.2 Laplace Transform of Fractional Derivatives 105
3.6.3 Laplace Transform of Caputo Derivative 106
3.7 Mellin Transforms 107
3.7.1 Mellin Transform of the nth Derivative 108
3.7.2 Illustrative Examples 108
3.8 Kober Operators 109
3.8.1 Erdélyi–Kober Operators 109
3.9 Generalized Kober Operators 112
3.10 Saigo Operators 114
3.10.1 Relations Among the Operators 117
3.10.2 Power Function Formulae 117
3.10.3 Mellin Transform of Saigo Operators 119
3.10.4 Representation of Saigo Operators 119
3.11 Multiple Erdélyi–Kober Operators 124
3.11.1 A Mellin Transform 125
3.11.2 Properties of the Operators 126
3.11.3 Mellin Transform of a Generalized Operator 127
4 Applications in Statistics 129
4.1 Introduction 129
4.2 General Structures 129
4.2.1 Product of Type-1 Beta Random Variables 131
4.2.2 Real Scalar Type-2 Beta Structure 134
4.2.3 A More General Structure 135
4.3 A Pathway Model 137
4.3.1 Independent Variables Obeying a Pathway Model 138
4.4 A Versatile Integral 141
4.4.1 Case of < 1 or <
4.4.2 Some Practical Situations 146
5 Functions of Matrix Argument 149
5.1 Introduction 149
5.2 Exponential Function of Matrix Argument 150
5.3 Jacobians of Matrix Transformations 153
5.4 Jacobians in Nonlinear Transformations 156
5.5 The Binomial Function 159
5.6 Hypergeometric Function and M-transforms 161
5.7 Meijer's G-Function of Matrix Argument 164
5.7.1 Some Special Cases 165
6 Applications in Astrophysics Problems 169
6.1 Introduction 169
6.2 Analytic Solar Model 169
6.3 Thermonuclear Reaction Rates 173
6.4 Gravitational Instability Problem 175
6.5 Generalized Entropies in Astrophysics Problems 178
6.5.1 Generalizations of Shannon Entropy 179
6.6 Input–Output Analysis 181
6.7 Application to Kinetic Equations 183
6.8 Fickean Diffusion 184
6.8.1 Application to Time-Fractional Diffusion 185
6.9 Application to Space-Fractional Diffusion 187
6.10 Application to Fractional Diffusion Equation 188
6.10.1 Series Representation of the Solution 190
6.11 Application to Generalized Reaction-Diffusion Model 192
6.11.1 Motivation 192
6.11.2 Mathematical Prerequisites 193
6.11.3 Fractional Reaction–Diffusion Equation 195
6.11.4 Some Special Cases 196
6.11.5 Fractional Order Moments 199
6.11.6 Some Further Applications 200
6.11.7 Background 201
6.11.8 Unified Fractional Reaction–Diffusion Equation 202
6.11.9 Some Special Cases 203
6.11.10 More Special Cases 208
Appendix 214
A.1 H-Function of Several Complex Variables 214
A.2 Kampé de Fériet Function and Lauricella Functions 216
A.2.1 Kampé de Fériet Series in the Generalized Form 216
A.2.2 Generalized Lauricella Function 217
A.3 Appell Series 220
A.3.1 Confluent Hypergeometric Function of Two Variables 221
A.4 Lauricella Functions of Several Variables 222
A.4.1 Confluent form of Lauricella Series 224
A.5 The Generalized H-Function (The -Function) 224
A.5.1 Special Cases of -Function 225
A.6 Representation of an H-Function in Computable Form 227
A.7 Further Generalizations of the H-Function 228
Bibliography 230
Glossary of Symbols 267
Author Index 269
Subject Index 275