Yakubovich S.B., Luchko Y.F. The Hypergeometric Approach to Integral Transforms and Convolutions

Amsterdam: Springer, 1994. — 334 p.The aim of this book is to develop a new approach which we called the hyper­ geometric one to the theory of various integral transforms, convolutions, and their applications to solutions of integro-differential equations, operational calculus, and evaluation of integrals. We hope that this simple approach, which will be explained below, allows students, post graduates in mathematics, physicists and technicians, and serious mathematicians and researchers to find in this book new interesting results in the theory of integral transforms, special functions, and convolutions. The idea of this approach can be found in various papers of many authors, but systematic discussion and development is realized in this book for the first time. Let us explain briefly the basic points of this approach. As it is known, in the theory of special functions and its applications, the hypergeometric functions play the main role. Besides known elementary functions, this class includes the Gauss's, Bessel's, Kummer's, functions et c. In general case, the hypergeometric functions are defined as a linear combinations of the Mellin-Barnes integrals. These ques­ tions are extensively discussed in Chapter 1. Moreover, the Mellin-Barnes type integrals can be understood as an inversion Mellin transform from the quotient of products of Euler's gamma-functions. Thus we are led to the general construc­ tions like the Meijer's G-function and the Fox's H-function.Preliminaries
Mellin Convolution Type Transforms With Arbitrary Kernels
H- and G-transforms
The Generalized H- and G-transforms
The Generating Operators of Generalized H-transforms
The Kontorovich-Lebedev Transform
General W-transform and its Particular Cases
Composition Theorems of Plancherel Type for Index Transforms
Some Examples of Index Transforms and Their New Properties
Applications to Evaluation of Index Integrals
Convolutions of Generalized H-transforms
Generalization of the Notion of Convolution
Leibniz Rules and Their Integral Analogues
Convolutions of Generating Operators
Convolution of the Kontorovich-Lebedev Transform
Convolutions of the General Index Transforms
Applications of the Kontorovich-Lebedev type Convolutions to Integral Equations
Convolutional Ring C α
The Fields of the Convolution Quotients
The Cauchy Problem for Erdelyi-Kober Operators
Operational Method of Solution of some Convolution Equations