Ebook: The Hypergeometric Approach to Integral Transforms and Convolutions

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.

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This volume deals with the theory and applications of integral transforms and convolutions of certain classes of integral, integrodifferential equations, and operational calculus. An extensive discussion is presented, based on the universal hypergeometric approach, i.e. many constructions of convolution and integral transforms are obtained using the theory of Mellin--Barnes integrals and the Mellin transforms of hypergeometric type functions. This approach is spread on so-called index transforms, in which the Kontorovich--Lebedev and the Mehler--Fock transforms play a very important part. The general constructions of index transforms are given and application to the evaluation of improper integral with respect to a parameter of special function (index) is considered. The operational calculus for general integrodifferential operators is constructed for both new types of convolutions. The book is self-contained, and includes a list of symbols with definitions, author and subject indices, and an up-to-date bibliography.
This work will be of interest to researchers and graduate students in the mathematical and physical sciences whose work involves integral transforms and convolutions.

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This volume deals with the theory and applications of integral transforms and convolutions of certain classes of integral, integrodifferential equations, and operational calculus. An extensive discussion is presented, based on the universal hypergeometric approach, i.e. many constructions of convolution and integral transforms are obtained using the theory of Mellin--Barnes integrals and the Mellin transforms of hypergeometric type functions. This approach is spread on so-called index transforms, in which the Kontorovich--Lebedev and the Mehler--Fock transforms play a very important part. The general constructions of index transforms are given and application to the evaluation of improper integral with respect to a parameter of special function (index) is considered. The operational calculus for general integrodifferential operators is constructed for both new types of convolutions. The book is self-contained, and includes a list of symbols with definitions, author and subject indices, and an up-to-date bibliography.
This work will be of interest to researchers and graduate students in the mathematical and physical sciences whose work involves integral transforms and convolutions.

Content:
Front Matter....Pages i-xi
Preliminaries....Pages 1-14
Mellin Convolution Type Transforms With Arbitrary Kernels....Pages 15-40
H- and G-transforms....Pages 41-58
The Generalized H- and G-transforms....Pages 59-68
The Generating Operators of Generalized H-transforms....Pages 69-78
The Kontorovich-Lebedev Transform....Pages 79-108
General W-transform and its Particular Cases....Pages 109-138
Composition Theorems of Plancherel Type for Index Transforms....Pages 139-148
Some Examples of Index Transforms and Their New Properties....Pages 149-166
Applications to Evaluation of Index Integrals....Pages 167-172
Convolutions of Generalized H-transforms....Pages 173-182
Generalization of the Notion of Convolution....Pages 183-188
Leibniz Rules and Their Integral Analogues....Pages 189-204
Convolutions of Generating Operators....Pages 205-212
Convolution of the Kontorovich-Lebedev Transform....Pages 213-228
Convolutions of the General Index Transforms....Pages 229-240
Applications of the Kontorovich-Lebedev type Convolutions to Integral Equations....Pages 241-252
The Fields of the Convolution Quotients....Pages 253-264
The Cauchy Problem for Erdelyi-Kober Operators....Pages 265-276
Back Matter....Pages 277-286
Operational Method of Solution of some Convolution Equations....Pages 295-324
....Pages 287-294