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.

Table of Contents:
1 Preliminaries.- 1.1 Some special functions.- 1.2 Integral transforms.- 2 Mellin Convolution Type Transforms With Arbitrary Kernels.- 2.1 General Fourier kernels.- 2.2 Examples of the Fourier kernels.- 2.3 Watson type kernels.- 2.4 Bilateral Watson transforms.- 2.5 Multidimensional Watson transforms.- 3 H- and G-transforms.- 3.1 Mellin convolution type transform with Fox’s H-function as a kernel.- 3.2 Mellin convolution type transforms with Meijer’s G-function as a kernel.- 3.3 The Erdelyi-Kober fractional integration operators.- 4 The Generalized H- and G-transforms.- 4.1 The generalized H-transform.- 4.2 The generalized G-transform.- 4.3 Composition structure of generalized H- and G-transforms.- 5 The Generating Operators of Generalized H-transforms.- 5.1 Generating operators in the space ?Mc,??1.- 5.2 Examples of the generating operators.- 6 The Kontorovich-Lebedev Transform.- 6.1 The Kontorovich-Lebedev transform: notion, existence and inversion theorems in Mc,??1 (L) spaces.- 6.2 The Kontorovich-Lebedev transform in weighted L-spaces.- 6.3 The Kontorovich-Lebedev transform in weighted L2 spaces.- 6.4 The Kontorovich-Lebedev transform of distributions.- 6.5 The Kontorovich-Lebedev transform in Lp-spaces.- 7 General W-transform and its Particular Cases.- 7.1 General G-transform with respect to an index of the Kontorovich-Lebedev type.- 7.2 General W-transform and its composition structure.- 7.3 Some particular cases of W-transform and their properties.- 7.4 F3-transform.- 7.5 L2-theory of the Kontorovich-Lebedev type index transforms.- 8 Composition Theorems of Plancherel Type for Index Transforms.- 8.1 Compositions with symmetric weight.- 8.2 Compositions with non-symmetric weight.- 8.3 Constructions of index transforms in terms of Mellin integrals.- 9Some Examples of Index Transforms and Their New Properties.- 9.1 The Kontorovich-Lebedev like composition transforms.- 9.2 Some index transforms with symmetric kernels.- 9.3 The $$ \Re $$ and $$ \Im - $$ index transforms.- 10 Applications to Evaluation of Index Integrals.- 10.1 Some useful representations and identities.- 10.2 Some general index integrals.- 11 Convolutions of Generalized H-transforms.- 11.1 H-convolutions.- 11.2 Examples of H-convolutions.- 12 Generalization of the Notion of Convolution.- 12.1 Generalized H-convolutions.- 12.2 Generalized G-convolutions.- 13 Leibniz Rules and Their Integral Analogues.- 13.1 General Leibniz rules.- 13.2 Modified Leibniz rule.- 13.3 Leibniz rule for the Erdelyi-Kober fractional differential operator.- 13.4 Modification of the Leibniz rule for the Erdelyi-Kober fractional differential operator.- 13.5 Integral analogues of Leibniz rules.- 14 Convolutions of Generating Operators.- 14.1 Convolutions in the Dimovski sense. General results.- 14.2 Examples of convolutions in the Dimovski sense.- 15 Convolution of the Kontorovich-Lebedev Transform.- 15.1 Definition and some properties of a convolution for the Kontorovich-Lebedev transform.- 15.2 The basic property of convolution. Analogues with the Parseval equality.- 15.3 On the inversion of the Kontorovich-Lebedev transform in the ring L?.- 15.4 The space L? as the commutative normed ring of functions with exponential growth.- 16 Convolutions of the General Index Transforms.- 16.1 Convolutions of the Kontorovich-Lebedev type transforms.- 16.2 The convolutions for the Mehler-Fock and the Lebedev-Skalskaya transforms.- 16.3 The convolution of the Wimp-Yakubovich type index transform.- 17 Applications of the Kontorovich-Lebedev type Convolutions to Integral Equations.- 17.1Kontorovich-Lebedev convolution equations of the second kind.- 17.2 General composition convolution equations.- 17.3 Some results on the homogeneous equation.- 18 Convolutional Ring C?.- 18.1 Multiple Erdelyi-Kober fractional integrodifferential operators.- 18.2 Convolutional ring C?.- 19 The Fields of the Convolution Quotients.- 19.1 Extension of the ring (C?,?*,+).- 19.2 Extension of the ring (L?,*,+).- 20 The Cauchy Problem for Erdelyi-Kober Operators.- 20.1 General scheme.- 20.2 Differential equations of fractional order.- 20.3 Differential equations of hyper-Bessel type.- 21 Operational Method of Solution of some Convolution Equations.- 21.1 Integral equations of Volterra type.- 21.2 Integral equations of second kind with Kontorovich-Lebedev convolution.- References.- Author Index.- Notations.