Integral transforms of the Mellin convolution type and their generating operators

In this article, the theory of the integral transforms of the Mellin convolution type is presented. The classical integral Laplace and Mellin transforms and their important properties, as well as a general schema for the applications of the integral transforms are first discussed. Based on the Mellin transform theory, a very general integral transform of the Mellin convolution type is introduced and investigated. An important case of this transform – the so-called H-transform with the Fox H-function as the kernel – is defined and considered in detail. Using the Parseval formula for the Mellin integral transform the generalized H-transform is introduced and investigated. For the generalized H-transform, the notion of the generating operator is defined and discussed. Numerous examples of the generating operators for the H-transforms together with the corresponding operational relations are presented.