Certain multidimensional integral transformations (I)

Abstract

The object of this paper is to introduce a general multiple integral transformation whose kernel involves the H -function of several complex variables, which was defined and studied elsewhere by the present authors ([25], [26] and [27]). This integral transform, defined by Equation (1.1) below, and its confluent form (1.15), not only provide interesting unifications (and extensions) of the various classes of known integral transformations whose kernels are expressible in terms of the familiar E, G and H functions of one and two variables, or the product of several such functions, but also offer the possibility of their appropriate further generalizations involving multiple integrals. Since a great variety of functions that occur rather frequently in problems of applied mathematics and mathematical analysis are special cases of the kernel used here, and since the need for a simultaneous operational calculus (based upon multidimensional integral transformations) presents itself quite naturally when problems dependent on several variables are to be treated operationally, a systematic study of the integral transform (1.1) and its confluent form (1.15) is believed to yield deeper, general and useful results.