LAPLACE TRANSFORM (PART 1) OF THE MULTI-INDEX MITTAG-LEFFLER-PRABHAKAR FUNCTIONS OF LE ROY TYPE

Abstract

: In this paper, the multi-index generalizations (with 3 and 4 indices, then with 3 m and 4 m indices) of the classical Le Roy function and its Mittag-Leffler analogues are considered on wider sets of the parameters. Thus, we extend our recent studies as continuation of the works on Le Roy type functions by Gerhold, Garra and Polito, Mainardi and Garrappa, Tomovski and Mehrez, Pogany, Gorska and Horzela, etc. The Laplace transforms of these multi-index Le Roy type functions are provided. In next Part 2, we will propose their images under the Erd´elyi-Kober operators of fractional calculus. The results are specified for the particular cases of the considered functions. Finally, we discuss some open problems about relation of the Le Roy type functions with special functions more general than the Fox H -functions.