ON THE MULTI-INDEX MITTAG-LEFFLER FUNCTIONS AND THEIR MELLIN TRANSFORMS

In this survey paper we consider the classes of the multi-index Mittag-Leffler functions, introduced and studied by the authors as extensions of the classical Mittag-Leffler functions Eα,β and of the Prabhakar function E γ α,β, by means of replacing the 2 parameters α, β, respectively the 3 parameters α, β, γ, by 2m-, resp. 3msets of parameters, m = 1, 2, 3, ...: α → (α1, α2, ..., αm), β → (β1, β2, ..., βm), γ → (γ1, γ2, ..., γm). Some of their basic properties are discussed, such as the order and type of these entire functions, their place among the special functions of fractional calculus and previously known classical special functions, especially their representations as Wright’s generalized hypergeometric functions and Fox’s H-functions. A very long list of interesting and useful special functions that appear as particular cases is provided. The importance of the Mellin integral transform is well known as a tool for development of the theories of the special functions and fractional calculus, in many problems for fractional order differential equations and systems whose solutions are usually presented in terms of Mittag-Leffler type functions, and in treating various mathematical models in stochastics, control theory, financial mathematics, etc., that are also widely exploring this kind of special functions. Therefore, in this survey we emphasize on the results for the Mellin-Barnes type Received: April 6, 2020 c © 2020 Academic Publications Correspondence author 550 J. Paneva-Konovska, V. Kiryakova contour integral representation of the multi-index Mittag-Leffler functions, and thus on their Mellin transform images. AMS Subject Classification: 30D20, 33E12, 44A20, 26A33