Multiple (multiindex) Mittag–Leffler functions and relations to generalized fractional calculus

The classical Mittag–Leffler (M–L) functions have already proved their efficiency as solutions of fractional-order differential and integral equations and thus have become important elements of the fractional calculus’ theory and applications. In this paper we introduce analogues of these functions, depending on two sets of multiple (m-tuple, m⩾2 is an integer) indices. The hint for this comes from a paper by Dzrbashjan (Izv. AN Arm. SSR 13 (3) (1960) 21–63) related to the case m=2. We study the basic properties and the relations of the multiindex M–L functions with the operators of the generalized fractional calculus. Corresponding generalized operators of integration and differention of the so-called Gelfond–Leontiev-type, as well as Borel–Laplace-type integral transforms, are also introduced and studied.