The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus

This paper is a short description of our recent results on an important class of the so-called “Special Functions of Fractional Calculus” (SF of FC), which became important as solutions of fractional order (or multi-order) differential and integral equations, control systems and refined mathematical models of various physical, chemical, economical, management, bioengineering phenomena. Basically, under “SF of FC” we mean the Wright generalized hypergeometric function pΨqpΨq, as a special case of the Fox HH-function. We have introduced and studied the multi-index Mittag-Leffler functions as their typical representatives, including many interesting special cases that have already proven their usefulness in FC and its applications. Some new results are also presented and open problems are discussed.