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The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus. (English) Zbl 1189.33034
Summary: 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 $p{\Psi }q$, as a special case of the Fox $H$-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.
##### MSC:
 33E10 Lamé, Mathieu, and spheroidal wave functions 26A33 Fractional derivatives and integrals (real functions) 33-02 Research monographs (special functions)