The Multi-index Mittag-Leffler Functions and Their Applications for Solving Fractional Order Problems in Applied Analysis.

During the last few decades, differential equations and systems of fractional order (that is arbitrary one, not necessarily integer) begun to play an important role in modeling of various phenomena of physical, engineering, automatization, biological and biomedical, chemical, earth, economics, social relations, etc. nature. The so-called Special Functions of Fractional Calculus (SF of FC) provide an important tool of Fractional Calculus (FC) and Applied Analysis (AA). In particular, they are often used to represent the solutions of fractional differential equations in explicit form. Among the most popular representatives of the SF of FC are: the Mittag-Leffler (ML) function, the Wright generalized hypergeometric function pΨq, the more general Fox H-function, and the Inayat-HussainH-function. The classical Special Functions (called also SF of Mathematical Physics), including the orthogonal polynomials, and the pFq-hypergeometric functions fall in this scheme as examples of the simpler Meijer G-function. In this survey talk, we overview the properties and some applications of an important class of SF of FC, introduced for the first time in our works. For integer m>1 and arbitrary real (or complex, under suitable restrictions) indices ρ1,...,ρm>0 and μ1,...,μm, we define the multi-index (vector-index) Mittag-Leffler functions by: <DFORMULA>E(1/ρi),(μi)(z) = E)1/ρi),(μi)(m)(z) = <SM><LU>K=0</LU><BU>∞</BU></SM>zk/Γ(μ1+k/ρ1)...Γ(μm+k/ρm) = 1Ψm<FENCE><ARRAY><ARRAYROW><ARRAYCEL>(1,1)</ARRAYCEL></ARRAYROW><ARRAYROW><ARRAYCEL>(μ1,1/ρi)1m;z</ARRAYCEL></ARRAYROW></ARRAY></FENCE> = H1,m+11,1<FENCE>-z|<ARRAY><ARRAYROW><ARRAYCEL>(0,1)</ARRAYCEL></ARRAYROW><ARRAYROW><ARRAYCEL>(0,1),(1-μi,1/ρi)1m</ARRAYCEL></ARRAYROW></ARRAY></FENCE>.</DFORMULA> We propose also a list of examples of SF of FC that are E(1/ρi),(μi)-functions and play important role in pure mathematics and in solving problems from natural, applied and social sciences, and state some open problems. [ABSTRACT FROM AUTHOR]