Gel’fond-Leont’ev integration operators of fractional (multi-)order generated by some special functions

Abstract

In this paper we survey some author’s results and developments relating the so-called Gel’fond-Leont’ev (G-L) operators of generalized integration and differentiation, classes of special functions (SF) of generalized hypergeometric type and the operators of generalized fractional calculus (GFC). The G-L operators have been introduced by Gel’fond-Leont’ev [9] in the classes of analytic functions in disks ΔR = {|z| < R}, by means of of multipliers’ sequences composed by the coeffcients of suitable entire (generating) functions. Introducing classes of SF related to Fractional Calculus (FC), as the Mittag-Leffer (ML) function, the multi-index Mittag-Leffer (multi-ML) function and its various particular cases ([16]–[18]), we specify the G-L operators generated by these entire functions. It is shown that in these cases, the G-L operators can be extended to analytic functions in wider complex domains Ω starlike with respect to the origin z = 0 and represented by operators of the Generalized Fractional Calculus (GFC), Kiryakova [14], i.e. operators of generalized integration and differentiation of arbitrary fractional multi-order. Illustrative examples and some open problems are proposed.