MULTI-INDEX MITTAG-LEFFLER FUNCTIONS, GENERALIZED FRACTIONAL CALCULUS AND LAPLACE TYPE TRANSFORM

Abstract

Recently, the interest in the Mittag-Leffler (M-L) functions has increased in view of their important role and applications in fractional calculus and fractional order differential and integral equations (FODIEs). We have introduced and studied analogues of these functions, depending on two sets of multi-indices. They generate operators of the generalized fractional calculus (Kiryakova, 1994: Generalized Fractional Calculus and Applications, Longman and J. Wiley), and Laplace-type integral transforms involving the Fox H-function. These new special functions are “fractional indices” analogues of Delerue's hyper-Bessel functions and the respective differential and integral equations are fractional (multi-)order analogues of the Bessel type equations arising often in problems of mathematical physics and engineering.