A study of fractional integral operators involving a certain generalized multi‐index Mittag‐Leffler function

Abstract

Motivated by the demonstrated potential for their applications in various research areas such as those in mathematical, physical, engineering, and statistical sciences, our main object in this paper is to introduce and investigate a fractional integral operator that contains a certain generalized multi‐index Mittag‐Leffler function in its kernel. In particular, we establish some interesting expressions for the composition of such well‐known fractional integral and fractional derivative operators as (for example) the Riemann‐Liouville fractional integral and fractional derivative operators, the Hilfer fractional derivative operator, and the above‐mentioned fractional integral operator with the generalized multi‐index Mittag‐Leffler function in its kernel. The main findings in this paper are shown to generalize the results that were derived earlier by Kilbas et al and Srivastava et al. Finally, in this paper, we derive integral representations for the product of 2 generalized multi‐index Mittag‐Leffler functions in terms of the familiar Fox‐Wright hypergeometric function.