On the fractional calculus of multivariate Mittag-Leffler functions

Abstract

Multivariate Mittag-Leffler functions are a strong generalisation of the univariate and bivariate Mittag-Leffler functions which are known to be important in fractional calculus. We consider the general functional operator defined by an integral transform with a multivariate Mittag-Leffler function in the kernel. We prove an expression for this operator as an infinite series of Riemann-Liouville integrals, thereby demonstrating that it fits into the established framework of fractional calculus, and we show the power of this series formula by straightforwardly deducing many facts, some new and some already known but now more quickly proved, about the original integral operator. We illustrate our work here by calculating some examples both analytically and numerically, and comparing the results on graphs. We also define fractional derivative operators corresponding to the established integral operator. As an application, we consider some Cauchy-type problems for fractional integro-differential equations involving this operator, where existence and uniqueness of solutions can be proved using fixed point theory. Finally, we generalise the theory by applying the same operators with respect to arbitrary monotonic functions.