Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

TL;DR

This paper introduces a new series representation for a fractional calculus operator involving generalized Mittag-Leffler functions, enabling the derivation of product and chain rules and exploring fractional iteration.

Contribution

It derives a novel series expression for the Prabhakar integral transform, connecting it to classical fractional integrals and expanding fractional calculus models.

Findings

New series formula for Prabhakar transform derived.

Product and chain rules extended to generalized fractional calculus.

Application to fractional iteration and semigroup properties.

Abstract

We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann-Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties.