Representation of the Mittag-Leffler function through the exponential functions in the case of rational derivatives

TL;DR

This paper presents a novel representation of the Mittag-Leffler function using exponential functions specifically for rational derivatives, avoiding integral formulas and expanding the analytical tools available for fractional calculus.

Contribution

It provides explicit formulas for the Mittag-Leffler function in terms of exponential functions for rational derivatives, without involving integrals, which is a new approach compared to prior work.

Findings

Formulas for Mittag-Leffler function without integrals

Applicable to rational derivatives of order m/n

Simplifies analytical and computational methods

Abstract

In this paper the Mittag-Leffler function is given through the exponential functions for any rational derivatives of m/n order, where m<n, n>1 are natural irreducible numbers (if n=1 then m is also equal to unity). Unlike the previous papers the given formulas do not contain integrals.