Why the Mittag-Leffler function can be considered the Queen function of the Fractional Calculus?

TL;DR

This survey highlights the central role of the Mittag-Leffler function in fractional calculus, emphasizing its analytical properties, applications in stochastic processes and diffusion equations, and potential for future scientific impact.

Contribution

It provides a comprehensive overview of the Mittag-Leffler function's properties, applications, and numerical methods, establishing it as the key function in fractional calculus.

Findings

Mittag-Leffler function as solution to fractional differential equations

Applications in stochastic processes and fractional diffusion-wave equations

Numerical methods for computing Mittag-Leffler functions

Abstract

In this survey we stress the importance of the higher transcendental Mittag-Leffler function in the framework of the Fractional Calculus. We first start with the analytical properties of the classical Mittag-Leffler function as derived from being the solution of the simplest fractional differential equation governing relaxation processes. Through the Sections of the text we plan to address the reader in this pathway towards the main applications of the Mittag-Leffler function that has induced us in the past to define it as the Queen Function of the Fractional Calculus. These applications concern some noteworthy stochastic processes and the time fractional diffusion-wave equation. We expect that in the next future this function will gain more credit in the science of complex systems. In Appendix A we sketch some historical aspects related to the author's acquaintance with this function. Finally, with respect to the published version in Entropy, we add Appendix B where we briefly refer to the numerical methods nowadays available to compute the functions of the Mittag-Leffler type.