On the Sheffer-type polynomials related to the Mittag-Leffler functions: applications to fractional evolution equations

TL;DR

This paper introduces fractional Hermite and Mittag-Leffler polynomials related to Mittag-Leffler functions, demonstrating their use in solving fractional Fokker-Planck equations with Caputo derivatives, and establishing their Sheffer-type generating functions.

Contribution

It develops new Sheffer-type polynomials based on Mittag-Leffler functions and applies them to fractional evolution equations, expanding the toolkit for fractional differential equations.

Findings

Derived generating functions for the polynomials.

Showed the polynomials solve fractional Fokker-Planck equations.

Established the Sheffer-type nature of the polynomials.

Abstract

We present two types of polynomials related to the Mittag-Leffler function namely the fractional Hermite polynomial and the Mittag-Leffler polynomial. The first modifies the Hermite polynomial and the second one is a refashioned Laguerre polynomial. The fractional Hermite and the Mittag-Leffler polynomials are used to solve {the Cauchy problems for} the fractional Fokker-Planck equation where the fractional derivative is taken in the Caputo sense with respect to time and/or space. The generating functions of these two kinds of polynomials are also calculated and they indicate that these polynomials belong to the Sheffer type.