Some properties of the Mittag-Leffler functions and their relation with the Wright functions

Muhammet Kurulay,Mustafa Bayram

doi:10.1186/1687-1847-2012-181

Muhammet Kurulay, Mustafa Bayram

Open Access

https://doi.org/10.1186/1687-1847-2012-181

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Abstract

This paper is a short description of our recent results on an important class of the so-called Mittag-Leffler functions, which became important as solutions of fractional order differential and integral equations, control systems and refined mathematical models of various physical, chemical, economical, management and bioengineering phenomena. We have studied the Mittag-Leffler functions as their typical representatives, including many interesting special cases that have already proven their usefulness in fractional calculus and its applications. We obtained a number of useful relationships between the Mittag-Leffler functions and the Wright functions. The Wright function plays an important role in the solution of a linear partial differential equation. The Wright function, which we denote by , is so named in honor of Wright who introduced and investigated this function in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions. MSC:33E12.

Highlights

1.1 The Mittag-Leffler function The Mittag-Leffler function is an important function that finds widespread use in the world of fractional calculus
Just as the exponential naturally arises out of the solution to integer order differential equations, the Mittag-Leffler function plays an analogous role in the solution of non-integer order differential equations
1.2 The Wright function The Wright function plays an important role in the solution of a linear partial differential equation

Summary

Introduction

1.1 The Mittag-Leffler function The Mittag-Leffler function is an important function that finds widespread use in the world of fractional calculus. Just as the exponential naturally arises out of the solution to integer order differential equations, the Mittag-Leffler function plays an analogous role in the solution of non-integer order differential equations. The exponential function itself is a very special form, one of an infinite set of these seemingly ubiquitous functions. The standard definition of Mittag-Leffler [ ] is given as follows:. A two-parameter function of the M-L (Mittag-Leffler) type is defined by the series expansion [ ]. The M-L function provides a simple generalization of the exponential function because of the substitution of n! E / , = ez erfc(–z), where erf (erfc) denotes the (complementary) error function defined as [ ]. By means of the series representation, a generalization of ( ) and ( ) is introduced by Prabhakar [ ] as.

Note that

Proof We have