Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach

Alexander Apelblat

doi:10.3390/math8050657

Abstract

In this work, properties of one- or two-parameter Mittag-Leffler functions are derived using the Laplace transform approach. It is demonstrated that manipulations with the pair direct–inverse transform makes it far more easy than previous methods to derive known and new properties of the Mittag-Leffler functions. Moreover, it is shown that sums of infinite series of the Mittag-Leffler functions can be expressed as convolution integrals, while the derivatives of the Mittag-Leffler functions with respect to their parameters are expressible as double convolution integrals. The derivatives can also be obtained from integral representations of the Mittag-Leffler functions. On the other hand, direct differentiation of the Mittag-Leffler functions with respect to parameters produces an infinite power series, whose coefficients are quotients of the digamma and gamma functions. Closed forms of these series can be derived when the parameters are set to be integers.

Highlights

At the beginning of the previous century, the exponential function was generalized by the Swedish mathematician G.M
The Mittag-Leffler functions play an important role in fractional calculus, solution of systems with fractional differential, and integral equations [9,10]
The differentiation operations will lead to power series with coefficients being quotients of psi and gamma functions. These series can be evaluated in a closed form, i.e., in terms of elementary and special functions

Summary

Introduction

At the beginning of the previous century, the exponential function was generalized by the Swedish mathematician G.M. In my monograph devoted to the Volterra functions [19], I presented in Appendix A some representations of the Mittag-Leffler functions in terms of other special functions They can be derived directly using the Laplace transform technique when applied to. The special forms of the Laplace transforms of Eα (±tα ) and Eα,β (±tα ) functions will be studied extensively to establish known properties of the Mittag-Leffler functions and to derive new functional relations. The differentiation operations will lead to power series with coefficients being quotients of psi and gamma functions In some cases, these series can be evaluated in a closed form, i.e., in terms of elementary and special functions. This results from the fact that the Mittag-Leffler functions are available as the build-in functions in the MATHEMATICA program

Properties of the Mittag-Leffler Functions in the Laplace Transform Approach

Conclusions