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. Finally, in an appendix we sketch some historical aspects related to the author’s acquaintance with this function.
Highlights
IntroductionThe special transcendental function known as the Mittag-Leffler function has attracted the increasing attention of researchers because of its key role in treating problems related to integral and differential equations of fractional order
Particular mention would be for the 1955 Handbook of High Transcendental Functions of the Bateman project [6], where this function was treated in Volume 3, in the chapter devoted to miscellaneous functions
We find some information on the Mittag-Leffler functions in any treatise on Fractional Calculus but for more details we refer the reader to the surveys of Haubold, Mathai and Saxena [12] and by Van
Summary
The special transcendental function known as the Mittag-Leffler function has attracted the increasing attention of researchers because of its key role in treating problems related to integral and differential equations of fractional order. We have not considered other applications of the Mittag-Leffler functions including, for example, anomalous diffusion theory in terms of fractional and generalized Langevin equations. On this respect we refer the readers to the articles of the author, see References [15,16], and to the recent book by Sandev and Tomovski [17] and references therein. For many items related to the Mittag-Leffler functions we refer again to the treatise by Gorenflo et al [14]
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