Abstract
Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, which is ideally suited for extending certain fractional-calculus operators into the complex plane. Complex analysis has been underused in combination with fractional calculus, especially with newly developed operators like those with Mittag-Leffler kernels. Here we show the natural analytic continuations of these operators using the modified Mittag-Leffler functions defined in this paper.
Highlights
The study of special functions has been a significant subfield of mathematical analysis for decades, connecting with other areas such as differential equations, fractional calculus, and mathematical physics [1,2,3]
We have already mentioned how Mittag-Leffler functions emerge naturally from the study of fractional calculus and fractional differential equations. They appear frequently as the kernels of fractional integral and derivative operators. Many such operators are special cases of the Prabhakar fractional calculus [14,25], which is based on the 3-parameter Mittag-Leffler function (3), and which itself can be seen as a special case of some even more general operators [17,26]
In the paper [36], a series formula is given for the Mittag-Leffler function Eα(z) which is valid for negative real numbers α, by using a functional equation that emerges from the complex integral representation
Summary
The study of special functions has been a significant subfield of mathematical analysis for decades, connecting with other areas such as differential equations, fractional calculus, and mathematical physics [1,2,3]. They appear frequently as the kernels of fractional integral and derivative operators Many such operators are special cases of the Prabhakar fractional calculus [14,25], which is based on the 3-parameter Mittag-Leffler function (3), and which itself can be seen as a special case of some even more general operators [17,26]. Among the most intensively used types of fractional calculus in the last few years are the so-called Atangana–Baleanu operators, defined in [27] using the 1-parameter Mittag-Leffler function (1). We shall perform a rigorous analysis of these modified Mittag-Leffler functions, their domains and convergence properties, and use them to extend the Atangana–Baleanu and Prabhakar fractional-calculus operators into the setting of complex variables.
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