A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

Abstract

We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional integral operator which has many interesting properties. The motivation for these definitions is twofold: firstly, their link with some fundamental fractional differential equations involving two independent fractional orders, and secondly, the fact that they emerge naturally from certain applications in bioengineering. Keywords Mittag-Leffler functions • Fractional integrals • Fractional derivatives • Fractional differential equations • Bivariate Mittag-Leffler functions ρ α,β (x) and E α,β,γ (x), the "two-parameter" and "three-parameter" Mittag-Leffler functions, being defined by power series similar to the Communicated by Roberto Garrappa.

On a Five-Parameter Mittag-Leffler Function and the Corresponding Bivariate Fractional Operators

“…A very important property of the bivariate fractional calculus defined in this paper is that it has a semigroup property in the variables β 1 , β 2 , γ, as expressed by the following theorem. There are several different ways to prove this result, as in [18], and we mention here two of them.…”

“…In this section, we move on from functions to operators. Having established the five-parameter Mittag-Leffler function and some of its properties, we now wish to define a fractional integral operator using this function as a kernel, following in the footsteps of other papers [10,18,46] which defined new models of fractional calculus by using various types of Mittag-Leffler functions as kernels.…”

“…• A Mittag-Leffler function of two variables with four parameters and a Mittag-Leffler function of three variables with five parameters were recently defined [18,19] and used to solve multi-order systems of fractional differential equations [20,21]. These are closely related to the general function considered by Luchko et al, but they are not special cases of it.…”

“…These are closely related to the general function considered by Luchko et al, but they are not special cases of it. Used as kernels, they also give rise to new models of fractional calculus with a semigroup property [18,21]. For other types of bivariate Mittag-Leffler function that have been defined in the literature, we refer to the papers [22][23][24].…”

On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative

“…Fractional calculus and its applications have been intensively investigated for a long time by many researchers in numerous disciplines and attention to this subject has grown tremendously. By making use of the concept of the fractional derivatives and integrals, various extensions of them have been introduced [27][28][29][30], and authors have gained different perspectives in many areas such as engineering, physics, economics, biology, statistics [31,32]. One of the generalizations of fractional derivatives is Riemann-Liouville k-fractional derivative operator studied in [24,25,33].…”