On Multi-Index Mittag–Leffler Function of Several Variables and Fractional Differential Equations

In this paper, we have studied a unified multi-index Mittag–Leffler function of several variables. An integral operator involving this Mittag–Leffler function is defined, and then, certain properties of the operator are established. *e fractional differential equations involving the multi-index Mittag–Leffler function of several variables are also solved. Our results are very general, and these unify many known results. Some of the results are concluded at the end of the paper as special cases of our primary results.

If we make c � K � 1 in (1) it reduces to the multi-index M-L function studied by Kiryakova [16,17].
A multivariable extension of Mittag-Leffler function widely studied by Gautam [18], and also by Saxena et al. ([19], p. 547, Equation (7.1)), is defined and represented as follows: where λ, c j , l j , Motivated by the work on these functions, we consider here the subsequent multivariable and multi-index Mittag-Leffler function: where We have also studied here, the integral operator involving the function defined by (3), as follows: with e Riemann-Liouville fractional derivative operator D α 0+ is defined as follows [20]: where (I α a+ ψ)(x) is the fractional integral operator defined by e elementary definitions are also required to be mentioned as follows.
e Laplace transform of fractional derivative (D α 0+ f)(x) is given as Also, the formula for Laplace transform is

Main Results
and . . , m; and i � 1, . . . , r with the initial condition (I 1−α 0+ y)(0+) � c (c is an arbitrary constant) and solution of differential equations existing in the space L(0, ∞), then eorems 2-4 are stated in the following form.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest regarding the publication of this article.