The aim of this paper is to study various properties of Mittag-Leffler (M-L) function. Here we establish two theorems which give the image of this M-L function under the generalized fractional integral operators involving Fox’s H-function as kernel. Corresponding assertions in terms of Euler, Mellin, Laplace, Whittaker, and K-transforms are also presented. On account of general nature of M-L function a number of results involving special functions can be obtained merely by giving particular values for the parameters.

We establish fractional integral and derivative formulas by using Marichev-Saigo-Maeda operators involving the S-function. The results are expressed in terms of the generalized Gauss hypergeometric functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals and derivatives are presented. Also we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.Hindawi Abstract and Applied Analysis Volume 2019, Article ID 6487687, 19 pages https://doi.org/10.1155/2019/6487687 . . . − / ] ] ] ] . (116) Proof. The proof of Theorem 38 is a similar manner of Theorem 37. Theorem 39. Let , , V, V , , , , , , ∈ C, ∈ R, , ∈ N, R( ) > R( ), > 0, and Abstract and Applied Analysis 15 < + 1, where ( = 1, . . . , ), such that R( ) > 0, R( ) > max{0, R( − − − V ), R(V − )}, then the following fractional derivative formula holds true: ((

The objective of this article is to present several new integral equalities involving the multivariate Mittag-Leffler functions which are associated with the Laguerre polynomials. To emphasize our main results, we also consider some important special cases. The main results of our paper are quite general in nature and yield a very large number of integral equalities involving polynomials occurring in problems of mathematical analysis and mathematical physics.

The aim of this paper is to evaluate two integral formulas involving a finite product of the generalized Bessel function of the first kind and multivariable polynomial functions which results are expressed in terms of the generalized Lauricella functions. The major results presented here are of general character and easily reducible to unique and well-known integral formulae.

In this paper, a beta operator is used with Caputo Marichev-Saigo-Maeda (MSM) fractional differentiation of extended Mittag-Leffler function in terms of beta function. Further in this paper, some corollaries and consequences are shown that are the special cases of our main findings. We apply the beta operator on the right-sided MSM fractional differential operator and on the left-sided MSM fractional differential operator. We also apply beta operator on the right-sided MSM fractional differential operator with Mittag-Leffler function and the left-sided MSM fractional differential operator with Mittag-Leffler function.