Certain Uniﬁed Integrals Involving a Multivariate Mittag–Lefﬂer Function

: A remarkably large number of uniﬁed integrals involving the Mittag–Lefﬂer function have been presented. Here, with the same technique as Choi and Agarwal, we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag–Lefﬂer function, which are expressed in terms of the generalized Lauricella series due to Srivastava and Daoust. We also present some interesting special cases.

The interested reader may refer to papers on the subject for more details [1,2].
Motivated by above works here, with the same technique as Choi and Agarwal [12], we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag-Leffler function, which are expressed in terms of the generalized Lauricella series due to Srivastava and Daoust [1] given in Equation (1).
We also require the generalized hypergeometric function p ψ q [z] (see [17,18]) defined by

Γ(1+η)
, we find that Finally, we interpret the multiple series in (15) as a special case of the general hypergeometric series in several variables defined by (1). Thus, we are led to the assertion (12). The assertion (13) of the Theorem 2.2 can be proved by a similar argument.

Special Cases
In this section, we derive certain new integral formulas involving Prabhakar-type Mittag-Leffler functions [15] in the integrands of (12) and (13), respectively.
By setting m = 1 in (12) and (13) and applying the expression in (1) to the identities, we obtain two integral formulas, as stated in Corollary 1 and 2, respectively.
with the convergence conditions followed by Theorem 1.

Corollary 2.
∞ 0 with the convergence conditions followed by Theorem 2.
It is easily seen that, if we set γ = 1 in (16) and (17), we obtain new integral formulas, as stated in Corollary 3 and 4, respectively.
with the convergence conditions followed by Theorem 1.

Corollary 4.
∞ 0 with the convergence conditions followed by Theorem 2.

Conclusions
We conclude our investigation by remarking that the results presented here can be easily converted in terms of the known and new integral formulas after small changes in parameters. We are investigating the main results to find potentially useful applications in a variety of areas.