Some Uniﬁed Integrals for Generalized Mittag-Lefﬂer Functions

: Here, we ascertain generalized integral formulas concerning the product of the generalized Mittag-Lefﬂer function. These integral formulas are described in the form of the generalized Lauricella series. Some special cases are also presented in terms of the Wright hypergeometric function.

The generalized Lauricella series is defined as [13]: where, for convenience, Here, we recommend [13,14], for the convergence conditions of the generalized Lauricella series.

Main Results
In the present paper, first, we introduce two main Theorems by using the product of the generalized Mittag-Leffler function in Equation (1). These Theorems are defined in the form of the series in Equation (4). We insert the product of the generalized Mittag-Leffler function into the integrand of Equation (6), to establish our main results.

Theorem 1.
For the product of the generalized Mittag-Leffler function, the following integral holds: where, (µ, ζ, Proof of Theorem 1. Let us denote the left-hand side of Equation (10), by I. Then use Equation (1), in the integrand of Equation (10). We have: By change of the order of summation and integration: By using (6), in (12), we get: Then ordering all the non variable terms and putting ζ + l 1 + ... + l m = Γ(ζ+l 1 +...+l m +1) Γ(ζ+l 1 +...+l m ) , we have: Now, multiply and divide the above equation with , we have: By using Equation (4), in the above equation and rearranging the terms, we get our desired result, Equation (10).
Proof of Theorem 2. By following the same rule that lead to the result in Equation (10), we get our desired result, Equation (16).
Taking i = 1 in Equations (10) and (16), following the same procedure as above, and then using Equation (8), we have the following result, which holds true for the Prabhakartype function [11].

Γ(ν)
. We get: Then, by using the definition of the Lauricella series (4), in the above equation and rearranging the terms, we get our desired result, Theorem 3.
Proof of Theorem 4. Let us denote the left-hand side of Equation (25) by K. Then, we use the generalized Mittag-Leffler function in the integrand of unified integral in Equation (7). We have: By the change of the order of summation and integration, we have: We obtain the following expression: Now, using the unified integral of Equation (7). We obtained the following: Multiply and divide the above equation with Γ(ν), Γ(ω),Γ(ν + ω), Γ(ζ 1 ) · · · Γ(ζ m ), and using the Gamma function property as (ν) l 1 +···+l m = Γ(ν+l 1 +···+l m ) Γ(ν) . We have: Then, we use the definition of the Lauricella series (Equation (4)) in the above equation, and, rearranging the terms, we get our desired result, Theorem 4. (19), following the same procedure as above, and using Equation (8), we have the following result, which holds true for the Prabhakar-type function [11], as follows:

Concluding Remark
We conclude our analysis by remarking that the results presented in this article are new and important for the class of Mittag-Leffler functions. By choosing different values of parameters, we can extract several sub-results from our main results. Further research will focus on basic applications and examples of these results for various research areas.
Funding: Shilpi Jain is very thankful to the funding agency SERB (project number: MTR/2017/000194) for providing necessary financial support for the present study.
Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.