Some Unified Integrals for Generalized Mittag-Leffler Functions

Abstract

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.

1. Introduction and Preliminaries

There are many properties like integral formulas and differential formulas concerning a diversity of special functions; specifically, hypergeometric functions and Mittag-Leffler functions have been discussed by numerous authors [1,2,3,4,5,6,7,8,9]. Recently, integral formulas concerning a generalized Mittag-Leffler function have been introduced by Jain et al. [10].

Currently, we are using a product of the generalized Mittag-Leffler function to ascertain some generalized integral formulas, which was described by Prabhakar [11]:

$$\begin{array}{c}\hfill \prod _{i=1}^m\left(E_{\left({\rho }_i\right),{\zeta }_i}^{\left({\gamma }_i\right)}\left(z_i\right)\right)\equiv E_{\left({\rho }_1\right),{\zeta }_1}^{\left({\gamma }_1\right)}\left(z_1\right)E_{\left({\rho }_2\right),{\zeta }_2}^{\left({\gamma }_2\right)}\left(z_2\right)\cdots E_{\left({\rho }_m\right),{\zeta }_m}^{\left({\gamma }_m\right)}\left(z_m\right)\\ \hfill =\sum _{l_1=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}{z}_1^{l_1}}{\mathsf{\Gamma }({\zeta }_1+{\rho }_1{l}_1)\left(l_1\right)!}\sum _{l_2=0}^{\infty }\frac{{\left({\gamma }_2\right)}_{l_2}{z}_2^{l_2}}{\mathsf{\Gamma }({\zeta }_2+{\rho }_2{l}_2)\left(l_2\right)!}\cdots \sum _{l_m=0}^{\infty }\frac{{\left({\gamma }_m\right)}_{l_m}{z}_m^{l_m}}{\mathsf{\Gamma }({\zeta }_m+{\rho }_m{l}_m)\left(l_m\right)!},\end{array}$$

(1)

where, (${\zeta }_i,{\gamma }_i,{\rho }_i,z_i\in \mathbb{C}$, $\Re \left({\rho }_i\right)>0$, $i=1,\cdots ,m).$

Remark 1.

If we set $i=1$ in (1.1), it becomes the generalized Mittag-Leffler function, due to Prabhakar [11]:

$$E_{\rho ,\zeta }^{\gamma }\left(z\right)=\sum _{l=0}^{\infty }\frac{{\left(\gamma \right)}_l{z}^l}{\mathsf{\Gamma }(\zeta +l\rho )l!},$$

(2)

where, ($\gamma ,\rho ,\zeta \in \mathbb{C},\Re \left(\rho \right),\Re \left(\zeta \right)>0,z\in \mathbb{C}$), with ${\left(\zeta \right)}_m$ defined by the Pochhammer symbol as (see [12]):

$$\begin{array}{ll}{\left(\zeta \right)}_m:& =\{\begin{array}{l}1(m=0)\\ \zeta (\zeta +1)\dots (\zeta +m-1)(m\in \mathbb{N}:=\{1,2,3,\dots \})\end{array}\\ & =\frac{\mathsf{\Gamma }(\zeta +m)}{\mathsf{\Gamma }\left(\zeta \right)}(\zeta \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}),\end{array}$$

(3)

and ${\mathbb{Z}}_0^{-}$ represents the set of non-positive integers.

The generalized Lauricella series is defined as [13]:

$$\begin{gathered} F_{C:D^{\left(1\right)};\cdots ;D^{\left(m\right)}}^{A:B^{\left(1\right)};\cdots ;B^{\left(m\right)}}\left(\begin{array}{c}{z}_1\\ ⋮\\ z_m\end{array}\right)=F_{C:D^{\left(1\right)};\cdots ;D^{\left(m\right)}}^{A:B^{\left(1\right)};\cdots ;B^{\left(m\right)}}(\begin{array}{c}[\left(p\right):{\eta }^{\left(1\right)},\dots ,{\eta }^{\left(m\right)}]:\\ [\left(r\right):{\psi }^{\left(1\right)},\dots ,{\psi }^{\left(m\right)}]:\end{array} \\ \begin{array}{c}[{\left(q\right)}^{\left(1\right)}:{\varphi }^{\left(1\right)}];\dots ;[{\left(q\right)}^{\left(m\right)}:{\varphi }^{\left(m\right)}];\\ [{\left(s\right)}^{\left(1\right)}:{\delta }^{\left(1\right)}];\dots ;[{\left(s\right)}^{\left(m\right)}:{\delta }^{\left(m\right)}];\end{array}{z}_1,\dots ,z_m), \\ =\sum _{l_1,\dots ,l_m=0}^{\infty }\Delta (l_1,\dots ,l_m)\frac{z_1^{l_1}}{l_1!}\cdots \frac{z_m^{l_m}}{l_m!}, \end{gathered}$$

(4)

where, for convenience,

$$\Delta (l_1,\dots ,l_m)=\frac{{\displaystyle \prod _{i=1}^A{\left(p_i\right)}_{l_1{\eta }_i^{\left(1\right)}+\cdots +l_m{\eta }_i^{\left(m\right)}}\prod _{i=1}^{B^{\left(1\right)}}{\left(q_i^{\left(1\right)}\right)}_{l_1{\varphi }_i^{\left(1\right)}}\cdots \prod _{i=1}^{B^{\left(m\right)}}{\left(q_i^{\left(m\right)}\right)}_{l_m{\varphi }_i^{\left(m\right)}}}}{{\displaystyle \prod _{i=1}^C{\left(r_i\right)}_{l_1{\psi }_i^{\left(1\right)}+\cdots +l_n{\psi }_i^{\left(m\right)}}\prod _{i=1}^{D^{\left(1\right)}}{\left(s_i^{\left(1\right)}\right)}_{l_1{\delta }_i^{\left(1\right)}}\cdots \prod _{i=1}^{D^{\left(m\right)}}{\left(s_i^{\left(m\right)}\right)}_{l_n{\delta }_i^{\left(m\right)}}}}.$$

(5)

Here, we recommend [13,14], for the convergence conditions of the generalized Lauricella series.

The Oberhettinger’s integral formula is given by [15]:

$${\int }_0^{\infty }{y}^{\mu -1}{\left(y+b+\sqrt{y^2+2by}\right)}^{-\zeta }dy=2\zeta b^{-\zeta }{\left(\frac{b}{2}\right)}^{\mu }\frac{\mathsf{\Gamma }\left(2\mu \right)\mathsf{\Gamma }(\zeta -\mu )}{\mathsf{\Gamma }(1+\zeta +\mu )},$$

(6)

here $0<\Re \left(\mu \right)<\Re \left(\zeta \right)$.

The unified integral is defined by Edward as [16]:

$${\int }_0^1{\int }_0^1{\left(v^{\prime }\right)}^{\nu }{\left(1-u^{\prime }\right)}^{\nu -1}{\left(1-v^{\prime }\right)}^{\omega -1}{\left(1-u^{\prime }{v}^{\prime }\right)}^{1-\nu -\omega }d{u}^{\prime }d{v}^{\prime }=\frac{\mathsf{\Gamma }\left(\nu \right)\mathsf{\Gamma }\left(\omega \right)}{\mathsf{\Gamma }(\nu +\omega )},$$

(7)

here, $\Re \left(\nu \right)>0$,$\Re \left(\omega \right)>0$.

We also require the generalized hypergeometric function ${}_r{\psi }_s\left[z\right]$ defined as [17]:

$${}_r{\psi }_s\left[z\right]=\sum _{t=0}^{\infty }\frac{{\prod }_{i^{\prime }=1}^r\mathsf{\Gamma }(a_{i^{\prime }}+{\alpha }_{i^{\prime }}t)}{{\prod }_{j^{\prime }=1}^s\mathsf{\Gamma }(b_{j^{\prime }}+{\beta }_{j^{\prime }}t)}\frac{z^t}{t!},$$

(8)

provided that,

$r,s\in N_0=N\bigcup \left\{0\right\}$; $a_{i^{\prime }},b_{j^{\prime }}\in \mathbb{C}$; ${\alpha }_{i^{\prime }},{\beta }_{j^{\prime }}\in \mathbb{R}$; ${\alpha }_{i^{\prime }},{\beta }_{j^{\prime }}\ne 0$; $i^{\prime }=1,....r$; $j^{\prime }=1,...,s$.

The generalized Mittag-Leffler function is [18]:

$$\begin{array}{c}\hfill E_{{\rho }_i,\zeta }^{{\gamma }_i}\left(\frac{a_j{y}^{\prime }(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)=\\ \hfill \frac{{\left({\gamma }_1\right)}_{l_1}\cdots {\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }(\zeta +{\rho }_1{l}_1+\cdots {\rho }_m{l}_m)}\left({\left(\frac{a_1{y}^{\prime }(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)}^{l_1}\frac{1}{l_1!}\cdots {\left(\frac{a_m{y}^{\prime }(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)}^{l_m}\frac{1}{l_m!}\right),\end{array}$$

(9)

here, $i=1,...,m.$

2. 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:

$$\begin{array}{c}{\int }_0^{\infty }{y}^{\mu -1}{\left(y+b+\sqrt{y^2+2by}\right)}^{-\zeta }\prod _{i=1}^m\left(E_{\left({\rho }_i\right),{\zeta }_i}^{\left({\gamma }_i\right)}\left(\frac{z_i}{y+b+\sqrt{y^2+2by}}\right)\right)dy\\ =2^{1-\mu }{b}^{\mu -\zeta }\mathsf{\Gamma }\left(2\mu \right)\frac{\mathsf{\Gamma }(1+\zeta )}{\mathsf{\Gamma }\left(\zeta \right)}\frac{\mathsf{\Gamma }(\zeta -\mu )}{\mathsf{\Gamma }(1+\zeta +\mu )}\frac{1}{\mathsf{\Gamma }\left({\zeta }_1\right)\cdots \mathsf{\Gamma }\left({\zeta }_m\right)}\times \\ F_{2:1;\dots ;1}^{2:1;\dots ;1}[\begin{array}{c}1+\zeta :1,\dots ,1,\left[\zeta -\mu :1,\dots ,1\right]:\\ \left[\zeta :1,\dots ,1\right],\left[1+\zeta +\mu :1,\dots ,1\right]:\end{array}\\ \begin{array}{c}[{\gamma }_1:1];\dots ;[{\gamma }_m;1];\\ \frac{}{[{\zeta }_1,{\rho }_1]};\dots ;\frac{}{[{\zeta }_m,{\rho }_m]};\end{array}\frac{z_1}{b},\dots ,\frac{z_m}{b}],\end{array}$$

(10)

where, ($\mu ,\zeta ,z_i\in \mathbb{C}$, $y>0$,( $i=1,\cdots ,m$)and$\Re \left(\mu \right)>0,\Re \left(\zeta \right)>0,\Re \left(\mu \right)<\Re \left(\zeta \right)$).

Proof of Theorem 1.

Let us denote the left-hand side of Equation (10), by $\mathcal{I}$. Then use Equation (1), in the integrand of Equation (10).

We have:

$$\begin{array}{c}\hfill \mathcal{I}={\int }_0^{\infty }{y}^{\mu -1}{\left(y+b+\sqrt{y^2+2by}\right)}^{-\zeta }\times \\ \hfill \sum _{l_1=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}}{\mathsf{\Gamma }({\zeta }_1+{\rho }_1{l}_1)}{\left(\frac{z_1}{(y+b+\sqrt{y^2+2by})}\right)}^{l_1}\frac{1}{l_1!}\cdots \times \\ \hfill \cdots \sum _{l_m=0}^{\infty }\frac{{\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }({\zeta }_m+{\rho }_m{l}_m)}{\left(\frac{z_m}{(y+b+\sqrt{y^2+2by})}\right)}^{l_n}\frac{1}{l_m!}dy.\end{array}$$

(11)

By change of the order of summation and integration:

$$\begin{gathered} \mathcal{I}=\sum _{l_1,\cdots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\dots {\left({\gamma }_l\right)}_{l_m}}{\mathsf{\Gamma }({\zeta }_1+{\rho }_1{l}_1)\mathsf{\Gamma }({\zeta }_2+{\rho }_2{l}_2)\cdots \mathsf{\Gamma }({\zeta }_m+{\rho }_m{l}_m)}\frac{z_1^{l_1}}{l_1!}\cdots \frac{z_m^{l_m}}{l_m!}\times \\ {\int }_0^{\infty }{y}^{\mu -1}{\left(y+b+\sqrt{y^2+2by}\right)}^{-(\zeta +l_1+\cdots +l_m)}dy. \end{gathered}$$

(12)

By using (6), in (12), we get:

$$\begin{array}{c}\hfill \mathcal{I}=\sum _{l_1,\dots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\dots {\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }({\zeta }_1+{\rho }_1{l}_1)\mathsf{\Gamma }({\zeta }_2+{\rho }_2{l}_2)\cdots \mathsf{\Gamma }({\zeta }_m+{\rho }_m{l}_m)}\times \\ \hfill 2(\zeta +l_1+...+l_m)b^{-(\zeta +l_1+....+l_m)}{\left(\frac{b}{2}\right)}^{\left(\mu \right)}\frac{\mathsf{\Gamma }\left(2\mu \right)\mathsf{\Gamma }(\zeta +l_1+....+l_m-\mu )}{\mathsf{\Gamma }(1+\zeta +\mu +l_1+...+l_m)}\frac{{\left(z_1\right)}^{l_1}}{l_1!}....\frac{{\left(z_m\right)}^{l_m}}{l_m!}.\end{array}$$

(13)

Then ordering all the non variable terms and putting $\zeta +l_1+...+l_m=\frac{\mathsf{\Gamma }(\zeta +l_1+...+l_m+1)}{\mathsf{\Gamma }(\zeta +l_1+...+l_m)}$, we have:

$$\begin{array}{c}\hfill \mathcal{I}=2^{1-\mu }{b}^{\mu -\zeta }\mathsf{\Gamma }\left(2\mu \right)\times \\ \hfill \sum _{l_1,\dots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\dots {\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }({\zeta }_1+{\rho }_1{l}_1)\mathsf{\Gamma }({\zeta }_2+{\rho }_2{l}_2)\cdots \mathsf{\Gamma }({\zeta }_m+{\rho }_m{l}_m)}\frac{\mathsf{\Gamma }(\zeta +l_1+\cdots +l_m+1)}{\mathsf{\Gamma }(\zeta +l_1+\cdots +l_m)}\times \\ \hfill \frac{\mathsf{\Gamma }(\zeta +l_1+\cdots +l_m-\mu )}{\mathsf{\Gamma }(1+\zeta +\mu +l_1+\cdots +l_m)}{\left(\frac{z_1}{b}\right)}^{l_1}\frac{1}{l_1!}\cdots {\left(\frac{z_m}{b}\right)}^{l_m}\frac{1}{l_m!}.\end{array}$$

(14)

Now, multiply and divide the above equation with $\mathsf{\Gamma }(\zeta +1)$, $\mathsf{\Gamma }\left(\zeta \right)$,$\mathsf{\Gamma }(\zeta -\mu )$, $\mathsf{\Gamma }(1+\zeta +\mu )$, $\mathsf{\Gamma }\left({\zeta }_1\right)\cdots \mathsf{\Gamma }\left({\zeta }_m\right)$ and using Gamma function property as ${(1+\zeta )}_{l_1+\cdots +l_m}=\frac{\mathsf{\Gamma }(1+\zeta +l_1+\cdots +l_m)}{\mathsf{\Gamma }(1+\zeta )}$, we have:

$$\begin{gathered} \mathcal{I}=2^{1-\mu }{b}^{\mu -\zeta }\mathsf{\Gamma }\left(2\mu \right)\frac{\mathsf{\Gamma }(1+\zeta )}{\mathsf{\Gamma }\left(\zeta \right)}\frac{1}{\mathsf{\Gamma }\left(\zeta \right)}\frac{\mathsf{\Gamma }(\zeta -\mu )}{\mathsf{\Gamma }(1+\zeta +\mu )}\frac{1}{\mathsf{\Gamma }\left({\zeta }_1\right)\cdots \mathsf{\Gamma }\left({\zeta }_m\right)}\times \\ \sum _{l_1,\cdots ,l_m=0}^{\infty }\frac{{(1+\zeta )}_{l_1+\cdots +l_m}{(\zeta -\mu )}_{l_1+\cdots +l_m}}{{\left(\zeta \right)}_{l_1+\cdots +l_m}{(1+\zeta +\mu )}_{l_1+\cdots +l_m}}\times \\ \frac{{\left(\gamma \right)}_{l_1}\cdots {\left(\gamma \right)}_{l_m}}{{\left({\zeta }_1\right)}_{{\rho }_1{l}_1}\cdots {\left({\zeta }_m\right)}_{{\rho }_m{l}_m}}\frac{{(z_1/b)}^{l_1}}{l_1!}\cdots \frac{{(z_m/b)}^{l_m}}{l_m!}. \end{gathered}$$

(15)

By using Equation (4), in the above equation and rearranging the terms, we get our desired result, Equation (10). □

Theorem 2.

For the product of the generalized Mittag-Leffler function, the following integral holds:

$$\begin{gathered} {\int }_0^{\infty }{y}^{\mu -1}{\left(y+b+\sqrt{y^2+2by}\right)}^{-\zeta }\prod _{i=1}^m\left(E_{\left({\rho }_i\right),{\zeta }_i}^{\left({\gamma }_i\right)}\left(\frac{y{z}_i}{y+b+\sqrt{y^2+2by}}\right)\right)dy \\ =2^{1-\mu }{b}^{\mu -\zeta }\mathsf{\Gamma }(\zeta -\mu )\frac{\mathsf{\Gamma }(1+\zeta )}{\mathsf{\Gamma }\left(\zeta \right)}\frac{\mathsf{\Gamma }\left(2\mu \right)}{\mathsf{\Gamma }(1+\zeta +\mu )}\frac{1}{\mathsf{\Gamma }\left({\zeta }_1\right)\cdots \mathsf{\Gamma }\left({\zeta }_m\right)}\times \\ {F}_{2:1;\dots ;1}^{2:1;\dots ;1}\left[\begin{array}{c}\left[1+\zeta :1,\dots ,1\right],\left[2\mu :2,\dots ,2\right]:\\ \left[\zeta :1,\dots ,1\right],\left[1+\zeta +\mu :2,\dots ,2\right]:\end{array} \\ \begin{array}{c}[{\gamma }_1:1];\dots ;[{\gamma }_m;1];\\ \frac{}{[{\zeta }_1,{\rho }_1]};\dots ;\frac{}{[{\zeta }_m,{\rho }_m]};\end{array}\frac{z_1}{2},\dots ,\frac{z_m}{2}\right], \end{gathered}$$

(16)

where, ($\mu ,\zeta ,z_i\in \mathbb{C}$, $y>0$,( $i=1,\cdots ,m$) and$\Re \left(\mu \right)>0,\Re \left(\zeta \right)>0,\Re \left(\mu \right)<\Re \left(\zeta \right)$).

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 Prabhakar-type function [11].

Corollary 1.

For the Prabhakar-type function [11], the following integral holds:

$$\begin{array}{c}\hfill {\int }_0^{\infty }{y}^{\mu -1}{\left(y+b+\sqrt{y^2+2by}\right)}^{-\zeta }{E}_{\rho ,\zeta }^{\gamma }\left(\frac{z}{y+b+\sqrt{y^2+2by}}\right)dy=\\ \hfill \frac{\mathsf{\Gamma }\left(2\mu \right)}{\mathsf{\Gamma }\left(\gamma \right)}{2}^{(1-\mu )}{b}^{(\mu -\zeta )}{}_3{\psi }_3\left[\begin{array}{c}(\gamma ,1),(\zeta -\mu ,1),(\zeta +1,1);\\ (\zeta ,\rho ),(1+\zeta +\mu ,1),(\zeta ,1);\end{array}\right]z/b.\end{array}$$

(17)

Corollary 2.

For the Prabhakar-type function [11], the following other integral also holds:

$$\begin{array}{c}\hfill {\int }_0^{\infty }{y}^{\mu -1}{\left(y+b+\sqrt{y^2+2by}\right)}^{-\zeta }{E}_{\rho ,\zeta }^{\gamma }\left(\frac{yz}{y+b+\sqrt{y^2+2by}}\right)dy=\\ \hfill \frac{\mathsf{\Gamma }(\zeta -\mu )}{\mathsf{\Gamma }\left(\gamma \right)}{2}^{(1-\mu )}{b}^{(\mu -\zeta )}{}_3{\psi }_3\left[\begin{array}{c}(\gamma ,1),(\zeta +1,1),(2\mu ,2);\\ (\zeta ,\rho ),(1+\zeta +\mu ,2),(\zeta ,1);\end{array}\right]z/2.\end{array}$$

(18)

Theorem 3.

The unified integral associated with the generalized Mittag-Leffler-type function holds:

$$\begin{gathered} {\int }_0^1{\int }_0^1{\left(y^{\prime }\right)}^{\nu }{\left(1-x^{\prime }\right)}^{\nu -1}{\left(1-y^{\prime }\right)}^{\omega -1}{\left(1-x^{\prime }{y}^{\prime }\right)}^{1-\nu -\omega }{E}_{{\rho }_i,\zeta }^{{\gamma }_i}\left(\frac{a_{i}y(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)d{x}^{\prime }d{y}^{\prime } \\ =\frac{\mathsf{\Gamma }\left(\nu \right)\mathsf{\Gamma }\left(\omega \right)}{\mathsf{\Gamma }(\nu +\omega )\mathsf{\Gamma }\left(\zeta \right)}{F}_{2:0;\dots ;0}^{2:1;\dots ;1}\left[\begin{array}{c}\left[\nu :1,\dots ,1\right],\left[\omega :1,\dots ,1\right]:\\ \left[\nu +\omega :2,\dots ,2\right],\left[\zeta :{\rho }_1,\dots ,{\rho }_m\right]:\end{array} \\ \begin{array}{c}[{\gamma }_1:1];\dots ;[{\gamma }_m;1];\\ -;\dots ;-;\end{array}{b}_1,\dots ,b_m,\right] \end{gathered}$$

(19)

here, ( $\Re \left(\nu \right)>0$, $\Re \left(\omega \right)>0$, $\zeta ,\gamma ,\rho ,z_i\in \mathbb{C}$, $y>0$,( $i=1,\cdots ,m$)and$0<\Re \left(\nu \right)<\Re \left(\omega \right)).$

Proof of Theorem 3.

Let us denote the left-hand side of Equation (19) by $\mathcal{J}$. Then, we use the generalized Mittag-Leffler function in the integrand of unified integral (7). We have:

$$\begin{array}{c}\hfill \mathcal{J}={\int }_0^1{\int }_0^1{\left(y^{\prime }\right)}^{\nu }{\left(1-x^{\prime }\right)}^{\nu -1}\times \\ \hfill {\left(1-y^{\prime }\right)}^{\omega -1}{\left(1-x^{\prime }{y}^{\prime }\right)}^{1-\nu -\omega }{E}_{{\rho }_i,\zeta }^{{\gamma }_i}\left(\frac{a_i{y}^{\prime }(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)d{x}^{\prime }d{y}^{\prime }.\end{array}$$

(20)

Then,

$$\begin{array}{c}\hfill \mathcal{J}={\int }_0^1{\int }_0^1{\left(y^{\prime }\right)}^{\nu }{\left(1-x^{\prime }\right)}^{\nu -1}{\left(1-y^{\prime }\right)}^{\omega -1}{\left(1-x^{\prime }{y}^{\prime }\right)}^{1-\nu -\omega }\times \\ \hfill \sum _{l_1,\cdots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\cdots {\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }(\zeta +{\rho }_1{l}_1+\cdots +{\rho }_m{l}_m)}{\left(\frac{a_1{y}^{\prime }(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)}^{l_1}\frac{1}{l_1!}\cdots \times \\ \hfill \cdots {\left(\frac{a_m{y}^{\prime }(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)}^{l_m}\frac{1}{l_m!}d{x}^{\prime }d{y}^{\prime }.\end{array}$$

(21)

Now, arranging the summation and integral part, we obtain:

$$\begin{array}{c}\hfill \mathcal{J}=\sum _{l_1,\dots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\dots {\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }({\zeta }_1+{\rho }_1{l}_1+\cdots +{\rho }_m{l}_m)}\frac{1}{l_1!}\cdots \frac{1}{l_m!}{a}_i^{l_1+\cdots +l_m}\times \\ \hfill {\int }_0^1{\int }_0^1{y}^{\nu +l_1+\cdots +l_m}{\left(1-x^{\prime }\right)}^{\nu +l_1+\cdots +l_m-1}\times \\ \hfill {\left(1-y^{\prime }\right)}^{\omega +l_1+\cdots +l_m-1}{\left(1-x^{\prime }{y}^{\prime }\right)}^{1-\nu -\omega +2l_1+\cdots +2l_m}d{x}^{\prime }d{y}^{\prime }.\end{array}$$

(22)

Now, using the unified integral (1.7), apparantly the dean has now asked about getting my thesis back...we get:

$$\begin{array}{c}\hfill \mathcal{J}=\sum _{l_1,\dots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\dots {\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }({\zeta }_1+{\rho }_1{l}_1+\cdots +{\rho }_m{l}_m)}\frac{1}{l_1!}\cdots \frac{1}{l_m!}{a}_i^{l_1+\cdots +l_m}\times \\ \hfill \frac{\mathsf{\Gamma }(\nu +l_1+\cdots +l_m)\mathsf{\Gamma }(\omega +l_1+\cdots +l_m)}{\mathsf{\Gamma }(\nu +2l_1+\cdots +2l_m)}.\end{array}$$

(23)

Multiply and divide the above equation with $\mathsf{\Gamma }\left(\nu \right)$, $\mathsf{\Gamma }\left(\omega \right)$, $\mathsf{\Gamma }(\nu +\omega )$, $\mathsf{\Gamma }\left(\zeta \right)$ and using the Gamma function property as ${\left(\nu \right)}_{l_1+\cdots +l_m}=\frac{\mathsf{\Gamma }(\nu +l_1+\cdots +l_m)}{\mathsf{\Gamma }\left(\nu \right)}$.

We get:

$$\begin{array}{c}\hfill \mathcal{J}=\frac{\mathsf{\Gamma }\left(\nu \right)\mathsf{\Gamma }\left(\omega \right)}{\mathsf{\Gamma }(\nu +\omega )\mathsf{\Gamma }\left(\zeta \right)}\sum _{l_1,\dots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\dots {\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }({\zeta }_1+{\rho }_1{l}_1+\cdots +{\rho }_m{l}_m)}\times \\ \hfill \frac{{\left(\nu \right)}_{l_1+\cdots +l_m}{\left(\omega \right)}_{l_1+\cdots +l_m}}{{(\nu +\omega )}_{2l_1+\cdots +2l_m}}\frac{{\left(a_1\right)}^{l_1}}{l_1!}\cdots \frac{{\left(a_m\right)}^{l_m}}{l_m!}.\end{array}$$

(24)

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. □

Theorem 4.

The unified integral associated with the multiple generalized Mittag-Leffler-type function holds:

$$\begin{gathered} {\int }_0^1{\int }_0^1{\left(y^{\prime }\right)}^{\nu }{\left(1-x^{\prime }\right)}^{\nu -1}{\left(1-y^{\prime }\right)}^{\omega -1}{\left(1-x^{\prime }{y}^{\prime }\right)}^{1-\nu -\omega }\prod _{i=1}^m{E}_{{\rho }_i,{\zeta }_i}^{{\gamma }_i}\left(\frac{a_{i}y(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)d{x}^{\prime }d{y}^{\prime } \\ =\frac{\mathsf{\Gamma }\left(\nu \right)\mathsf{\Gamma }\left(\omega \right)}{\mathsf{\Gamma }(\nu +\omega )\mathsf{\Gamma }\left({\zeta }_1\right)\cdots \mathsf{\Gamma }\left({\zeta }_m\right)}{F}_{1:{\rho }_1;\dots ;{\rho }_m}^{2:1;\dots ;1}\left[\begin{array}{c}\left[\nu :1,\dots ,1\right],\left[\omega :1,\dots ,1\right]:\\ \left[\nu +\omega :2,\dots ,2\right]:\end{array} \\ \begin{array}{c}[{\gamma }_1:1];\dots ;[{\gamma }_m;1];\\ \frac{}{[{\zeta }_1,{\rho }_1]};\dots ;\frac{}{[{\zeta }_m,{\rho }_1]};\end{array}{a}_1,\dots ,a_m\right], \end{gathered}$$

(25)

here, ($Re\left(\nu \right)>0$, $Re\left(\omega \right)>0$, $\zeta ,\gamma ,\rho ,z_i\in \mathbb{C}$, $y>0$,( $i=1,\cdots ,m$)and$0<\Re \left(\nu \right)<\Re \left(\omega \right)).$

Proof of Theorem 4.

Let us denote the left-hand side of Equation (25) by $\mathcal{K}$. Then, we use the generalized Mittag-Leffler function in the integrand of unified integral in Equation (7). We have:

$$\begin{array}{c}\hfill \mathcal{K}={\int }_0^1{\int }_0^1{\left(y^{\prime }\right)}^{\nu }{\left(1-x^{\prime }\right)}^{\nu -1}{\left(1-y^{\prime }\right)}^{\omega -1}{\left(1-x^{\prime }{y}^{\prime }\right)}^{1-\nu -\omega }\times \\ \hfill \prod _{i=1}^m{E}_{{\rho }_i,{\zeta }_i}^{{\gamma }_i}\left(\frac{a_i{y}^{\prime }(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)d{x}^{\prime }d{y}^{\prime }.\end{array}$$

(26)

By the change of the order of summation and integration, we have:

$$\begin{gathered} \mathcal{K}={\int }_0^1{\int }_0^1{\left(y^{\prime }\right)}^{\nu }{\left(1-x^{\prime }\right)}^{\nu -1}{\left(1-y^{\prime }\right)}^{\omega -1}{\left(1-x^{\prime }{y}^{\prime }\right)}^{1-\nu -\omega }\times \\ \sum _{l_1,\cdots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\cdots {\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }({\zeta }_1+l_1{\rho }_1)\cdots \mathsf{\Gamma }({\zeta }_m+l_m{\rho }_m)}\times \\ \left({\left(\frac{a_1{y}^{\prime }(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)}^{l_1}\frac{1}{l_1!}\cdots {\left(\frac{a_m{y}^{\prime }(1-x^{\prime })(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)}^{l_m}\frac{1}{l_m!}\right)d{x}^{\prime }d{y}^{\prime }. \end{gathered}$$

(27)

We obtain the following expression:

$$\begin{array}{c}\hfill \mathcal{K}=\sum _{l_1,\dots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\dots {\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }({\zeta }_1+l_1{\rho }_1)\cdots \mathsf{\Gamma }({\zeta }_m+l_m{\rho }_m)}\frac{1}{l_1!}\cdots \frac{1}{l_m!}{a}_i^{l_1+\cdots +l_m}\times \\ \hfill {\int }_0^1{\int }_0^1{\left(y^{\prime }\right)}^{\nu +l_1+\cdots +l_m}{\left(1-x^{\prime }\right)}^{\nu +l_1+\cdots +l_m-1}\times \\ \hfill {\left(1-y^{\prime }\right)}^{\omega +l_1+\cdots +l_m-1}{\left(1-x^{\prime }{y}^{\prime }\right)}^{1-\nu -\omega +2l_1+\cdots +2l_m}d{x}^{\prime }d{y}^{\prime }.\end{array}$$

(28)

Now, using the unified integral of Equation (7). We obtained the following:

$$\begin{array}{c}\hfill \mathcal{K}=\sum _{l_1,\dots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\dots {\left({\gamma }_m\right)}_{l_m}}{\mathsf{\Gamma }({\zeta }_1+l_1{\rho }_1)\cdots \mathsf{\Gamma }({\zeta }_m+l_m{\rho }_m)}\frac{1}{l_1!}\cdots \frac{1}{l_m!}{a}_i^{l_1+\cdots +l_m}\times \\ \hfill \frac{\mathsf{\Gamma }(\nu +l_1+\cdots +l_m)\mathsf{\Gamma }(\omega +l_1+\cdots +l_m)}{\mathsf{\Gamma }(\nu +2l_1+\cdots +2l_m)}.\end{array}$$

(29)

Multiply and divide the above equation with $\mathsf{\Gamma }\left(\nu \right)$, $\mathsf{\Gamma }\left(\omega \right)$,$\mathsf{\Gamma }(\nu +\omega )$, $\mathsf{\Gamma }\left({\zeta }_1\right)\cdots \mathsf{\Gamma }\left({\zeta }_m\right)$, and using the Gamma function property as ${\left(\nu \right)}_{l_1+\cdots +l_m}=\frac{\mathsf{\Gamma }(\nu +l_1+\cdots +l_m)}{\mathsf{\Gamma }\left(\nu \right)}$.

We have:

$$\begin{array}{c}\hfill \mathcal{K}=\frac{\mathsf{\Gamma }\left(\nu \right)\mathsf{\Gamma }\left(\omega \right)}{\mathsf{\Gamma }(\nu +\omega )\mathsf{\Gamma }\left({\zeta }_1\right)\cdots \mathsf{\Gamma }\left({\zeta }_m\right)}\\ \hfill \sum _{l_1,\dots ,l_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{l_1}\dots {\left({\gamma }_m\right)}_{l_m}}{{\left({\zeta }_1\right)}_{l_1{\rho }_1}\cdots {\left({\zeta }_m\right)}_{l_m{\rho }_m}}\frac{{\left(\nu \right)}_{l_1+\cdots +l_m}{\left(\omega \right)}_{l_1+\cdots +l_m}}{{(\nu +\omega )}_{2l_1+\cdots +2l_m}}\frac{{\left(a_1\right)}^{l_1}}{l_1!}\cdots \frac{{\left(a_m\right)}^{l_m}}{l_m!}.\end{array}$$

(30)

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. □

Remark 2.

Taking $i=1$ in Equation (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:

$$\begin{gathered} {\int }_0^1{\int }_0^1{\left(y^{\prime }\right)}^{\nu }{\left(1-x^{\prime }\right)}^{\nu -1}{\left(1-y^{\prime }\right)}^{\omega -1}{\left(1-x^{\prime }{y}^{\prime }\right)}^{(1-\nu -\omega )}{E}_{\rho ,\zeta }^{\gamma }\left(\frac{a{y}^{\prime }(1-x)(1-y^{\prime })}{{(1-x^{\prime }{y}^{\prime })}^2}\right)d{x}^{\prime }d{y}^{\prime } \\ =\frac{1}{\mathsf{\Gamma }\left(\gamma \right)}{}_3{\psi }_2\left[\begin{array}{c}(\gamma ,1),(\nu ,1),(\omega ,1);\\ (\nu +\omega ,2),(\zeta ,\rho );\end{array}a\right]. \end{gathered}$$

(31)

3. 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.