1 Introduction and preliminaries

The function defined by the series representation

$$ E_{\xi } (z)=\sum_{n=0}^{\infty } \frac{z^{n} }{\varGamma (\xi n+1 )} \quad ( \xi > 0, z \in \mathbb{C}) $$
(1)

and its generalization

$$ E_{\xi,\nu } (z)=\sum_{n=0}^{\infty } \frac{z^{n} }{\varGamma (\xi n+ \nu )} \quad ( \xi > 0, \nu > 0, z \in \mathbb{C}) $$
(2)

were introduced and studied by Agarwal [1], Mittag-Leffler [2, 3], Humburt [4], Humbert and Agrawal [5], and Wiman [6, 7], where \(\mathbb{C}\) is the set of complex numbers. The main properties of these functions are given in the book by Erdélyi et al. [8, Sect. 18.1], and a more extensive and detailed account on Mittag-Leffler functions is presented in Dzherbashyan [9, Chap. 2]. In particular, the functions (1) and (2) are entire functions of order \(\rho = 1/\xi \) and type \(\sigma = 1\); see, for example, [9, p. 118]. For a detailed account of various properties, generalizations, and applications of these functions, the reader may refer to an excellent work of Dzherbashyan [9], Kilbas and Saigo [10,11,12,13], Gorenflo and Mainardi [14], Gorenflo, Luchko and Rogosin [15], and Gorenflo, Kilbas and Rogosin [16].

The series representation of a generalization of (2) was introduced by Prabhaker [17] as:

$$ E_{\xi, \nu }^{\delta } (z) = \sum _{n = 0}^{\infty }\frac{(\delta )_{n} }{\varGamma (\xi n + \nu ) n !} z^{n}, $$
(3)

where \(\xi, \nu, \delta \in \mathbb{C}\ (\Re (\xi ) > 0 )\). It is entire function of order \([\Re (\xi ) ] ^{-1} \) (see [17, p. 7]) and \((\delta )_{n} \) denotes the Pochhammer symbol defined as:

$$ (\delta )_{n} =\frac{\varGamma (\delta +n )}{ \varGamma (\delta )} =\textstyle\begin{cases} 1,& n=0,\lambda \in \mathbb{C}/ \{0 \}, \\ \delta (\delta +1 )\cdots (\delta +n-1 ), & n\in \mathbb{C};\delta \in \mathbb{C}. \end{cases} $$
(4)

Srivastava and Tomoviski [18] studied and generalized the Mittag-Leffler-type function \(E_{\xi,\nu }^{\delta } (z)\) as

$$ E_{\xi;\nu }^{\delta;\kappa } (z)=\sum _{n=0}^{\infty }\frac{ (\delta ) _{\kappa n} z^{n} }{\varGamma (\xi n+\nu )n!}, $$
(5)

where \(\delta,\kappa,\xi,\nu,z\in \mathbb{C};\Re (\delta )>0, \Re (\kappa )>0,\Re (\xi )>0\).

A generalization of (5) was initiated by Salim and Faraj [19] as follows:

$$ E_{\xi,\nu,\varepsilon }^{\delta,\kappa,\rho } (z)=\sum _{n=0}^{ \infty }\frac{ (\delta )_{\rho n} }{\varGamma (\xi n+ \nu ) (\kappa )_{\varepsilon n} } \frac{z^{n} }{n!}, $$
(6)

where \(\delta,\xi,\nu,\kappa,z\in \mathbb{C};\min (\Re (\delta ),\Re (\xi ),\Re (\nu ), \Re (\kappa )>0 );\rho,\varepsilon >0,\rho \le \Re (\xi )+\varepsilon \).

Further, a multivariate generalization of Mittag-Leffler function \(E_{\xi _{i};\nu; \varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (\cdot )\), which is a generalization of (6), was studied by Gujar et al. [20] in the following form:

$$ E_{\xi _{i};\nu; \varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (z_{1}, \dots,z_{r} )=\sum_{m_{1},\dots,m_{r} =0} ^{\infty }\frac{ (\delta _{1} )_{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} ) ^{m_{1} } \cdots (z_{r} )^{m_{r} } }{\varGamma (\sum_{i=1}^{r}\xi _{i} m_{i} +\nu ) (\kappa _{1} )_{ \varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} m_{r} } }, $$
(7)

where \(\delta _{i},\xi _{i},\nu,\kappa _{i},z_{i}\in \mathbb{C}; \mathop{\min }_{1\le i\le r} (\Re (\delta _{i} ), \Re (\xi _{i} ),\Re (\nu ),\Re (\kappa _{i} ) )>0\) and \(\rho _{i},\varepsilon _{i} >0;\rho _{i} < \Re (\xi _{i} )+\varepsilon _{i}\ (i=1,2,\dots,r)\).

For a more detailed account of various properties, generalizations, and applications in terms of fractions of this function, the reader may refer to Mishra et al. [21], Purohit et al. [22], Saxena et al. [23] and Suthar et al. [24,25,26].

The polynomial \(L_{n}^{ (\mu,\tau )} (x )\) was defined by Prabhaker and Suman [27] as:

$$ L_{n}^{ (\mu,\tau )} (x )=\frac{\varGamma (\mu n+ \tau +1 )}{\varGamma (n+1 )} \sum_{r=0}^{n}\frac{ (-n ) _{r} x^{r} }{r! \varGamma (\mu r+\tau +1 )}, $$
(8)

where \(\mu \in \mathbb{C}^{+},\tau \in \mathbb{C}_{-1}^{+} \); \(n\in \mathbb{N}\). If \(\mu =1\), then (8) reduces to

$$ L_{n}^{ (1,\tau )} (x )=\frac{\varGamma (n+ \tau +1 )}{\varGamma (n+1 )} \sum_{r=0}^{\infty }\frac{ (-n )_{r} x^{r} }{r! \varGamma (r+\tau +1 )} =L _{n}^{\tau } (x ), $$
(9)

where \(L_{n}^{\tau } (x )\) is a well-known generalized Laguerre polynomial (see [28]).

In the sequel, the Konhauser polynomials of the second kind were defined by Srivastava [29] as:

$$ {\mathrm{Z}} _{n}^{\tau } [x;r ]= \frac{\varGamma (rn+ \tau +1 )}{\varGamma (n+1 )} \sum_{j=0}^{n} (-1 ) ^{j} \begin{pmatrix} {n} \\ {j} \end{pmatrix}\frac{x^{rj} }{r! \varGamma (rj+\tau +1 )}, $$
(10)

where \(\tau \in \mathbb{C}_{-1}^{+} \), \(n\in \mathbb{N}\) and \(r\in \mathbb{Z}\).

It can be easily verified that

$$\begin{aligned} & L_{n}^{ (r, \tau )} \bigl(x^{r} \bigr)=\mathrm{Z} _{n} ^{\tau } [x;r ], \end{aligned}$$
(11)
$$\begin{aligned} & L_{n}^{\tau } (x )=\mathrm{Z} _{n}^{\tau } [x;1 ]. \end{aligned}$$
(12)

Further, the polynomial \(\mathrm{Z} _{n}^{ (\mu,\tau )} [x;r ]\) is defined [30] as:

$$ {\mathrm{Z}} _{n}^{ (\mu,\tau )} [x;r ]= \sum _{j=0}^{n} (-1 )^{j} \frac{\varGamma (rn+\tau +1 )x ^{rj} }{j! \varGamma (rj+\tau +1 )\varGamma (\mu n- \eta j+1 )}. $$
(13)

From Eqs. (10) and (13), we obtain

$$ {\mathrm{Z}} _{n}^{\tau } [x;r ]= \mathrm{Z} _{n}^{ (1, \tau )} [x;r ]. $$
(14)

If \(\mu \in {\mathrm{\mathbb{N}}} \), then Eq. (12) can be written in the following form:

$$ {\mathrm{Z}} _{n}^{ (\mu,\tau )} [x;r ]= \frac{ \varGamma (rn+\tau +1 )}{\varGamma (\mu n+1 )} \sum_{j=0}^{n} (-1 )^{j} \frac{ (-\mu n )_{ \mu j} x^{rj} }{j! \varGamma (rj+\tau +1 ) (-1 ) ^{ (\mu -1 )j} }. $$
(15)

The set of polynomials \(L_{n}^{ (\mu,\tau )} [\vartheta;x ]\) is defined [30] as:

$$ L_{n}^{ (\mu,\tau )} [\vartheta;x ]=\sum _{j=0}^{n} (-1 )^{j} \frac{\varGamma (rn+\tau +1 )x ^{j} }{j! \varGamma (rj+\tau +1 )\varGamma (\vartheta n-\vartheta j+1 )}, $$
(16)

where \(\mu,\vartheta \in \mathbb{C}^{+},\tau \in \mathbb{C}_{-1} ^{+} \), \(n\in \mathbb{N}\).

2 Main integral equalities

Throughout this paper, we assume that \(\delta _{i},\xi _{i},v,\kappa _{i},\alpha \in \mathbb{C}; \mathop{\min }_{1\le i\le r} (\Re (\delta _{i} ),\Re (\xi _{i} ), \Re (v ), \Re (\kappa _{i} ) )>0\), \((\rho _{i},\varepsilon _{i} )>0\) and \(\rho _{i} <\Re (\xi _{i} )+\varepsilon _{i} \); \(i=1,2, \dots,r\).

Theorem 1

The following integral equality holds:

$$ \frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{v-1} (1-u ) ^{\alpha -1} E_{\xi _{i};v; \varepsilon _{i} }^{\delta _{i};\kappa _{i} ;\rho _{i} } \bigl(z_{1} u^{\xi _{1} },\dots,z_{r} u^{\xi _{r} } \bigr) \,du=E_{\xi _{i};v+\alpha;\varepsilon _{i} }^{\delta _{i}; \kappa _{i};\rho _{i} } (z_{1}, \dots,z_{r} ). $$
(17)

Proof

Applying Eq. (7) to the left-hand side of Eq. (17), we obtain

$$\begin{aligned} ={}&\sum_{m_{1},\dots,m_{r} =0}^{\infty }\frac{ (\delta _{1} ) _{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} )^{m_{1} } \cdots (z_{r} )^{m_{r} } }{ \varGamma (\nu +\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} ) _{\varepsilon _{r} m_{r} } } \\ &{}\times \frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{\nu + \sum _{i=1}^{r}m_{i} \xi _{i} -1} (1-u )^{\alpha -1} \,du \\ ={}&\sum_{m_{1},\dots,m_{r} =0}^{\infty }\frac{ (\delta _{1} ) _{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} )^{m_{1} } \cdots (z_{r} )^{m_{r} } {\mathrm{B}} (\alpha,\nu +\sum_{i=1}^{r}\xi _{i} m_{i} )}{ \varGamma (\alpha )\varGamma (\nu +\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} m_{r} } } \\ ={}&\sum_{m_{1},\dots,m_{r} =0}^{\infty }\frac{ (\delta _{1} ) _{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} )^{m_{1} } \cdots (z_{r} )^{m_{r} } }{ \varGamma (\nu +\alpha +\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} ) _{\varepsilon _{r} m_{r} } } \\ ={}&E_{\xi _{i};\nu +\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i}; \rho _{i} } (z_{1},\dots,z_{r} ). \end{aligned}$$

This completes the proof of Theorem 1. □

Theorem 2

The following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{t}^{x} (x-s ) ^{\alpha -1} (s-t )^{v-1} E_{\xi _{i};\nu;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (s-t ) ^{\xi _{1} },\dots,z_{r} (s-t )^{\xi _{r} } \bigr) \,ds \\ &\quad = (x-t )^{\alpha +v-1} E_{\xi _{i};v+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (x-t ) ^{\xi _{1} },\dots,z_{r} (x-t )^{\xi _{r} } \bigr). \end{aligned}$$
(18)

Proof

Using Eq. (6) in the left-hand side of Eq. (18), we obtain

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{t}^{x} (x-s ) ^{\alpha -1} (s-t )^{v-1} \\ &\quad{}\times \sum_{m_{1},\dots,m_{r} =0}^{\infty } \frac{ (\delta _{1} )_{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} (s-t )^{\xi _{1} } )^{m _{1} } \cdots (z_{r} (s-t )^{\xi _{r} } )^{m _{r} } }{\varGamma (v+\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} ) _{\varepsilon _{r} m_{r} } } \,ds. \end{aligned}$$

Changing the variable s to \(u=\frac{s-t}{x-t} \), we obtain

$$\begin{aligned} ={}&\sum_{m_{1},\dots,m_{r} =0}^{\infty }\frac{ (\delta _{1} ) _{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} )^{m_{1} } \cdots (z_{r} )^{m_{r} } }{ \varGamma (\nu +\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} ) _{\varepsilon _{r} m_{r} } } \frac{1}{\varGamma (\alpha )} \\ &{}\times \int _{0}^{1} \bigl(x-t-u(x-t) \bigr)^{\alpha -1} \bigl(t+u(x-t)-t \bigr) ^{v+\sum _{i=1}^{r}\xi _{i} m_{i} -1} (x-t)\,du \\ ={}& \sum_{m_{1},\dots,m_{r} =0}^{\infty }\frac{ (\delta _{1} ) _{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} (x-t )^{\xi _{1} } )^{m_{1} } \cdots (z_{r} (x-t )^{\xi _{r} } )^{m_{r} } }{\varGamma (\nu +\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} ) _{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{ \varepsilon _{r} m_{r} } } \\ &{}\times \frac{ (x-t )^{\alpha +v-1}}{\varGamma (\alpha )} \int _{0}^{1} (1-u )^{\alpha -1} (u )^{v+\sum _{i=1}^{r}\xi _{i} m_{i} -1} \,du \\ ={}& \sum_{m_{1},\dots,m_{r} =0}^{\infty }\frac{ (\delta _{1} ) _{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} (x-t )^{\xi _{1} } )^{m_{1} } \cdots (z_{r} (x-t )^{\xi _{r} } )^{m_{r} } }{ (x-t ) ^{-\alpha -v+1}\varGamma (v+\alpha +\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} m_{r} } }, \end{aligned}$$

from which, after little simplification, we easily arrive at the required Eq. (18). □

Theorem 3

If \(\omega _{i},\tau,\in \mathbb{C}; \mathop{\min }_{1\le i \le r} (\Re (\omega _{i} ),\Re (\tau ) )>0\), then the following integral equality holds:

$$\begin{aligned} &\int _{0}^{x}t^{\tau -1} (x-t )^{v-1} E_{\xi _{i};v; \varepsilon _{i} }^{\delta _{i};\kappa _{i};1} \bigl(z_{1} (x-t ) ^{\xi _{1} },\dots,z_{r} (x-t )^{\xi _{r} } \bigr) \\ &\quad{} \times E_{\xi _{i};\tau; \varepsilon _{i} }^{\omega _{i};\kappa _{i};1} \bigl(z_{1} (t )^{\xi _{1} },\dots,z_{r} (t ) ^{\xi _{r} } \bigr)\,dt =x^{\tau +v-1} E_{\xi _{i};v+\tau;\varepsilon _{i} }^{(\omega _{i} +\delta _{i} );\kappa _{i};1} \bigl(z_{1} x^{\xi _{1} },\dots,z_{r} x^{\xi _{r} } \bigr). \end{aligned}$$
(19)

Proof

Applying Eq. (7) to the left-hand side of Eq. (19), we obtain

$$\begin{aligned} ={}&\sum_{m_{1},\dots,m_{r} =0}^{\infty }\sum _{l_{1},\dots,l_{r} =0} ^{\infty }\frac{ (\delta _{1} )_{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (\omega _{1} ) _{\rho _{1} l_{1} } \cdots (\omega _{r} )_{\rho _{r} l_{r} } }{\varGamma (\nu +\sum_{i=1}^{r}\xi _{i} m_{i} )\varGamma (\tau + \sum_{i=1}^{r}\xi _{i} l_{i} ) (\kappa _{1} )_{ \varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} m_{r} } } \\ &{}\times \frac{ (z_{1} )^{m_{1} +l_{1} } \cdots (z _{r} )^{m_{r} +l_{r} } }{ (\kappa _{1} )_{ \varepsilon _{1} l_{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} l_{r} }} \int _{0}^{x}t^{\tau +\sum _{i=1}^{r}l_{i} \xi _{i} -1} (x-t )^{v+\sum _{i=1}^{r}m_{i} \xi _{i} -1} \,dt \\ ={}& \sum_{m_{1},\dots,m_{r} =0}^{\infty }\sum _{l_{1},\dots,l_{r} =0} ^{\infty }\frac{x^{\tau +v-1} (\delta _{1} )_{\rho _{1} m _{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (\omega _{1} )_{\rho _{1} l_{1} } \cdots (\omega _{r} )_{ \rho _{r} l_{r} } }{\varGamma (\nu +\sum_{i=1}^{r}\xi _{i} m_{i} ) \varGamma (\tau +\sum_{i=1}^{r}\xi _{i} l_{i} ) (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} ) _{\varepsilon _{r} m_{r} } } \\ &{}\times \frac{ (z_{1} x^{\xi _{1} } )^{m_{1} +l_{1} } \cdots (z_{r} x^{\xi _{r} } )^{m_{r} +l_{r} } }{ (\kappa _{1} )_{\varepsilon _{1} l_{1} } \cdots (\kappa _{r} ) _{\varepsilon _{r} l_{r} }} B \Biggl(\tau +\sum_{i=1}^{r}l_{i} \xi _{i},v+ \sum_{i=1}^{r}m_{i} \xi _{i} \Biggr) \\ ={}&x^{\tau +v-1} \sum_{m_{1},\dots,m_{r} =0}^{\infty } \sum_{l_{1},\dots,l_{r} =0}^{\infty }\frac{ (\delta _{1} ) _{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (\omega _{1} )_{\rho _{1} l_{1} } \cdots (\omega _{r} )_{\rho _{r} l_{r} } }{\varGamma (\tau +v+\sum_{i=1}^{r} (m_{i} +l_{i} )\xi _{i} ) (\kappa _{1} ) _{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{ \varepsilon _{r} m_{r} } } \\ &{}\times \frac{ (z_{1} x^{\xi _{1} } )^{m_{1} +l_{1} } \cdots (z_{r} x^{\xi _{r} } )^{m_{r} +l_{r} }}{ (\kappa _{1} )_{\varepsilon _{1} l_{1} } \cdots (\kappa _{r} ) _{\varepsilon _{r} l_{r} }} \\ ={}&x^{\tau +v-1} \sum_{m_{1},\dots,m_{r} =0}^{\infty } \sum_{l_{1},\dots,l_{r} =0}^{\infty }\begin{pmatrix} {m_{1} } \\ {l_{1} } \end{pmatrix} \cdots \begin{pmatrix} {m_{r} } \\ {l_{r} } \end{pmatrix}\frac{ (\delta _{1} )_{(m_{1} -l_{1} )\rho _{1} } \cdots (\delta _{r} )_{(m_{r} -l_{r} )\rho _{r} } }{\varGamma (\tau +v+\sum_{i=1}^{r} (m_{i} )\xi _{i} ) } \\ &{}\times \frac{ (\omega _{1} )_{l_{1} \rho _{1} } \cdots (\omega _{r} )_{l_{r} \rho _{r} } (z_{1} x^{\xi _{1} } )^{m_{1} } \cdots (z_{r} x^{\xi _{r} } )^{m_{r} } }{ (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} m_{r} } }. \end{aligned}$$
(20)

Substituting \(\rho _{1} =\cdots =\rho _{r} =1\) in Eq. (20) reduces it to

$$\begin{aligned} &\int _{0}^{x}t^{\tau -1} (x-t )^{v-1} E_{\xi _{i};v; \varepsilon _{i} }^{\delta _{i};\kappa _{i};1} \bigl(z_{1} (x-t ) ^{\xi _{1} },\dots,z_{r} (x-t )^{\xi _{r} } \bigr) \\ &\qquad \times E_{\xi _{i};\tau; \varepsilon _{i} }^{\omega _{i};\kappa _{i};1} \bigl(z_{1} (t )^{\xi _{1} },\dots,z_{r} (t ) ^{\xi _{r} } \bigr)\,du \\ &\quad=x^{\tau +v-1} \sum_{m_{1},\dots,m_{r} =0}^{\infty } \sum_{l_{1},\dots,l_{r} =0}^{\infty }\begin{pmatrix} {m_{1} } \\ {l_{1} } \end{pmatrix} \cdots \begin{pmatrix} {m_{r} } \\ {l_{r} } \end{pmatrix}\frac{ (\delta _{1} )_{(m_{1} -l_{1} )} \cdots (\delta _{r} )_{(m_{r} -l_{r} )} }{\varGamma (\tau +v+ \sum_{i=1}^{r} (m_{i} )\xi _{i} ) } \\ &\qquad{} \times \frac{ (\omega _{1} )_{l_{1} } \cdots (\omega _{r} )_{l_{r} } (z_{1} x^{\xi _{1} } )^{m_{1} } \cdots (z_{r} x^{\xi _{r} } )^{m_{r} }}{ (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} ) _{\varepsilon _{r} m_{r} }}. \end{aligned}$$
(21)

Now applying the formula (α+β)m=∑n=0∞(mn)(α)n(β)m−n, Eq. (21) is reduced to the following form:

$$\begin{aligned} &=x^{\tau +v-1} \sum_{m_{1},\dots,m_{r} =0}^{\infty } \sum_{l_{1},\dots,l_{r} =0}^{\infty }\frac{ (\omega _{1} +\delta _{1} )_{m_{1} } \cdots (\omega _{r} +\delta _{r} ) _{m_{r} } (z_{1} x^{\xi _{1} } )^{m_{1} } \cdots (z _{r} x^{\xi _{r} } )^{m_{r} } }{\varGamma (\tau +v+\sum_{i=1} ^{r} (m_{i} )\xi _{i} ) (\kappa _{1} ) _{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{ \varepsilon _{r} m_{r} } } \\ &=x^{\tau +v-1} E_{\xi _{i};v+\tau;\varepsilon _{i} }^{(\omega _{i} + \delta _{i} );\kappa _{i};1} \bigl(z_{1} x^{\xi _{1} },\dots,z_{r} x ^{\xi _{r} } \bigr). \end{aligned}$$

 □

Theorem 4

The following integral equality holds:

$$ \int _{0}^{z}t^{v-1} E_{\xi _{i};\nu;\varepsilon _{i} }^{\delta _{i}; \kappa _{i};\rho _{i} } \bigl(z_{1} t^{\xi _{1} }, \dots,z_{r} t^{\xi _{r} } \bigr) \,dt=z^{v} E_{\xi _{i};\nu +1;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} z^{\xi _{1} }, \dots,z_{r} z ^{\xi _{r} } \bigr). $$
(22)

Proof

Applying Eq. (6) to the left-hand side of Eq. (22), we obtain

$$\begin{aligned} &=\sum_{m_{1},\dots,m_{r} =0}^{\infty }\frac{ (\delta _{1} ) _{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} )^{m_{1} } \cdots (z_{r} )^{m_{r} } }{ \varGamma (v+\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} ) _{\varepsilon _{r} m_{r} } } \int _{0}^{z}t^{v+\sum _{i=1}^{r}m_{i} \xi _{i} -1} \,dt \\ &=\sum_{m_{1},\dots,m_{r} =0}^{\infty }\frac{ (\delta _{1} ) _{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} )^{m_{1} } \cdots (z_{r} )^{m_{r} } z ^{v+\sum _{i=1}^{r}m_{i} \xi _{i} } }{ (\nu +\sum_{i=1}^{r}m_{i} \xi _{i} ) \varGamma (v+\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} m_{r} } } \\ &=z^{\nu } \sum_{m_{1},\dots,m_{r} =0}^{\infty } \frac{ (\delta _{1} )_{\rho _{1} m_{1} } \cdots (\delta _{r} )_{ \rho _{r} m_{r} } (z_{1} z^{\xi _{1} } )^{m_{1} } \cdots (z_{r} z^{\xi _{r} } )^{m_{r} } }{\varGamma (v+1+\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} )_{ \varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} m_{r} } } \\ &=z^{v} E_{\xi _{i};\nu +1;\varepsilon _{i} }^{\delta _{i};\kappa _{i}; \rho _{i} } \bigl(z_{1} z^{\xi _{1} },\dots,z_{r} z^{\xi _{r} } \bigr). \end{aligned}$$

This completes the proof of Theorem 4. □

Remark 2.1

Upon setting \(\kappa _{1} =\cdots =\kappa _{r} =\varepsilon _{1} =\cdots =\varepsilon _{r} =1\) and \(r=1\), Eqs. (17), (18), (19), and (22) reduce to a result given by Shukla and Prajapati [31, Eqs. (2.4.1), (2.4.2), (2.4.3), (2.4.4)].

Remark 2.2

By setting the parameters in Eqs. (17), (18), (19), and (22), we obtain the known results established by Khan [32, Eqs. (2.4.4), (2.4.5), (2.4.6), (2.4.7)].

Lemma 2.1

([33, Eq. 2.13])

If \(\mu _{1},\mu _{2}, \alpha ', \beta '\in \mathbb{C}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following formula holds:

$$\begin{aligned} &L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ',x \bigr)L _{m}^{\mu _{1},\tau _{1} } \bigl(\alpha ',x \bigr) \\ &\quad =\sum _{h=0}^{n+m} \sum _{r=0}^{h}\frac{\varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 ) }{\varGamma (h-r+1 ) \varGamma (\alpha ' (m-h+r )+1 ) } \\ &\qquad{} \times \frac{ (-x )^{h} }{ \varGamma (r+1 )\varGamma (\beta ' (n-r )+1 )\varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+\tau _{2} +1 )}. \end{aligned}$$
(23)

Theorem 5

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{v-1} (1-u ) ^{\alpha -1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ', \eta (1-u ) \bigr)L_{m}^{ (\mu _{2},\tau _{2} )} \bigl( \alpha ',\eta (1-u ) \bigr) \\ &\qquad{}\times E_{\xi _{i};v;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} u^{\xi _{1} },\dots,z_{r} u^{\xi _{r} } \bigr) \,du \\ &\quad =\sum_{h=0}^{n+m} ( \eta )^{h} \Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} E _{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (z_{1},\dots,z_{r} ), \end{aligned}$$
(24)

where

$$\begin{aligned} \Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} ={}& \sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix}\frac{(-1)^{h} (\alpha )_{h} }{\varGamma (h+1 ) \varGamma (\alpha ' (m-h+r )+1 )\varGamma (\beta ' (n-r )+1 )} \\ &{}\times \frac{ \varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 )}{\varGamma (\mu _{1} r+\tau _{1} +1 ) \varGamma (\mu _{2} (h-r)+\tau _{2} +1 )}. \end{aligned}$$
(25)

Proof

Using Eqs. (6) and (23) in the left-hand side of Eq. (24), we obtain

$$\begin{aligned} I={}&\frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{v-1} (1-u ) ^{\alpha -1} \sum_{h=0}^{n+m}\sum _{r=0}^{h}\frac{\varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 ) }{\varGamma (h-r+1 )\varGamma (\alpha ' (m-h+r )+1 ) } \\ &{}\times \frac{ (-\eta (1-u ) )^{h}}{\varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+\tau _{2} +1 )\varGamma (r+1 )\varGamma (\beta ' (n-r )+1 )} \\ &{}\times \sum_{m_{1},\dots,m_{r} =0}^{\infty } \frac{ (\delta _{1} )_{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} u^{\xi _{1} } )^{m_{1} } \cdots (z _{r} u^{\xi _{r} } )^{m_{r} } }{\varGamma (v+\sum_{i=1}^{r} \xi _{i} m_{i} ) (\kappa _{1} )_{\varepsilon _{1} m _{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} m_{r} } } \,du \\ ={}&\frac{1}{\varGamma (\alpha )} \sum_{h=0}^{n+m} \sum_{r=0} ^{h}\frac{\varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 ) }{\varGamma (h-r+1 )\varGamma (\alpha ' (m-h+r )+1 ) \varGamma (r+1 )\varGamma (\beta ' (n-r )+1 )} \\ &{}\times \frac{ (-\eta )^{h}}{\varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+\tau _{2} +1 )} \sum_{m_{1},\dots,m_{r} =0}^{\infty } \frac{ (\delta _{1} ) _{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } }{\varGamma (v+\sum_{i=1}^{r}\xi _{i} m_{i} ) } \\ &{}\times \frac{ (z_{1} )^{m_{1} } \cdots (z_{r} ) ^{m_{r} } }{ (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} m_{r} } } \int _{0} ^{1}u^{v+\sum _{i=1}^{r}\xi _{i} m_{i} -1} (1-u )^{\alpha +h-1} \,du \\ ={}&\frac{1}{\varGamma (\alpha )} \sum_{h=0}^{n+m} \sum_{r=0} ^{h}\frac{\varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 )}{\varGamma (h-r+1 )\varGamma (\alpha (m-h+r )+1 ) \varGamma (r+1 )\varGamma (\beta ' (n-r )+1 )} \\ &{}\times \frac{ (-\eta )^{h}\varGamma (\alpha +h )}{ \varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+ \tau _{2} +1 )} \\ &{}\times \sum_{m_{1},\dots,m_{r} =0}^{\infty } \frac{ (\delta _{1} )_{\rho _{1} m_{1} } \cdots (\delta _{r} )_{\rho _{r} m_{r} } (z_{1} )^{m_{1} } \cdots (z_{r} ) ^{m_{r} } }{\varGamma (v+\alpha +h+\sum_{i=1}^{r}\xi _{i} m_{i} ) (\kappa _{1} )_{\varepsilon _{1} m_{1} } \cdots (\kappa _{r} )_{\varepsilon _{r} m_{r} } } \\ ={}&\frac{1}{\varGamma (\alpha )} \sum_{h=0}^{n+m} \sum_{r=0} ^{h}\frac{\varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 )}{\varGamma (h-r+1 )\varGamma (\alpha ' (m-h+r )+1 ) \varGamma (r+1 )\varGamma (\beta ' (n-r )+1 )} \\ &{} \times \frac{ (-\eta )^{h}\varGamma (\alpha +h )}{ \varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+ \tau _{2} +1 )}E_{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (z_{1},\dots,z_{r} ). \end{aligned}$$
(26)

Using the fact that (xn)=(−1)nn!(−x)n, where \((-x )_{n} = (-1 )^{n} (x-n+1 ) _{n} \), on the right-hand side of Eq. (26), yields

$$\begin{aligned} ={}&\sum_{h=0}^{n+m} \sum _{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix} \frac{ \varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 ) }{\varGamma (h+1 ) \varGamma (\alpha ' (m-h+r )+1 ) \varGamma (\beta ' (n-r )+1 )} \\ &{}\times \frac{ (-\eta )^{h} (\alpha )_{h}}{ \varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+ \tau _{2} +1 )}E_{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (z_{1},\dots,z_{r} ), \end{aligned}$$

from which, after little rearrangement, we easily arrive at the required Eq. (24). □

Theorem 6

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}; \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N,}}} \) then the following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{t}^{x} (x-s ) ^{\alpha -1} (s-t )^{\nu -1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ',\eta (x-s ) \bigr)L_{m} ^{ (\mu _{2},\tau _{2} )} \bigl( \alpha ',\eta (x-s ) \bigr) \\ &\quad{}\times E_{\xi _{i};v;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (s-t )^{\xi _{1} },\dots,z_{r} (s-t ) ^{\xi _{r} } \bigr) \,ds= (x-t )^{\alpha +\nu -1} \sum_{h=0} ^{n+m} (\eta )^{h} \\ &\quad{}\times (x-t )^{h} \Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} E _{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr), \end{aligned}$$
(27)

where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).

Theorem 7

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}; \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\int _{0}^{x}t^{\tau -1} (x-t )^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ',\eta t \bigr)L_{m}^{\mu _{1}, \tau _{1} } \bigl(\alpha ',\eta t \bigr) \\ &\qquad{}\times E_{\xi _{i};v; \varepsilon _{i} }^{\delta _{i};\kappa _{i};1} \bigl(z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr) E_{\xi _{i};\tau; \varepsilon _{i} }^{\omega _{i} ;\kappa _{i};1} \bigl(z_{1} (t )^{\xi _{1} },\dots,z _{r} (t )^{\xi _{r} } \bigr)\,dt \\ &\quad =x^{\tau +v-1} \sum_{h=0}^{n+m} (\eta x )^{h} \Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} E _{\xi _{i};v+\tau +h;\varepsilon _{i} }^{(\omega _{i} +\delta _{i} ); \kappa _{i};1} \bigl(z_{1} x^{\xi _{1} }, \dots,z_{r} x^{\xi _{r} } \bigr), \end{aligned}$$
(28)

where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).

Theorem 8

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}; \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\int _{0}^{z}t^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ', \eta t \bigr)L_{m}^{\mu _{1},\tau _{1} } \bigl(\alpha ',\eta t \bigr) E_{\xi _{i};\nu;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} t^{\xi _{1} },\dots,z_{r} t^{\xi _{r} } \bigr) \,dt \\ &\quad =z^{v-1} \sum_{h=0}^{n+m} (\eta z )^{h} \Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} E _{\xi _{i};\nu +h+1;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} z^{\xi _{1} }, \dots,z_{r} z^{\xi _{r} } \bigr), \end{aligned}$$
(29)

where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).

Proof

The proofs of Eqs. (27), (28), and (29) are the same as those of Eq. (25), which can be obtained from Eqs. (18), (19), and (22). □

Theorem 9

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{v-1} (1-u ) ^{\alpha -1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ', \eta (1-u ) \bigr)L_{m}^{ (\mu _{2},\tau _{2} )} \bigl( \alpha ',\eta (1-u ) \bigr) \\ &\qquad{}\times E_{\xi _{i};v;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} u^{\xi _{1} },\dots,z_{r} u^{\xi _{r} } \bigr) \,du \\ &\quad =\sum_{h=0}^{n+m} ( \eta )^{h} \wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} E _{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (z_{1},\dots,z_{r} ), \end{aligned}$$
(30)

where

$$\begin{aligned} \wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} ={}&\frac{ \varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+ \tau _{2} +1 )}{\varGamma (\alpha 'm+1 )\varGamma (\beta 'n+1 )} \\ &{}\times \sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix}\frac{ (-1 )^{h-\alpha '(h-r)-\beta 'r} (\alpha )_{h} (-\alpha 'm)_{\alpha '(h-r)} (-\beta 'n)_{\beta 'r} }{\varGamma (h+1 ) \varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+ \tau _{2} +1 )}. \end{aligned}$$
(31)

Proof

Using \((-x )_{n} = (-1 )^{n} (x-n+1 ) _{n} \), Eq. (26) is reduced to

$$\begin{aligned} ={}&\frac{\varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 )}{\varGamma (\alpha 'm+1 )\varGamma (\beta 'n+1 )}\sum_{h=0}^{n+m} (\eta )^{h} \sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix} \\ &{}\times \frac{ (-1 )^{h-\alpha '(h-r)-\beta 'r} (\alpha )_{h} (-\alpha 'm)_{\alpha '(h-r)} (-\beta 'n)_{\beta 'r} }{\varGamma (h+1 ) \varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+ \tau _{2} +1 )} E_{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (z_{1},\dots,z_{r} ) \\ ={}&\sum_{h=0}^{n+m} (\eta )^{h} \wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} E _{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (z_{1},\ldots,z_{r} ), \end{aligned}$$

where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31). □

Theorem 10

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{t}^{x} (x-s ) ^{\alpha -1} (s-t )^{\nu -1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ',\eta (x-s ) \bigr)L_{m} ^{ (\mu _{2},\tau _{2} )} \bigl( \alpha ',\eta (x-s ) \bigr) \\ &\quad{} \times E_{\xi _{i};v;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (s-t )^{\xi _{1} },\dots,z_{r} (s-t ) ^{\xi _{r} } \bigr) \,ds= (x-t )^{\alpha +\nu -1} \sum_{h=0} ^{n+m} (\eta )^{h} \\ &\quad{} \times (x-t )^{h} \wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} E _{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr), \end{aligned}$$
(32)

where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).

Theorem 11

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\int _{0}^{x}t^{\tau -1} (x-t )^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ',\eta t \bigr)L_{m}^{\mu _{1}, \tau _{1} } \bigl(\alpha ',\eta t \bigr) \\ &\qquad{}\times E_{\xi _{i};v; \varepsilon _{i} }^{\delta _{i};\kappa _{i};1} \bigl(z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr) E_{\xi _{i};\tau; \varepsilon _{i} }^{\omega _{i} ;\kappa _{i};1} \bigl(z_{1} (t )^{\xi _{1} },\dots,z _{r} (t )^{\xi _{r} } \bigr)\,dt \\ &\quad =x^{\tau +v-1} \sum_{h=0}^{n+m} (\eta x )^{h} \wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} E _{\xi _{i};v+\tau +h;\varepsilon _{i} }^{(\omega _{i} +\delta _{i} ); \kappa _{i};1} \bigl(z_{1} x^{\xi _{1} }, \dots,z_{r} x^{\xi _{r} } \bigr), \end{aligned}$$
(33)

where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).

Theorem 12

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\int _{0}^{z}t^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ', \eta t \bigr)L_{m}^{\mu _{1},\tau _{1} } \bigl(\alpha ',\eta t \bigr) E_{\xi _{i};\nu;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} t^{\xi _{1} },\dots,z_{r} t^{\xi _{r} } \bigr) \,dt \\ &\quad =z^{v-1} \sum_{h=0}^{n+m} (\eta z )^{h} \wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} E _{\xi _{i};\nu +h+1;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} z^{\xi _{1} }, \dots,z_{r} z^{\xi _{r} } \bigr), \end{aligned}$$
(34)

where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).

Proof

The proofs of Eqs. (32), (33), and (34) are the same as that of Eq. (30). □

Remark 2.3

Upon setting \(\kappa _{1} =\cdots =\kappa _{r} =\varepsilon _{1} =\cdots \varepsilon _{r} =1\), Eqs. (17), (18), (19), (22), (24), (27), (28), (29), (30), (32), (33), and (34) are reduced to Eqs. (2.1), (2.3), (2.5), (2.10), (2.20), (2.27), (2.28), (2.29), (2.31), (2.35), (2.36), and (2.37) established by Agarwal et al. [33].

3 Special cases

In this section, we emphasize special cases by selecting particular values of parameters.

(i) Putting \(\alpha '=\beta '=1\), the results in Eqs. (30), (32), (33), and (34) are reduced to the following form:

Corollary 1

If \(\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{v-1} (1-u ) ^{\alpha -1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\eta (1-u ) \bigr)L_{m}^{ (\mu _{2},\tau _{2} )} \bigl(\eta (1-u ) \bigr) \\ &\qquad{}\times E_{\xi _{i};v;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} u^{\xi _{1} },\dots,z_{r} u^{\xi _{r} } \bigr) \,du \\ &\quad =\sum_{h=0}^{n+m} (\eta )^{h} \frac{\varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 )}{ \varGamma (m+1 )\varGamma (n+1 )}\sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix} \\ &\qquad{} \times \frac{(\alpha )_{h} (-m)_{(h-r)} (-n)_{r} }{\varGamma (h+1 ) \varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+ \tau _{2} +1 )} E_{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (z_{1},\dots,z_{r} ). \end{aligned}$$
(35)

Corollary 2

If \(\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{t}^{x} (x-s ) ^{\alpha -1} (s-t )^{\nu -1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\eta (x-s ) \bigr)L_{m}^{ (\mu _{2},\tau _{2} )} \bigl(\eta (x-s ) \bigr) \\ &\qquad{}\times E_{\xi _{i};v;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (s-t )^{\xi _{1} },\dots,z_{r} (s-t ) ^{\xi _{r} } \bigr) \,ds \\ &\quad = (x-t )^{\alpha +\nu -1} \sum_{h=0}^{n+m} (\eta ) ^{h} (x-t )^{h} \frac{\varGamma (\mu _{1} n+\tau _{1} +1 ) \varGamma (\mu _{2} m+\tau _{2} +1 )}{\varGamma (m+1 ) \varGamma (n+1 )} \\ &\qquad{}\times \sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix}\frac{(\alpha )_{h} (-m)_{(h-r)} (-n)_{r} }{\varGamma (h+1 ) \varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+ \tau _{2} +1 )} \\ &\qquad{} \times E_{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (x-t )^{\xi _{1} },\dots,z _{r} (x-t )^{\xi _{r} } \bigr). \end{aligned}$$
(36)

Corollary 3

If \(\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \),; \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\int _{0}^{x}t^{\tau -1} (x-t )^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} (\eta t )L_{m}^{\mu _{1},\tau _{1} } (\eta t ) \\ &\qquad{}\times E_{\xi _{i};v; \varepsilon _{i} }^{\delta _{i};\kappa _{i};1} \bigl(z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr) E_{\xi _{i};\tau; \varepsilon _{i} }^{\omega _{i} ;\kappa _{i};1} \bigl(z_{1} (t )^{\xi _{1} },\dots,z _{r} (t )^{\xi _{r} } \bigr)\,dt \\ &\quad =x^{\tau +v-1} \sum_{h=0}^{n+m} ( \eta x )^{h} \frac{\varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 )}{ \varGamma (m+1 )\varGamma (n+1 )} \sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix} \\ &\qquad{}\times \frac{(\alpha )_{h} (-m)_{(h-r)} (-n)_{r} }{\varGamma (h+1 ) \varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+ \tau _{2} +1 )} E_{\xi _{i};v+\tau +h;\varepsilon _{i} }^{(\omega _{i} +\delta _{i} );\kappa _{i};1} \bigl(z_{1} x^{\xi _{1} },\dots,z _{r} x^{\xi _{r} } \bigr). \end{aligned}$$
(37)

Corollary 4

If \(\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\int _{0}^{z}t^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} (\eta t )L _{m}^{\mu _{1},\tau _{1} } (\eta t ) E_{\xi _{i};\nu; \varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} t ^{\xi _{1} },\dots,z_{r} t^{\xi _{r} } \bigr) \,dt \\ &\quad{}=z^{v-1} \sum_{h=0}^{n+m} ( \eta z )^{h} \frac{\varGamma (\mu _{1} n+\tau _{1} +1 )\varGamma (\mu _{2} m+\tau _{2} +1 )}{ \varGamma (m+1 )\varGamma (n+1 )} \sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix} \\ &\qquad{}\times \frac{(\alpha )_{h} (-m)_{(h-r)} (-n)_{r} }{\varGamma (h+1 ) \varGamma (\mu _{1} r+\tau _{1} +1 )\varGamma (\mu _{2} (h-r)+ \tau _{2} +1 )} E_{\xi _{i};\nu +h+1;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} z^{\xi _{1} },\dots,z_{r} z ^{\xi _{r} } \bigr). \end{aligned}$$
(38)

(ii) Putting \(\mu _{1} =\mu _{2} =\alpha '=\beta '=1\) and using Eq. (12), the results in Eqs. (30), (32), (33), and (34) reduced to the following form:

Corollary 5

If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{v-1} (1-u ) ^{\alpha -1} L_{n}^{ (1,\tau _{1} )} \bigl(\eta (1-u ); 1 \bigr)L_{m}^{ (1,\tau _{2} )} \bigl(\eta (1-u ); 1 \bigr) \\ &\qquad{}\times E_{\xi _{i};v;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} u^{\xi _{1} },\dots,z_{r} u^{\xi _{r} } \bigr) \,du \\ &\quad =\sum_{h=0}^{n+m} (\eta )^{h} \frac{\varGamma (n+\tau _{1} +1 )\varGamma (m+\tau _{2} +1 )}{\varGamma (m+1 ) \varGamma (n+1 )} \\ &\qquad{}\times \sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix}\frac{(\alpha )_{h} (-m)_{(h-r)} (-n)_{r} }{\varGamma (h+1 ) \varGamma (r+\tau _{1} +1 )\varGamma ((h-r)+\tau _{2} +1 )} E_{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i}; \kappa _{i};\rho _{i} } (z_{1},\dots,z_{r} ). \end{aligned}$$
(39)

Corollary 6

If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{t}^{x} (x-s ) ^{\alpha -1} (s-t )^{\nu -1} L_{n}^{ (1,\tau _{1} )} \bigl(\eta (x-s );1 \bigr)L_{m}^{ (1,\tau _{2} )} \bigl(\eta (x-s );1 \bigr) \\ &\qquad{}\times E_{\xi _{i};v;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (s-t )^{\xi _{1} },\dots,z_{r} (s-t ) ^{\xi _{r} } \bigr) \,ds \\ &\quad= (x-t )^{\alpha +\nu -1} \sum_{h=0}^{n+m} (\eta ) ^{h} (x-t )^{h} \frac{\varGamma (n+\tau _{1} +1 ) \varGamma (m+\tau _{2} +1 )}{\varGamma (m+1 )\varGamma (n+1 )} \\ &\qquad{}\times \sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix}\frac{(\alpha )_{h} (-m)_{(h-r)} (-n)_{r} }{\varGamma (h+1 ) \varGamma (r+\tau _{1} +1 )\varGamma ((h-r)+\tau _{2} +1 )} \\ &\qquad {}\times E_{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i}; \rho _{i} } \bigl(z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t )^{\xi _{r} } \bigr). \end{aligned}$$
(40)

Corollary 7

If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \), \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\int _{0}^{x}t^{\tau -1} (x-t )^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} (\eta t;1 )L_{m}^{\mu _{1},\tau _{1} } (\eta t;1 ) \\ &\qquad{}\times E_{\xi _{i};v; \varepsilon _{i} }^{\delta _{i};\kappa _{i};1} \bigl(z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr) E_{\xi _{i};\tau; \varepsilon _{i} }^{\omega _{i} ;\kappa _{i};1} \bigl(z_{1} (t )^{\xi _{1} },\dots,z _{r} (t )^{\xi _{r} } \bigr) \,dt \\ &\quad =x^{\tau +v-1} \sum_{h=0}^{n+m} ( \eta x )^{h} \frac{\varGamma (n+\tau _{1} +1 )\varGamma (m+\tau _{2} +1 )}{ \varGamma (m+1 )\varGamma (n+1 )} \sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix} \\ &\qquad{} \times \frac{(\alpha )_{h} (-m)_{(h-r)} (-n)_{r} }{\varGamma (h+1 ) \varGamma (r+\tau _{1} +1 )\varGamma ((h-r)+\tau _{2} +1 )} E_{\xi _{i};v+\tau +h;\varepsilon _{i} }^{(\omega _{i} + \delta _{i} );\kappa _{i};1} \bigl(z_{1} x^{\xi _{1} },\dots,z_{r} x ^{\xi _{r} } \bigr). \end{aligned}$$
(41)

Corollary 8

If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \), \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\int _{0}^{z}t^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} (\eta t;1 )L _{m}^{\mu _{1},\tau _{1} } (\eta t;1 ) E_{\xi _{i};\nu; \varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} t ^{\xi _{1} },\dots,z_{r} t^{\xi _{r} } \bigr) \,dt \\ &\quad =z^{v-1} \sum_{h=0}^{n+m} ( \eta z )^{h} \frac{\varGamma (n+ \tau _{1} +1 )\varGamma (m+\tau _{2} +1 )}{\varGamma (m+1 ) \varGamma (n+1 )} \sum_{r=0}^{h} \begin{pmatrix} {h} \\ {r} \end{pmatrix} \\ &\qquad{} \times \frac{(\alpha )_{h} (-m)_{(h-r)} (-n)_{r} }{\varGamma (h+1 ) \varGamma (r+\tau _{1} +1 )\varGamma ((h-r)+\tau _{2} +1 )} E_{\xi _{i};\nu +h+1;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} z^{\xi _{1} },\dots,z_{r} z^{\xi _{r} } \bigr). \end{aligned}$$
(42)

(iii) Putting \(\mu _{1} =\mu _{1} =0, \alpha '=\beta '=1\), the results in Eqs. (30), (32), (33), and (34) are reduced to the following form:

Corollary 9

If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{v-1} (1-u ) ^{\alpha -1} \bigl(1-\eta (1-u ) \bigr)^{m} E_{\xi _{i};v; \varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} u ^{\xi _{1} },\dots,z_{r} u^{\xi _{r} } \bigr) \,du \\ &\quad =\sum_{h=0}^{m} ( \eta )^{h} (-m )_{h} (\alpha )_{h} E_{\xi _{i};v+h+\alpha;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (z_{1},\dots,z_{r} ). \end{aligned}$$
(43)

Corollary 10

If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{t}^{x} (x-s ) ^{\alpha -1} (s-t )^{\nu -1} \bigl(1-\eta (x-s ) \bigr) ^{m} \\ &\quad{}\times E_{\xi _{i};v;\varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (s-t )^{\xi _{1} },\dots,z_{r} (s-t ) ^{\xi _{r} } \bigr) \,ds= (x-t )^{\alpha +\nu -1} \sum_{h=0} ^{m} \bigl(\eta (x-t ) \bigr)^{h} \\ &\quad{}\times (-m )_{h} (\alpha )_{h} E_{\xi _{i};v+h+\alpha; \varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t )^{\xi _{r} } \bigr). \end{aligned}$$
(44)

Corollary 11

If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\int _{0}^{x}t^{\tau -1} (x-t )^{v-1} (1-\eta t ) ^{m} E_{\xi _{i};v; \varepsilon _{i} }^{\delta _{i};\kappa _{i};1} \bigl(z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr) \\ &\qquad{}\times E_{\xi _{i};\tau; \varepsilon _{i} }^{\omega _{i};\kappa _{i};1} \bigl(z_{1} (t )^{\xi _{1} },\dots,z_{r} (t ) ^{\xi _{r} } \bigr)\,dt \\ &\quad =x^{\tau +v-1} \sum_{h=0}^{m} (\eta x )^{h} (-m ) _{h} (\alpha )_{h} E_{\xi _{i};v+\tau +h;\varepsilon _{i} }^{(\omega _{i} +\delta _{i} );\kappa _{i};1} \bigl(z_{1} x^{\xi _{1} }, \dots,z _{r} x^{\xi _{r} } \bigr). \end{aligned}$$
(45)

Corollary 12

If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then

$$\begin{aligned} &\int _{0}^{z}t^{v-1} (1-\eta t )^{m} E_{\xi _{i};\nu; \varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } \bigl(z_{1} t ^{\xi _{1} },\dots,z_{r} t^{\xi _{r} } \bigr) \,dt \\ &\quad =z^{v-1} \sum_{h=0}^{m} (\eta z )^{h} (-m ) _{h} (\alpha )_{h} E_{\xi _{i};\nu +h+1;\varepsilon _{i} }^{\delta _{i} ;\kappa _{i};\rho _{i} } \bigl(z_{1} z^{\xi _{1} }, \dots,z_{r} z^{\xi _{r} } \bigr). \end{aligned}$$
(46)

Remark 3.1

If we put \(\kappa _{1} =\cdots =\kappa _{r} =\varepsilon _{1} =\cdots = \varepsilon _{r} =1\), then Eqs. (35)–(46) are reduced to Eqs. (3.1), (3.2), (3.3), (3.4), (3.6), (3.7), (3.8), (3.9), (3.11), (3.16), (3.17), (3.18) established in Agarwal et al. [33].

(iv) Setting \(\rho _{i} =\kappa _{i} =\varepsilon _{i} =1, \xi _{1} =\cdots =\xi _{r} =1\), the multivariate Mittag-Leffler function reduces to the confluent hypergeometric series, and we have the following results, which are obtained from the main results in (17), (18), (19), (22), (24), (27), (28), (29), (30), (32), (33), and (34).

Corollary 13

The following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{v-1} (1-u ) ^{\alpha -1} \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z_{1} u^{\xi _{1} },\dots,z_{r} u^{\xi _{r} } \bigr] \,du \\ &\quad =\varphi _{2}^{(r)} [\delta _{1},\dots,\delta _{r}; v+\alpha; z _{1}, \dots,z_{r} ]. \end{aligned}$$
(47)

Corollary 14

Then following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{t}^{x} (x-s ) ^{\alpha -1} (s-t )^{v-1} \\ &\qquad{}\times\varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z_{1} (s-t )^{\xi _{1} }, \dots,z_{r} (s-t )^{\xi _{r} } \bigr] \,ds \\ &\quad = (x-t )^{\alpha +v-1} \varphi _{2}^{(r)} \bigl[\delta _{1} ,\dots,\delta _{r}; v+\alpha; z_{1} (s-t )^{\xi _{1} }, \dots,z_{r} (s-t )^{\xi _{r} } \bigr]. \end{aligned}$$
(48)

Corollary 15

Then following integral equality holds:

$$\begin{aligned} &\int _{0}^{x}t^{\tau -1} (x-t )^{v-1} \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z_{1} (x-t ) ^{\xi _{1} },\dots,z_{r} (x-t )^{\xi _{r} } \bigr] \\ &\qquad{}\times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z _{1} t^{\xi _{1} },\dots,z_{r} t^{\xi _{r} } \bigr]\,dt \\ &\quad =x^{\tau +v-1} \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r} ; v+\tau; z_{1} t^{\xi _{1} },\dots,z_{r} t^{\xi _{r} } \bigr]. \end{aligned}$$
(49)

Corollary 16

Then following integral equality holds:

$$\begin{aligned} &\int _{0}^{z}t^{v-1} \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z_{1} t^{\xi _{1} },\dots,z_{r} t^{\xi _{r} } \bigr] \,dt \\ &\quad =z^{v-1} \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v+1; z_{1} z^{\xi _{1} },\dots,z_{r} z^{\xi _{r} } \bigr]. \end{aligned}$$
(50)

Corollary 17

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{t}^{x} (x-s ) ^{\alpha -1} (s-t )^{\nu -1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ',\eta (x-s ) \bigr)L_{m} ^{ (\mu _{2},\tau _{2} )} \bigl( \alpha ',\eta (x-s ) \bigr) \\ &\qquad{}\times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; _{1} (s-t )^{\xi _{1} },\dots,z_{r} (s-t ) ^{\xi _{r} } \bigr] \,ds \\ &\quad= (x-t )^{\alpha +\nu -1} \sum_{h=0}^{n+m} \bigl(\eta (x-t ) \bigr)^{h} \Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} \\ &\qquad{} \times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r};v+h+ \alpha; z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr], \end{aligned}$$
(51)

where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).

Corollary 18

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{v-1} (1-u ) ^{\alpha -1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ', \eta (1-u ) \bigr)L_{m}^{ (\mu _{2},\tau _{2} )} \bigl( \alpha ',\eta (1-u ) \bigr) \\ &\qquad{}\times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z _{1} u^{\xi _{1} },\dots,z_{r} u^{\xi _{r} } \bigr] \,du \\ &\quad =\sum_{h=0}^{n+m} ( \eta )^{h} \Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} \varphi _{2}^{(r)} [\delta _{1},\dots,\delta _{r}; v+\alpha; z _{1},\dots,z_{r} ], \end{aligned}$$
(52)

where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).

Corollary 19

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\int _{0}^{x}t^{\tau -1} (x-t )^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ',\eta t \bigr)L_{m}^{\mu _{1}, \tau _{1} } \bigl(\alpha ',\eta t \bigr) \\ &\qquad{}\times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z _{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr] \\ &\qquad{}\times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z _{1} (t )^{\xi _{1} },\dots,z_{r} (t )^{\xi _{r} } \bigr]\,dt \\ &\quad =x^{\tau +v-1} \sum_{h=0}^{n+m} (\eta x )^{h} \Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r};v+\tau +h; z _{1} x^{\xi _{1} }, \dots,z_{r} x^{\xi _{r} } \bigr], \end{aligned}$$
(53)

where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).

Corollary 20

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\int _{0}^{z}t^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ', \eta t \bigr)L_{m}^{\mu _{1},\tau _{1} } \bigl(\alpha ',\eta t \bigr) \varphi _{2}^{(r)} \bigl[\delta _{1}, \dots,\delta _{r}; v; z_{1} t ^{\xi _{1} }, \dots,z_{r} t^{\xi _{r} } \bigr] \,dt \\ &\quad =z^{v-1} \sum_{h=0}^{n+m} (\eta z )^{h} \Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; \nu +h+1; z _{1} t^{\xi _{1} }, \dots,z_{r} t^{\xi _{r} } \bigr], \end{aligned}$$
(54)

where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).

Corollary 21

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{0}^{1}u^{v-1} (1-u ) ^{\alpha -1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ', \eta (1-u ) \bigr)L_{m}^{ (\mu _{2},\tau _{2} )} \bigl( \alpha ',\eta (1-u ) \bigr) \\ &\qquad{}\times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z _{1} u^{\xi _{1} },\dots,z_{r} u^{\xi _{r} } \bigr] \,du \\ &\quad =\sum_{h=0}^{n+m} ( \eta )^{h} \wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} \varphi _{2}^{(r)} [\delta _{1},\dots,\delta _{r}; v+h+\alpha; z_{1},\dots,z_{r} ], \end{aligned}$$
(55)

where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).

Corollary 22

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int _{t}^{x} (x-s ) ^{\alpha -1} (s-t )^{\nu -1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ',\eta (x-s ) \bigr)L_{m} ^{ (\mu _{2},\tau _{2} )} \bigl( \alpha ',\eta (x-s ) \bigr) \\ &\qquad{}\times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z _{1} (s-t )^{\xi _{1} },\dots,z_{r} (s-t ) ^{\xi _{r} } \bigr] \,ds \\ &\quad = (x-t )^{\alpha +\nu -1} \sum_{h=0}^{n+m} \bigl(\eta (x-t ) \bigr)^{h} \wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} \\ &\qquad{} \times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v+h+ \alpha; z_{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr], \end{aligned}$$
(56)

where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).

Corollary 23

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\int _{0}^{x}t^{\tau -1} (x-t )^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ',\eta t \bigr)L_{m}^{\mu _{1}, \tau _{1} } \bigl(\alpha ',\eta t \bigr) \\ &\qquad{}\times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z _{1} (x-t )^{\xi _{1} },\dots,z_{r} (x-t ) ^{\xi _{r} } \bigr] \\ &\qquad{}\times \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v; z _{1} (t )^{\xi _{1} },\dots,z_{r} (t )^{\xi _{r} } \bigr]\,dt \\ &\quad =x^{\tau +v-1} \sum_{h=0}^{n+m} (\eta x )^{h} \wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; v+\tau +h; z _{1} x^{\xi _{1} }, \dots,z_{r} x^{\xi _{r} } \bigr], \end{aligned}$$
(57)

where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).

Corollary 24

If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:

$$\begin{aligned} &\int _{0}^{z}t^{v-1} L_{n}^{ (\mu _{1},\tau _{1} )} \bigl(\beta ', \eta t \bigr)L_{m}^{\mu _{1},\tau _{1} } \bigl(\alpha ',\eta t \bigr) \varphi _{2}^{(r)} \bigl[\delta _{1}, \dots,\delta _{r}; v; z_{1} t ^{\xi _{1} }, \dots,z_{r} t^{\xi _{r} } \bigr] \,dt \\ &\quad =z^{v-1} \sum_{h=0}^{n+m} (\eta z )^{h} \wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ',\beta '} \varphi _{2}^{(r)} \bigl[\delta _{1},\dots,\delta _{r}; \nu +h+1; z _{1} z^{\xi _{1} }, \dots,z_{r} z^{\xi _{r} } \bigr], \end{aligned}$$
(58)

where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).

(v) Setting \(\rho _{i} =\kappa _{i} =\varepsilon _{i} =\delta _{i} =\xi _{i} =v=r=1\), the multivariate Mittag-Leffler function is reduced to the exponential function \(E_{1;1; 1}^{1;1;1} (z )=E_{1,1} (z )=\exp (z)\); and we can find a similar line of results associated with exponential function from Eqs. (47)–(58).