Analytical Evaluation of Fractional Calculus and Transforms Involving the Generalized Srivastava Triple Hypergeometric Function H_B,q,a^η,ξ

1. Introduction and Preliminaries

Srivastava and Karlsson [23] introduced and examined triple hypergeometric functions, offering a table of 205 different functions of this type. In [19,20], Srivastava presented the triple hypergeometric functions $H_A$, $H_B$, and $H_C$ of the second order. $H_A$ is the generalization of both F1 and F2, whereas $H_B$, and $H_C$ are known to be generalizations of Appell’s hypergeometric functions $F_1$ and $F_2$.

This study will concentrate on Srivastava’s triple hypergeometric function $H_B$, which is provided as [23], p. 43, 1.5(11) to 1.5(13) (see also [19] and [22], p. 68)

$${\mathbf{H}}_B({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3):=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{\left({\chi }_1\right)}_{i+k}{\left({\chi }_2\right)}_{i+j}{\left({\chi }_3\right)}_{j+k})}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{B({\chi }_1+i+k,{\chi }_2+i+j)}{B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}.$$

(1)

In [6], p. 43, the convergence region for the hypergeometric series ${\mathbf{H}}_B(·)$ is given as ${\Phi }_1<r_1$, ${\Phi }_2<r_2$, ${\Phi }_3<r_3$ , with the condition

$$r_1+r_2+r_3+2\sqrt{r_1{r}_2{r}_3}=1.$$

(2)

Exton’s function $X_4$ is a different type of hypergeometric function defined as

$$X_4({\chi }_1,{\chi }_2;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3):=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{\left({\chi }_1\right)}_{2i+j+k}{\left({\chi }_2\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}},$$

(3)

with the convergence region for the series $2\sqrt{|{\Phi }_1|}+{(\sqrt{|{\Phi }_2|}+\sqrt{|{\Phi }_3|})}^2<1$.

Now, we find it suitable to introduce a new parameter b into $H_B(·)$ as follows

$${\mathbf{H}}_B^{\left(b\right)}({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{B({\chi }_1+b+i+k,{\chi }_2+b+i+j)}{B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}},$$

(4)

which reduces to (1) when $b=0$.

Where ${(\Phi )}_i$ is the Pochhammer symbol defined by

$${(\Phi )}_i={\displaystyle \frac{\Gamma (\Phi +i)}{\Gamma (\Phi )}}=\left\{\begin{array}{c}1,(i=0,\Phi \in \mathsf{C}\setminus \{0\left\}\right),\hfill \\ \\ \Phi (\Phi +1)...(\Phi +i-1),(i\in \mathsf{N},\Phi \in \mathsf{C})\hfill \end{array}\right\}.$$

(5)

Here, the classical Gamma function [13] is defined as

$$\Gamma (\Phi )={\int }_0^{\infty }{e}^{-t}{t}^{\Phi -1}dt,(R(\Phi )>0),$$

(6)

and the classical Beta function $B({\Phi }_1,{\Phi }_2)$ is defined as [11], (5.12.1)

$$B({\Phi }_1,{\Phi }_2)=\left\{\begin{array}{cc}{\displaystyle {\int }_0^1{t}^{{\Phi }_1-1}{(1-t)}^{{\Phi }_2-1}dt,}\hfill & (R\left({\Phi }_1\right)>0,R\left({\Phi }_2\right)>0)\hfill \\ \\ {\displaystyle \frac{\Gamma \left({\Phi }_1\right)\Gamma \left({\Phi }_2\right)}{\Gamma ({\Phi }_1+{\Phi }_2)},}\hfill & (({\Phi }_1,{\Phi }_2)\in \mathsf{C}\setminus {\mathsf{Z}}_0^{-}).\hfill \end{array}\right.$$

(7)

In 1997, Chaudhry et al. [2], q.(1.7) introduced a q-extension of $B({\Phi }_1,{\Phi }_2)$ given by

$$B_q({\Phi }_1,{\Phi }_2)={\int }_0^1{t}^{{\Phi }_1-1}{(1-t)}^{{\Phi }_2-1}exp\left[{\displaystyle \frac{-q}{t(1-t)}}\right]dt,(R\left(q\right)\ge 0).$$

(8)

In 2018, Shadab et al. [15] has given a further extension of the Beta function as

$$B_q^{\eta }({\Phi }_1,{\Phi }_2)={\int }_0^1{t}^{{\Phi }_1-1}{(1-t)}^{{\Phi }_2-1}{E}_{\eta }\left[{\displaystyle \frac{-q}{t(1-t)}}\right]dt,(R\left(q\right)\ge 0),$$

(9)

which reduces to (8) when $\eta =1$ and $E_{\eta }\left(z\right)$ denotes the Mittag-Leffler function defined by [10],

$$E_{\eta }\left(z\right)=\sum _{i=0}^{\infty }{\displaystyle \frac{z^i}{\Gamma (\eta i+1)}},(z\in \mathsf{C};\eta >0).$$

In 2020, Oraby et al. [12] introduced a generalized Beta function in the form

$$B_{q,a}^{\eta ,\xi }({\Phi }_1,{\Phi }_2)={\int }_0^1{t}^{{\Phi }_1-1}{(1-t)}^{{\Phi }_2-1}{E}_{\eta ,a}\left[{\displaystyle \frac{-q}{t^{\xi }{(1-t)}^{\xi }}}\right]dt,$$

(10)

$$(R\left({\Phi }_1\right)>0,R\left({\Phi }_2\right)>0,R\left(q\right)\ge 0,R\left(\xi \right)>0),$$

which reduces to (9) when $a=\xi =1$.

Where $E_{\eta ,a}\left(w\right)$ is the generalized Mittag-Leffler function introduced in 1905 by Wiman [24] defined as

$$E_{\eta ,a}\left(z\right)=\sum _{i=0}^{\infty }{\displaystyle \frac{z^i}{\Gamma (\eta i+a)}},(z,a\in \mathsf{C};\eta >0,R\left(a\right)>0).$$

Abubakar [1] introduced a generalized Beta function in 2021, as

$${}^{\Psi }B_{\tau }^{\xi }({\Phi }_1,{\Phi }_2)={}^{\Psi }B_{\tau }^{\xi }\left.\left[\begin{array}{c}{(c_m,{\acute{c}}_m)}_{1,{\mu }_1}\\ {(d_n,{\acute{d}}_n)}_{1,{\mu }_2}\end{array}\right|{\Phi }_1,{\Phi }_2\right]$$

$$={\int }_0^1{t}^{{\Phi }_1-1}{(1-t)}^{{\Phi }_2-1}{}_{{\mu }_1}\Psi _{{\mu }_2}\left(-{\displaystyle \frac{\tau }{t^{\xi }{(1-t)}^{\xi }}}\right)dt,$$

(11)

(${\Phi }_1,{\Phi }_2,\tau ,\xi ,c_m,d_n\in \mathsf{C};$ ${\acute{c}}_m,{\acute{d}}_n\in \mathsf{R},$$R\left({\Phi }_1\right)>0,R\left({\Phi }_2\right)>0,R\left(\tau \right)>0,R\left(\xi \right)>0$ $andm=1,...,{\mu }_1,n=1,...,{\mu }_2$)

${}_{{\mu }_1}\Psi _{{\mu }_2}$ is the Fox-Wright function ([7,8]) defined as

$${}_{{\mu }_1}\Psi _{{\mu }_2}\left.\left[\begin{array}{c}{(c_m,{\acute{c}}_m)}_{1,{\mu }_1}\\ {(d_n,{\acute{d}}_n)}_{1,{\mu }_2}\end{array}\right|z\right]={}_{{\mu }_1}\Psi _{{\mu }_2}\left[\begin{array}{c}(c_1,{\acute{c}}_1),...,(c_{{\mu }_1},{\acute{c}}_{{\mu }_1});\\ (d_1,{\acute{d}}_1),...,(d_{{\mu }_2},{\acute{d}}_{{\mu }_2});\end{array}z\right]$$

$$=\sum _{k=0}^{\infty }{\displaystyle \frac{{\displaystyle \prod _{m=1}^{{\mu }_1}\Gamma (c_m+{\acute{c}}_{m}k)}}{{\displaystyle \prod _{n=1}^{{\mu }_2}\Gamma (d_n+{\acute{d}}_{n}k)}}}{\displaystyle \frac{z^k}{k!}},$$

(12)

where $z,c_m,d_n\in \mathsf{C}$, ${\acute{c}}_m\ge 0$, ${\acute{d}}_n\ge 0$ ($m=1,...,{\mu }_1;n=1,...,{\mu }_2$). By taking

$$\Delta =\sum _{n=1}^{{\mu }_2}{\acute{d}}_n-\sum _{m=1}^{{\mu }_1}{\acute{c}}_m,\delta =\prod _{m=1}^{{\mu }_1}{\acute{c}}_m^{-{\acute{c}}_m}\prod _{n=1}^{{\mu }_2}{\acute{d}}_n^{{\acute{d}}_n},{\mu }^{*}=\sum _{n=1}^{{\mu }_2}{d}_n-\sum _{m=1}^{{\mu }_1}{c}_m+\frac{1}{2}({\mu }_1-{\mu }_2),$$

(12) converges for (i) $\left|z\right|<\infty$ when $\Delta >-1$,

(ii) $\left|z\right|<\delta$ when $\Delta =-1$, and

(iii) $\left|z\right|=\delta$ if, in addition, $R\left({\mu }^{*}\right)>\frac{1}{2}$.

If ${\acute{c}}_1=...={\acute{c}}_{{\mu }_1}={\acute{d}}_1=...={\acute{d}}_{{\mu }_2}=1$ eq. (12) becomes the generalized hypergeometric function ${}_{{\mu }_1}F_{{\mu }_2}$ [26]

$${}_{{\mu }_1}\Psi _{{\mu }_2}\left[\begin{array}{c}(c_1,1),...,(c_{{\mu }_1},1);\\ (d_1,1),...,(d_{{\mu }_2},1);\end{array}z\right]={\displaystyle \frac{{\displaystyle \prod _{m=1}^{{\mu }_1}\Gamma \left(c_m\right)}}{{\displaystyle \prod _{n=1}^{{\mu }_2}\Gamma \left(d_n\right)}}}{}_{{\mu }_1}F_{{\mu }_2}\left[\begin{array}{c}{c}_1,...,c_{{\mu }_1};\\ d_1,...,d_{{\mu }_2};\end{array}z\right].$$

(13)

The layout of this paper is as follows. In Section 2, we define the generalized Srivastava triple hypergeometric function ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$ using the generalized Beta function given in (10) and provide a numerical approximation table. Section 3 presents several integral representations involving the generalized Mittag-Leffler function and Exton’s function $X_4$, along with some differential properties. In Section 4, we examine classical integral transforms, including the Euler–Beta, Laplace, Mellin, and Whittaker transforms. Section 5 focuses on the application of Riemann-Liouville fractional integral and differential operators to ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$, leading to new results in the context of fractional calculus. Finally, in Section 6 we establishes several recurrence relations to further characterize the generalized function. This work is also motivated by previous studies on Srivastava’s triple hypergeometric function, as discussed in [3,5].

2. The Generalized Srivastava Triple Hypergeometric Function ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

Here, we define the generalized Srivastava triple ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$ through the generalized Beta function defined in (10)

$${\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{B_{q,a}^{\eta ,\xi }({\chi }_1+i+k,{\chi }_2+i+j)}{B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}},$$

(14)

where, ${\chi }_1,{\chi }_2,{\chi }_3\in \mathsf{C}$ and ${\varrho }_1,{\varrho }_2,{\varrho }_3\in \mathsf{C}\setminus {\mathsf{Z}}_0^{-}$ and the region of convergence is $|{\Phi }_1|<r_1$, $|{\Phi }_2|<r_2$, $|{\Phi }_3|<r_3$, with $r_1$+ $r_2$+ $r_3$+$2\sqrt{r_1{r}_2{r}_3}=1$.

In terms of the generalized Mittag-Leffler function, we now obtain ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$ using the generalized beta function provided in (10) in (14) as follows:

$${\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$\times {\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}{E}_{\eta ,a}\left[-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right]dt.$$

(15)

We use of the above eq. (15) throughout the paper.

Example 1.Table 1 provides numerical approximation values of the generalized Srivastava triple hypergeometric function $H_{B,q,a}^{\eta ,\xi }(·)$ as defined in (14), evaluated for various values of the parameters q, a, $\eta$, and $\xi$, with the indices considered up to the nth term; for example, $i=j=k=7$.

3. Certain Properties of ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

We go over a few integral representations and derivative properties in this section.

1. Integral representations

The integral representations for the ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$ function, which involves the product of Exton’s function $X_4$ and the Mittag-Leffler function with two parameters, are obtained as follows.

Theorem 1.

For $R\left(q\right)\ge 0$, $R\left({\chi }_n\right)>0$ ($n=1,2,3$) and $min\{R\left({\chi }_1\right),R\left({\chi }_2\right)\}>0,$ the integral representation of ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$ is as follows :

$${\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)={\displaystyle \frac{\Gamma ({\chi }_1+{\chi }_2)}{\Gamma \left({\chi }_1\right)\Gamma \left({\chi }_2\right)}}{\int }_0^1{t}^{{\chi }_1-1}{(1-t)}^{{\chi }_2-1}$$

$$\times E_{\eta ,a}\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)X_4({\chi }_1+{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1(1-t)t,{\Phi }_2(1-t),{\Phi }_{3}t)dt.$$

(16)

Proof: By changing the order of integration and summation (with uniform convergence of the integral) in (15) and using the relation (7), after simplification, utilizing the Exton’s triple hypergeometric function (3), we get the result (16).

Remark 1. Through appropriate changes of the integration variable, we now discuss some special cases. To obtain the following integral representations:

$$\left(i\right){\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$={\displaystyle \frac{\Gamma ({\chi }_1+{\chi }_2)}{\Gamma \left({\chi }_1\right)\Gamma \left({\chi }_2\right)}}{\displaystyle \frac{{(x_2-x_3)}^{{\chi }_1}{(x_1-x_3)}^{{\chi }_2}}{{(x_2-x_1)}^{{\chi }_1+{\chi }_2-1}}}{\int }_{x_1}^{x_2}{\displaystyle \frac{{(x_4-x_1)}^{{\chi }_1-1}{(x_2-x_4)}^{{\chi }_2-1}}{{(x_4-x_3)}^{{\chi }_1+{\chi }_2}}}$$

$$\times E_{\eta ,a}\left(-{\displaystyle \frac{q}{{\Lambda }_1^{\xi }{\Lambda }_2^{\xi }}}\right)X_4({\chi }_1+{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1{\Lambda }_1{\Lambda }_2,{\Phi }_2{\Lambda }_1,{\Phi }_3{\Lambda }_2)d{x}_4,$$

(17)

where

$${\Lambda }_1={\displaystyle \frac{(x_1-x_3)(x_2-x_4)}{(x_2-x_1)(x_4-x_3)}},{\Lambda }_2={\displaystyle \frac{(x_2-x_3)(x_4-x_1)}{(x_2-x_1)(x_4-x_3)}}(x_3<x_1<x_2);$$

$$\left(ii\right){\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)={\displaystyle \frac{2\Gamma ({\chi }_1+{\chi }_2)}{\Gamma \left({\chi }_1\right)\Gamma {\chi }_2}}{\int }_0^{{\displaystyle \frac{\pi }{2}}}{\left({sin}^{2}x\right)}^{{\chi }_1-{\displaystyle \frac{1}{2}}}{\left({cos}^{2}x\right)}^{{\chi }_2-{\displaystyle \frac{1}{2}}}$$

$$\times E_{\eta ,a}\left(-{\displaystyle \frac{q}{{\Lambda }_1^{\xi }{\Lambda }_2^{\xi }}}\right)X_4({\chi }_1+{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1{\Lambda }_1{\Lambda }_2,{\Phi }_2{\Lambda }_1,{\Phi }_3{\Lambda }_2)dx,$$

(18)

where,

$${\Lambda }_1={cos}^{2}x,{\Lambda }_2={sin}^{2}x;$$

$$\left(iii\right){\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$={\displaystyle \frac{2\Gamma ({\chi }_1+{\chi }_2){(1+x_3)}^{{\chi }_1}}{\Gamma \left({\chi }_1\right)\Gamma \left({\chi }_2\right)}}{\int }_0^{{\displaystyle \frac{\pi }{2}}}{\displaystyle \frac{{\left({sin}^2{x}_4\right)}^{{\chi }_1-\frac{1}{2}}{\left({cos}^2{x}_4\right)}^{{\chi }_2-\frac{1}{2}}}{{(1+x_3{sin}^2{x}_4)}^{{\chi }_1+{\chi }_2}}}$$

$$\times E_{\eta ,a}\left(-{\displaystyle \frac{q}{{\Lambda }_1^{\xi }{\Lambda }_2^{\xi }}}\right)X_4({\chi }_1+{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1{\Lambda }_1{\Lambda }_2,{\Phi }_2{\Lambda }_1,{\Phi }_3{\Lambda }_2)d{x}_4,$$

(19)

where

$${\Lambda }_1={\displaystyle \frac{{cos}^2{x}_4}{1+x_3{sin}^2{x}_4}},{\Lambda }_2={\displaystyle \frac{(1+x_3){sin}^2{x}_4}{1+x_3{sin}^2{x}_4}}(x_3>-1);$$

and

$$\left(iv\right){\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$={\displaystyle \frac{2\Gamma ({\chi }_1+{\chi }_2)x_3^{{\chi }_1-1}}{\Gamma \left({\chi }_1\right)\Gamma \left({\chi }_2\right)}}{\int }_0^{{\displaystyle \frac{\pi }{2}}}{\displaystyle \frac{{\left({sin}^2{x}_4\right)}^{{\chi }_1-\frac{1}{2}}{\left({cos}^2{x}_4\right)}^{{\chi }_2-\frac{1}{2}}}{{({cos}^2{x}_4+x_3{sin}^2{x}_4)}^{{\chi }_1+{\chi }_2}}}$$

$$\times E_{\eta ,a}\left(-{\displaystyle \frac{q}{{\Lambda }_1^{\xi }{\Lambda }_2^{\xi }}}\right)X_4({\chi }_1+{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1{\Lambda }_1{\Lambda }_2,{\Phi }_2{\Lambda }_1,{\Phi }_3{\Lambda }_2)d{x}_4,$$

(20)

where

$${\Lambda }_1={\displaystyle \frac{{cos}^2{x}_4}{{cos}^2{x}_4+x_3{sin}^2{x}_4}},{\Lambda }_2={\displaystyle \frac{x_3{sin}^2{x}_4}{{cos}^2{x}_4+x_3{sin}^2{x}_4}}(x_3>0),$$

we put

$$\begin{array}{ccc}\hfill \left(i\right)t& =& {\displaystyle \frac{(x_2-x_3)(x_4-x_1)}{(x_2-x_1)(x_4-x_3)}},{\displaystyle \frac{dt}{d{x}_4}}={\displaystyle \frac{(x_2-x_3)(x_1-x_3)}{(x_2-x_1){(x_4-x_3)}^2}},\hfill \\ \hfill \left(ii\right)t& =& {sin}^{2}x,{\displaystyle \frac{dt}{dx}}=2sinxcosx,\hfill \\ \hfill \left(iii\right)t& =& {\displaystyle \frac{(1+x_3){sin}^2{x}_4}{1+x_3{sin}^2{x}_4}},{\displaystyle \frac{dt}{d{x}_4}}={\displaystyle \frac{2(1+x_3)sin{x}_{4}cos{x}_4}{{(1+x_3{sin}^2{x}_4)}^2}},\hfill \\ \hfill \left(iv\right)t& =& {\displaystyle \frac{x_3{sin}^2{x}_4}{{cos}^2{x}_4+x_3{sin}^2{x}_4}},{\displaystyle \frac{dt}{d{x}_4}}={\displaystyle \frac{2x_{3}sin{x}_{4}cos{x}_4}{{({cos}^2{x}_4+x_3{sin}^2{x}_4)}^2}}.\hfill \end{array}$$

2. Derivative properties

Here we discuss some derivative propeties of ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$.

Theorem 2.

$.$ The following derivative formula holds for ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

$${\left({\displaystyle \frac{d}{dq}}\right)}^m\left[q^{a-1}{\mathbf{H}}_{B,q^{\eta },a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)\right]$$

$$=q^{a-m-1}{\mathbf{H}}_{B,q^{\eta },a-m}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3),$$

(21)

where $a,{\chi }_n\in \mathsf{C}$, ${\varrho }_1,{\varrho }_2,{\varrho }_3\in \mathsf{C}\setminus {\mathsf{Z}}_0^{-}$; $m\in \mathsf{N}$ and $R(a-m)>0,R\left(a\right)>0,\eta >0,R\left({\chi }_n\right)>0;$$n=1,2,3$.

Proof: Using (15) and employing term-wise differentiation m times in L.H.S of (21), we obtain

$${\left({\displaystyle \frac{d}{dq}}\right)}^m\left[q^{a-1}{\mathbf{H}}_{B,q^{\eta },a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)\right]$$

$$={\left({\displaystyle \frac{d}{dq}}\right)}^m\left[q^{a-1}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}\right.$$

$$\left.\times {\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}{E}_{\eta ,a}\left\{-{\displaystyle \frac{q^{\eta }}{t^{\xi }{(1-t)}^{\xi }}}\right\}dt\right]$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}{\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}$$

$$\times \sum _{l=0}^{\infty }{\displaystyle \frac{1}{\Gamma (l\eta +a)}}{\left(-{\displaystyle \frac{1}{t^{\xi }{(1-t)}^{\xi }}}\right)}^l{\left({\displaystyle \frac{d}{dq}}\right)}^m\left[q^{l\eta +a-1}\right]dt$$

$$=q^{a-m-1}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}{\int }_0^1{t}^{{\chi }_1+i+k-1}$$

$$\times {(1-t)}^{{\chi }_2+i+j-1}\sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{q^{\eta }}{t^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{1}{\Gamma (l\eta +a-m)}}dt$$

$$=q^{a-m-1}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}{\int }_0^1{t}^{{\chi }_1+i+k-1}$$

$$\times {(1-t)}^{{\chi }_2+i+j-1}{E}_{\eta ,a-m}\left(-{\displaystyle \frac{q^{\eta }}{t^{\xi }{(1-t)}^{\xi }}}\right)dt$$

$$=q^{a-m-1}{\mathbf{H}}_{B,q^{\eta },a-m}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3).$$

Theorem 3.

$.$ Let $a\in \mathsf{C},$${\varrho }_1,{\varrho }_2,{\varrho }_3\in \mathsf{C}\setminus {\mathsf{Z}}_0^{-},$ and $\eta >0,R\left(a\right)>0$, then ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$ satisfies the following differentiation formula

$${\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$=a{\mathbf{H}}_{B,q,a+1}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)+\eta q{\displaystyle \frac{d}{dq}}{\mathbf{H}}_{B,q,a+1}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3).$$

(22)

Proof: Using (15) in the R.H.S of (22), yields

$$a{\mathbf{H}}_{B,q,a+1}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)+\eta q{\displaystyle \frac{d}{dq}}{\mathbf{H}}_{B,q,a+1}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$=a{\mathbf{H}}_{B,q,a+1}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)+\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$\times {\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}\sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{1}{t^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{{\eta }^l{q}^l}{\Gamma (l\eta +a+1)}}dt$$

(23)

Now using the relation $\Gamma (a+1)=a\Gamma \left(a\right)$ in (23), we get

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}{\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}$$

$$\times \sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{1}{\Gamma (l\eta +a)}}dt$$

$$={\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3).$$

4. Integral Transform of ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

The focus here is on analyzing integral transforms of the function ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$, specifically the Euler-Beta, Laplace, Mellin, and Whittaker transforms.

1. Euler-Beta Transform

The Euler-Beta transform of the function $f\left(q\right)$ is defined as [18]

$$\mathbf{B}\left\{f\left(q\right);{\alpha }_1,{\alpha }_2\right\}={\int }_0^1{q}^{{\alpha }_1-1}{(1-q)}^{{\alpha }_2-1}f\left(q\right)dq.$$

(24)

Theorem 4.

For $a,{\alpha }_1,{\alpha }_2,\xi ,({\chi }_1+i+k),({\chi }_2+i+j)\in \mathsf{C};$$\eta >0,R\left(a\right)>0,R\left({\alpha }_1\right)>0,R\left({\alpha }_2\right)>0,R({\chi }_1+i+k)>0,R({\chi }_2+i+j)>0$, we have the following Euler-Beta transform of ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

$$\mathbf{B}\left\{{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3);{\alpha }_1,{\alpha }_2\right\}=\Gamma \left({\alpha }_2\right)\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}$$

$$\times {\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}{}^{\Psi }B_1^{\xi }\left.\left[\begin{array}{c}(1,1),({\alpha }_1,1)\\ (a,\eta ),({\alpha }_1+{\alpha }_2,1)\end{array}\right|({\chi }_1+i+k),({\chi }_2+i+j)\right].$$

(25)

Proof: Using (24) in (15), we obtain

$$\mathbf{B}\left\{{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3);{\alpha }_1,{\alpha }_2\right\}$$

$$={\int }_0^1{q}^{{\alpha }_1-1}{(1-q)}^{{\alpha }_2-1}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)dq$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$\times \left({\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}\left\{{\int }_0^1{q}^{{\alpha }_1-1}{(1-q)}^{{\alpha }_2-1}{E}_{\eta ,a}\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)dq\right\}dt\right).$$

(26)

By evaluating the integral as

$$\begin{array}{ccc}\hfill & & {\int }_0^1{q}^{{\alpha }_1-1}{(1-q)}^{{\alpha }_2-1}{E}_{\eta ,a}\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)dq\hfill \\ \hfill & & =\sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{1}{t^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{1}{\Gamma (l\eta +a)}}{\int }_0^1{q}^{{\alpha }_1+l-1}{(1-p)}^{{\alpha }_2-1}dq\hfill \\ \hfill & & =\Gamma \left({\alpha }_2\right)\sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{1}{t^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{\Gamma ({\alpha }_1+l)\Gamma (l+1)}{\Gamma (l\eta +a)\Gamma ({\alpha }_1+{\alpha }_2+l)l!}},\hfill \end{array}$$

using the Fox-Wright function (12) in the above sum and employing in (26), we obtain,

$$\mathbf{B}\left\{{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3);{\alpha }_1,{\alpha }_2\right\}=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}$$

$$\times {\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}\Gamma \left({\alpha }_2\right)\left({\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}{}_2\Psi _2\left[\begin{array}{c}(1,1),({\alpha }_1,1)\\ (a,\eta ),({\alpha }_1+{\alpha }_2,1)\end{array};-{\displaystyle \frac{1}{t^{\xi }{(1-t)}^{\xi }}}\right]dt\right).$$

The above integral is evaluated using the generalized Beta function given in (11) to reach the required result (25).

2. Laplace Transform

The Laplace transform of a function $f\left(q\right)$ is defined as [18]

$$\mathbf{L}\left\{f\left(q\right)\right\}={\int }_0^{\infty }{e}^{-sq}f\left(q\right)dq$$

(27)

Theorem 5.

For $\eta ,\alpha ,a,({\chi }_1+i+k),({\chi }_2+i+j)\in \mathsf{C};$ $R\left(\eta \right)>0,R\left(\alpha \right)>0,R\left(a\right)>0,R\left(s\right)>0,R({\chi }_1+i+k)>0,R({\chi }_2+i+j)>0$ and $|-{\displaystyle \frac{1}{s{t}^{\xi }{(1-t)}^{\xi }}}|<1$, we have the following Laplace transform of ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

$${\int }_0^{\infty }{q}^{\alpha -1}{e}^{-sq}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)dq={\displaystyle \frac{1}{s^{\alpha }}}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}$$

$$\times {\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}{}^{\Psi }B_{\left({\displaystyle \frac{1}{s}}\right)}^{\xi }\left.\left[\begin{array}{c}(1,1),(\alpha ,1)\\ (a,\eta )\end{array}\right|({\chi }_1+i+k),({\chi }_2+i+j)\right].$$

(28)

Proof: Employing (15) in the integral on the L.H.S. of (28), yields

$${\int }_0^{\infty }{q}^{\alpha -1}{e}^{-sq}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)dq$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$\times \left({\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}\left\{{\int }_0^{\infty }{e}^{-sq}{q}^{\alpha -1}{E}_{\eta ,a}\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)dq\right\}dt\right).$$

(29)

By evaluating the integral as

$$\begin{array}{ccc}\hfill & & {\int }_0^{\infty }{e}^{-sq}{q}^{\alpha -1}{E}_{\eta ,a}\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)dq\hfill \\ \hfill & & =\sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{1}{t^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{1}{\Gamma (l\eta +a)}}{\int }_0^{\infty }{e}^{-sq}{q}^{\alpha +l-1}dq\hfill \\ \hfill & & ={\displaystyle \frac{1}{s^{\alpha }}}\sum _{l=0}^{\infty }{\displaystyle \frac{\Gamma (\alpha +l)\Gamma (l+1)}{\Gamma (l\eta +a)l!}}{\left(-{\displaystyle \frac{1}{s{t}^{\xi }{(1-t)}^{\xi }}}\right)}^l,\hfill \end{array}$$

using the Fox-Wright function (12) in the above sum and employing in (29), we find,

$${\int }_0^{\infty }{q}^{\alpha -1}{e}^{-sq}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)dq=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}$$

$$\times {\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}{\displaystyle \frac{1}{s^{\alpha }}}\left({\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}{}_2\Psi _1\left[\begin{array}{c}(1,1),(\alpha ,1)\\ (a,\eta )\end{array};-{\displaystyle \frac{1}{s{t}^{\xi }{(1-t)}^{\xi }}}\right]dt\right).$$

The above integral is evaluated using the generalized Beta function given in (11) to obtain the required result (28).

3. Mellin Transform

The Mellin transform of the function $f\left(q\right)$ is defined as [18],

$$f^{*}\left(s\right)=\mathbf{M}\left\{f\left(q\right);s\right\}={\int }_0^{\infty }{q}^{s-1}f\left(q\right)dq,R\left(s\right)>0,$$

(30)

and the inverse Mellin transform is given by

$$f\left(q\right)={\mathbf{M}}^{-\mathbf{1}}\left\{f^{*}\left(s\right);q\right\}={\displaystyle \frac{1}{2\pi i}}{\int }_{c-i\infty }^{c+i\infty }{q}^{-s}{f}^{*}\left(s\right)ds,c\in \mathsf{R}.$$

(31)

Theorem 6.

$.$ For $a,s\in \mathsf{C}$; $R\left(s\right)>a>0$, $R\left(a\right)>0,\eta >0,$${\varrho }_1,{\varrho }_2,{\varrho }_3\in \mathsf{C}\setminus {\mathsf{Z}}_0^{-}$, we have the following Mellin transform of ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

$$\mathbf{M}\left\{{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3);s\right\}$$

$$={\int }_0^{\infty }{q}^{s-1}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)dq$$

$$={\displaystyle \frac{\Gamma \left(s\right)\Gamma (1-s)}{\Gamma (a-\eta s)}}{\mathbf{H}}_B^{\left(\xi s\right)}\left({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3\right),$$

(32)

where ${\mathbf{H}}_B^{\left(\xi s\right)}(·)$ is given in (4).

Proof: Employing (15) in the integral on the L.H.S. of (32), we obtain

$$\mathbf{M}\left\{{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3);s\right\}$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$\times {\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}\left\{{\int }_0^{\infty }{q}^{s-1}{E}_{\eta ,a}\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)dq\right\}dt.$$

(33)

Now, by taking $\lambda =p=1$ and $y=-q/t^{\xi }{(1-t)}^{\xi }$ in the result ([16], Th. 4.1) as follows

$$E_{\eta ,a}^{\lambda ,p}\left(y\right)dy={\displaystyle \frac{1}{2\pi i\Gamma \left(\lambda \right)}}{\int }_L{\displaystyle \frac{\Gamma \left(s\right)\Gamma (\lambda -ps)}{\Gamma (a-\eta s)}}{(-y)}^{-s}ds,$$

where L is the contour of integration that begins at $c-i\infty$ and ends at $c+i\infty$, $c\in \mathsf{R},$

yields,

$$E_{\eta ,a}\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)={\displaystyle \frac{1}{2\pi i}}{\int }_L{\displaystyle \frac{\Gamma \left(s\right)\Gamma (1-s)}{\Gamma (a-\eta s)}}{\left({\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)}^{-s}ds$$

$$={\displaystyle \frac{1}{2\pi i}}{\int }_L{f}^{*}\left(s\right)q^{-s}ds,$$

where

$$f^{*}\left(s\right)={\displaystyle \frac{\Gamma \left(s\right)\Gamma (1-s)}{\Gamma (a-\eta s)}}{\left[t^{\xi }{(1-t)}^{\xi }\right]}^s.$$

(34)

Using (30), (31), and (34), we get

$${\int }_0^{\infty }{q}^{s-1}{E}_{\eta ,a}\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)dq={\displaystyle \frac{\Gamma \left(s\right)\Gamma (1-s)}{\Gamma (a-\eta s)}}{\left[t^{\xi }{(1-t)}^{\xi }\right]}^s$$

(35)

Now, put (35) in (33), leads to

$$\mathbf{M}\left\{{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3);s\right\}=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}$$

$$\times {\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}{\displaystyle \frac{\Gamma \left(s\right)\Gamma (1-s)}{\Gamma (a-\eta s)}}{\int }_0^1{t}^{{\chi }_1+i+k+\xi s-1}{(1-t)}^{{\chi }_2+i+j+\xi s-1}dt.$$

Evaluate the integral using the classical Beta function defined in (7), yields

$$\mathbf{M}\left\{{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3);s\right\}$$

$$={\displaystyle \frac{\Gamma \left(s\right)\Gamma (1-s)}{\Gamma (a-\eta s)}}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{B({\chi }_1+\xi s+i+k,{\chi }_2+\xi s+i+j)}{B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}.$$

Identifying the above sum as ${\mathbf{H}}_B^{\left(\xi s\right)}({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$ given in (4), we get the desired result (32).

Corollary 1:

For $c>a,$ the inverse Mellin transform formula for ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$ is as follows:

$${\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)={\mathbf{M}}^{-1}\left\{f^{*}\left(s\right);q\right\}$$

$$={\displaystyle \frac{1}{2\pi i}}{\int }_{c-i\infty }^{c+i\infty }{q}^{-s}{\displaystyle \frac{\Gamma \left(s\right)\Gamma (1-s)}{\Gamma (a-\eta s)}}{H}_B^{\left(\xi s\right)}\left({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3\right)ds.$$

(36)

5. Whittaker Transform

Theorem 7.

For ${\alpha }_1,a,({\chi }_1+i+k),({\chi }_2+i+j)\in \mathsf{C}$; $\eta >0,R\left({\alpha }_1\right)>0,R\left(a\right)>0,R({\chi }_1+i+k)>0,R({\chi }_2+i+j)>0$, we have the following Whittaker transform of ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

$${\int }_0^{\infty }{q}^{{\alpha }_1-1}{e}^{-{\displaystyle \frac{1}{2}}{\alpha }_{2}q}{\mathbf{W}}_{{\tau }_1,{\tau }_2}\left({\alpha }_{2}q\right){\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)dq$$

$$={\displaystyle \frac{1}{{\alpha }_2^{{\alpha }_1}}}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$\times {}^{\Psi }B_{\left({\displaystyle \frac{1}{{\alpha }_2}}\right)}^{\xi }\left.\left[\begin{array}{c}(1,1),({\displaystyle \frac{1}{2}}\pm {\tau }_2+{\alpha }_1,1)\\ (a,\eta ){,}_{(}1-{\tau }_1+{\alpha }_1,1)\end{array}\right|({\chi }_1+i+k),({\chi }_2+i+j)\right].$$

(37)

Proof: Applying (37) in (15), yields

$${\int }_0^{\infty }{q}^{{\alpha }_1-1}{e}^{-{\displaystyle \frac{1}{2}}{\alpha }_{2}q}{\mathbf{W}}_{{\tau }_1,{\tau }_2}\left({\alpha }_{2}q\right){\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)dq$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$\times \left({\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}\left\{{\int }_0^{\infty }{q}^{{\alpha }_1-1}{e}^{-{\displaystyle \frac{1}{2}}{\alpha }_{2}q}{\mathbf{W}}_{{\tau }_1,{\tau }_2}\left({\alpha }_{2}q\right)E_{\eta ,a}\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)dq\right\}dt\right).$$

(38)

Now consider,

$$\begin{array}{ccc}\hfill I& =& {\int }_0^{\infty }{q}^{{\alpha }_1-1}{e}^{-{\displaystyle \frac{1}{2}}{\alpha }_{2}q}{\mathbf{W}}_{{\tau }_1,{\tau }_2}\left({\alpha }_{2}q\right)E_{\eta ,a}\left(-{\displaystyle \frac{q}{t^{\xi }{(1-t)}^{\xi }}}\right)dq\hfill \end{array}$$

put ${\alpha }_{2}q=u$ in the above equation, yields

$$\begin{array}{ccc}\hfill I& =& {\int }_0^{\infty }{\left({\displaystyle \frac{u}{{\alpha }_2}}\right)}^{{\alpha }_1-1}{e}^{{\displaystyle \frac{-u}{2}}}{\mathbf{W}}_{{\tau }_1,{\tau }_2}\left(u\right)\sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{u}{{\alpha }_2{t}^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{1}{\Gamma (l\eta +a)}}{\displaystyle \frac{1}{{\alpha }_2}}du\hfill \end{array}$$

$$={\displaystyle \frac{1}{{\alpha }_2^{{\alpha }_1}}}\sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{1}{{\alpha }_2{t}^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{1}{\Gamma (l\eta +a)}}{\int }_0^{\infty }{u}^{l+{\alpha }_1-1}{e}^{-{\displaystyle \frac{u}{2}}}{\mathbf{W}}_{{\tau }_1,{\tau }_2}\left(u\right)du.$$

(39)

Now using the relation

$${\int }_0^{\infty }{u}^{{\alpha }_1-1}{e}^{-{\displaystyle \frac{u}{2}}}{\mathbf{W}}_{{\tau }_1,{\tau }_2}\left(u\right)du={\displaystyle \frac{\Gamma (\frac{1}{2}+{\tau }_2+{\alpha }_1)\Gamma (\frac{1}{2}-{\tau }_2+{\alpha }_1)}{\Gamma (1-{\tau }_1+{\alpha }_1)}},\left(R\left({\tau }_2\pm {\alpha }_1\right)>{\displaystyle \frac{-1}{2}}\right),$$

where ${\mathbf{W}}_{{\tau }_1,{\tau }_2}\left(u\right)$ is the Whittaker function [25], in (39), we get

$$I={\displaystyle \frac{1}{{\alpha }_2^{{\alpha }_1}}}\sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{1}{{\alpha }_2{t}^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{\Gamma (\frac{1}{2}+{\tau }_2+{\alpha }_1+l)\Gamma (\frac{1}{2}-{\tau }_2+{\alpha }_1+l)\Gamma (l+1)}{\Gamma (l\eta +a)\Gamma (1-{\tau }_1+{\alpha }_1+l)l!}}$$

Using (12), we get

$$I={\displaystyle \frac{1}{{\alpha }_2^{{\alpha }_1}}}{}_3\Psi _2\left[\begin{array}{c}(1,1),({\displaystyle \frac{1}{2}}+{\tau }_2+{\alpha }_1,1),({\displaystyle \frac{1}{2}}-{\tau }_2+{\alpha }_1,1)\\ (a,\eta ),(1-{\tau }_1+{\alpha }_1,1)\end{array};-{\displaystyle \frac{1}{{\alpha }_2{t}^{\xi }{(1-t)}^{\xi }}}\right].$$

(40)

Applying (40) in (38), we obtain

$${\int }_0^{\infty }{q}^{{\alpha }_1-1}{e}^{-{\displaystyle \frac{1}{2}}{\alpha }_{2}q}{\mathbf{W}}_{{\tau }_1,{\tau }_2}\left({\alpha }_{2}q\right){\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)dq$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$\times {\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}{\displaystyle \frac{1}{{\alpha }_2^{{\alpha }_1}}}{}_3\Psi _2\left[\begin{array}{c}(1,1),({\displaystyle \frac{1}{2}}\pm {\tau }_2+{\alpha }_1,1)\\ (a,\eta ),(1-{\tau }_1+{\alpha }_1,1)\end{array};-{\displaystyle \frac{1}{{\alpha }_2{t}^{\xi }{(1-t)}^{\xi }}}\right]dt.$$

The above integral is evaluated using the generalized Beta function given in (11) to obtain the required result (37).

6. Fractional Calculus of ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

In this part, we examine various findings related to the right-sided Riemann-Liouville fractional integral operator $I_{\alpha +}^{\theta }$ and derivative operator $D_{\alpha +}^{\theta }$, which are defined as follows, respectively [9,14]:

$$\left(I_{\alpha +}^{\theta }f\right)\left(w\right)={\displaystyle \frac{1}{\Gamma \left(\theta \right)}}{\int }_{\alpha }^w{\displaystyle \frac{f\left(q\right)}{{(w-q)}^{1-\theta }}}dq,(R\left(\theta \right)>0,\theta \in \mathsf{C}),$$

(41)

and

$$\left(D_{\alpha +}^{\theta }f\right)\left(w\right)={\left({\displaystyle \frac{d}{dw}}\right)}^m\left(I_{\alpha +}^{m-\theta }f\right)\left(w\right),(m=\left[R\left(\theta \right)\right]+1;R\left(\theta \right)>0,\alpha \in \mathsf{C}),$$

(42)

where $\left[w\right]$ is the greatest integer.

A generalized Riemann-Liouville fractional derivative operator $D_{\alpha +}^{\theta ,\lambda }$ of order $0<\theta <1$ and type $0\le \lambda \le 1$ with respect to w was introduced by Hilfer [4] as follows:

$$\left(D_{\alpha +}^{\theta ,\lambda }f\right)\left(w\right)=\left(I_{\alpha +}^{\lambda (1-\theta )}{\displaystyle \frac{d}{dw}}\right)\left(I_{\alpha +}^{(1-\lambda )(1-\theta )}f\right)\left(w\right),(m=\left[R\left(\theta \right)\right]+1;R\left(\theta \right)>0,\theta \in \mathsf{C}).$$

(43)

Theorem 8.

Let $\alpha \in {\mathsf{R}}_{+}$, then for $w>\alpha$, the function ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$ satisfies the following relations

$$\left(i\right)\left(I_{\alpha +}^{\theta }\left[{(q-\alpha )}^{a-1}{\mathbf{H}}_{B,{(q-\alpha )}^{\eta },a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)\right]\right)\left(w\right)$$

$$={(w-\alpha )}^{a+\theta -1}{\mathbf{H}}_{B,{(w-\alpha )}^{\eta },a+\theta }^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3),$$

(44)

$$\left(ii\right)\left(D_{\alpha +}^{\theta }\left[{(q-\alpha )}^{a-1}{\mathbf{H}}_{B,{(q-\alpha )}^{\eta },a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)\right]\right)\left(w\right)$$

$$={(w-\alpha )}^{a-\theta -1}{\mathbf{H}}_{B,{(w-\alpha )}^{\eta },a-\theta }^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3),$$

(45)

and

$$\left(iii\right)\left(D_{\alpha +}^{\theta ,\lambda }\left[{(q-\alpha )}^{a-1}{\mathbf{H}}_{B,{(q-\alpha )}^{\eta },a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)\right]\right)\left(w\right)$$

$$={(w-\alpha )}^{a-\theta -1}{\mathbf{H}}_{B,{(w-\alpha )}^{\eta },a-\theta }^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3),$$

(46)

where $a,\theta \in \mathsf{C};$$\eta >0,R\left(a\right)>0,R\left(\theta \right)>0,$ and ${\varrho }_1,{\varrho }_2,{\varrho }_3\in \mathsf{C}\setminus {\mathsf{Z}}_0^{-}$.

Proof. Use (15) in (41), and with the help of the relation [14]

$$\left(I_{\alpha +}^{\theta }\left[{(q-\alpha )}^{\nu -1}\right]\right)\left(w\right)={\displaystyle \frac{\Gamma \left(\nu \right)}{\Gamma (\nu +\theta )}}{(w-\alpha )}^{\theta +\nu -1},$$

for $w>\alpha$ it follows that,

$$\left(I_{\alpha +}^{\theta }\left[{(q-\alpha )}^{a-1}{\mathbf{H}}_{B,{(q-\alpha )}^{\eta },a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)\right]\right)\left(w\right)$$

$$=\left(I_{\alpha +}^{\theta }\left[{(q-\alpha )}^{a-1}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}\right.\right.$$

$$\left.\left.\times {\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}{E}_{\eta ,a}\left\{-{\displaystyle \frac{{(q-\alpha )}^{\eta }}{t^{\xi }{(1-t)}^{\xi }}}\right\}dt\right]\right)\left(w\right)$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}{\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}$$

$$\times \sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{1}{t^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{1}{\Gamma (l\eta +a)}}\left(I_{\alpha +}^{\theta }\left[{(q-\alpha )}^{l\eta +a-1}\right]dt\right)\left(w\right)$$

$$={(w-\alpha )}^{a+\theta -1}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$\times {\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}{E}_{\eta ,a+\theta }\left\{-{\displaystyle \frac{{(w-\alpha )}^{\eta }}{t^{\xi }{(1-t)}^{\xi }}}\right\}dt$$

$$={(w-\alpha )}^{a+\theta -1}{\mathbf{H}}_{B,{(w-\alpha )}^{\eta },a+\theta }^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3).$$

Now, using (15) in (42), yields

$$\left(D_{\alpha +}^{\theta }\left[{(q-\alpha )}^{a-1}{\mathbf{H}}_{B,{(q-\alpha )}^{\eta },a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)\right]\right)\left(w\right)$$

$$={\left({\displaystyle \frac{d}{dw}}\right)}^m\left(I_{\alpha +}^{m-\theta }\left[{(q-\alpha )}^{a-1}{\mathbf{H}}_{B,{(q-\alpha )}^{\eta },a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)\right]\right)\left(w\right)$$

$$={\left({\displaystyle \frac{d}{dw}}\right)}^m\left[{(w-\alpha )}^{a+m-\theta -1}{\mathbf{H}}_{B,{(w-\alpha )}^{\eta },a+m-\theta }^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)\right].$$

Using (21), reaches the result (45).

Furthermore, after using the eqs. (15) and (43), follows that

$$\left(D_{\alpha +}^{\theta ,\lambda }\left[{(q-\alpha )}^{a-1}{\mathbf{H}}_{B,{(q-\alpha )}^{\eta },a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)\right]\right)\left(w\right)$$

$$=\left(D_{\alpha +}^{\theta ,\lambda }\left[{(q-\alpha )}^{a-1}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}\right.\right.$$

$$\left.\left.\times {\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}{E}_{\eta ,a}\left\{-{\displaystyle \frac{{(q-\alpha )}^{\eta }}{t^{\xi }{(1-t)}^{\xi }}}\right\}dt\right]\right)\left(w\right)$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_{k}B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}{\int }_0^1{t}^{{\chi }_1+i+k-1}{(1-t)}^{{\chi }_2+i+j-1}$$

$$\times \sum _{l=0}^{\infty }{\left(-{\displaystyle \frac{1}{t^{\xi }{(1-t)}^{\xi }}}\right)}^l{\displaystyle \frac{1}{\Gamma (l\eta +a)}}\left(D_{\alpha +}^{\theta ,\lambda }\left[{(q-\alpha )}^{l\eta +a-1}\right]dt\right)\left(w\right).$$

(47)

Applying the result of Srivastava and Tomovski ([21], p.203, eq.(2.18))

$$\left(D_{\alpha +}^{\theta ,\lambda }\left[{(q-\alpha )}^{\beta -1}\right]\right)\left(w\right)={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma (\beta -\theta )}}{(w-\alpha )}^{\beta -\theta -1}$$

$$(w>\alpha ;0<\theta <1;0\le \lambda \le 1;R(\beta )>0),$$

in (47), (46) is established.

7. Recurrence Relations for ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

In this section, we derive some recurrence relations for ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$.

Theorem 9.

$.$ With respect to the numerator parameter ${\chi }_3$, we have the following recurrences for ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$

$${\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3+1;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)={\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$+{\displaystyle \frac{{\Phi }_2{\chi }_2}{{\varrho }_2}}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2+1,{\chi }_3+1;{\varrho }_1,{\varrho }_2+1,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$+{\displaystyle \frac{{\Phi }_3{\chi }_1}{{\varrho }_3}}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1+1,{\chi }_2,{\chi }_3+1;{\varrho }_1,{\varrho }_2,{\varrho }_3+1;{\Phi }_1,{\Phi }_2,{\Phi }_3),$$

(48)

and

$${\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3-1;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)={\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$-{\displaystyle \frac{{\Phi }_2{\chi }_2}{{\varrho }_2}}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2+1,{\chi }_3-1;{\varrho }_1,{\varrho }_2+1,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$-{\displaystyle \frac{{\Phi }_3{\chi }_1}{{\varrho }_3}}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1+1,{\chi }_2,{\chi }_3-1;{\varrho }_1,{\varrho }_2,{\varrho }_3+1;{\Phi }_1,{\Phi }_2,{\Phi }_3).$$

(49)

Proof. With the help of the identity ${({\chi }_3+1)}_{j+k}={\left({\chi }_3\right)}_{j+k}(1+j/{\chi }_3+k/{\chi }_3)$ and from (14), we find

$${\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3+1;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$=\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{({\chi }_3+1)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{B_{q,a}^{\eta ,\xi }({\chi }_1+i+k,{\chi }_2+i+j)}{B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$={\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$+{\displaystyle \frac{{\Phi }_2}{{\chi }_3}}\sum _{i=0}^{\infty }\sum _{j=1}^{\infty }\sum _{k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{B_{q,a}^{\eta ,\xi }({\chi }_1+i+k,{\chi }_2+i+j)}{B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^{j-1}}{(j-1)!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$+{\displaystyle \frac{{\Phi }_3}{{\chi }_3}}\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }\sum _{k=1}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{B_{q,a}^{\eta ,\xi }({\chi }_1+i+k,{\chi }_2+i+j)}{B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^{k-1}}{(k-1)!}}.$$

(50)

Let the first sum in (50) as S. Following the index shift $j\to j+1$, we apply the identity ${\left(\varrho \right)}_{j+1}=\varrho {(\varrho +1)}_j$, which yields

$$S={\displaystyle \frac{{\Phi }_2}{{\chi }_3}}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2)}_{2i+j+k}{\left({\chi }_3\right)}_{j+k}}{{\left({\varrho }_1\right)}_i{\left({\varrho }_2\right)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{B_{q,a}^{\eta ,\xi }({\chi }_1+i+k,{\chi }_2+i+j)}{B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$={\displaystyle \frac{({\chi }_1+{\chi }_2){\Phi }_2}{{\varrho }_2}}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2+1)}_{2i+j+k}{({\chi }_3+1)}_{j+k}}{{\left({\varrho }_1\right)}_i{({\varrho }_2+1)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{B_{q,a}^{\eta ,\xi }({\chi }_1+i+k,{\chi }_2+i+j)}{B({\chi }_1,{\chi }_2)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}.$$

Now, using the relation

$$B({\chi }_1,{\chi }_2)={\displaystyle \frac{{\chi }_1+{\chi }_2}{{\chi }_2}}B({\chi }_1,{\chi }_2+1),$$

we get

$$S={\displaystyle \frac{{\chi }_2{\Phi }_2}{{\varrho }_2}}\sum _{i,j,k=0}^{\infty }{\displaystyle \frac{{({\chi }_1+{\chi }_2+1)}_{2i+j+k}{({\chi }_3+1)}_{j+k}}{{\left({\varrho }_1\right)}_i{({\varrho }_2+1)}_j{\left({\varrho }_3\right)}_k}}{\displaystyle \frac{B_{q,a}^{\eta ,\xi }({\chi }_1+i+k,{\chi }_2+i+j)}{B({\chi }_1,{\chi }_2+1)}}{\displaystyle \frac{{\Phi }_1^i}{i!}}{\displaystyle \frac{{\Phi }_2^j}{j!}}{\displaystyle \frac{{\Phi }_3^k}{k!}}$$

$$={\displaystyle \frac{{\chi }_2{\Phi }_2}{{\varrho }_2}}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2+1,{\chi }_3+1;{\varrho }_1,{\varrho }_2+1,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3).$$

(51)

Similarly, analyzing the second summation in (50) with the subsitution $k\to k+1$, we express it as

$${\displaystyle \frac{{\Phi }_3{\chi }_1}{{\varrho }_3}}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1+1,{\chi }_2,{\chi }_3+1;{\varrho }_1,{\varrho }_2,{\varrho }_3+1;{\Phi }_1,{\Phi }_2,{\Phi }_3).$$

(52)

Combining (51) and (52) with (50) yields the desired recurrence relation (48). Replacing ${\chi }_3$ with ${\chi }_3-1$ in (48), leads directly to the result (49).

Corollary 2:

For positive integer N, the recurrence formulas for ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$ from (48) are

$${\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3+N;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)={\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$+{\displaystyle \frac{{\Phi }_2{\chi }_2}{{\varrho }_2}}\sum _{l=1}^N{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2+1,{\chi }_3+l;{\varrho }_1,{\varrho }_2+1,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$+{\displaystyle \frac{{\Phi }_3{\chi }_1}{{\varrho }_3}}\sum _{l=1}^N{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1+1,{\chi }_2,{\chi }_3+l;{\varrho }_1,{\varrho }_2,{\varrho }_3+1;{\Phi }_1,{\Phi }_2,{\Phi }_3),$$

and

$${\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3-N;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)={\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2,{\chi }_3;{\varrho }_1,{\varrho }_2,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$-{\displaystyle \frac{{\Phi }_2{\chi }_2}{{\varrho }_2}}\sum _{l=1}^{N-1}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1,{\chi }_2+1,{\chi }_3-l;{\varrho }_1,{\varrho }_2+1,{\varrho }_3;{\Phi }_1,{\Phi }_2,{\Phi }_3)$$

$$-{\displaystyle \frac{{\Phi }_3{\chi }_1}{{\varrho }_3}}\sum _{l=1}^{N-1}{\mathbf{H}}_{B,q,a}^{\eta ,\xi }({\chi }_1+1,{\chi }_2,{\chi }_3-l;{\varrho }_1,{\varrho }_2,{\varrho }_3+1;{\Phi }_1,{\Phi }_2,{\Phi }_3).$$

8. Conclusions

In this paper, we presented generalized Srivastava triple hypergeometric function ${\mathbf{H}}_{B,q,a}^{\eta ,\xi }(·)$, as introduced in (14). Several integral representations involving the generalized Mittag-Leffler function and Exton’s function $X_4$ were established, alongside several analytical properties. We also derived integral transforms of the generalized function, including the Euler-Beta, Laplace, Mellin, and Whittaker transform. The action of Riemann-Liouville fractional integral and differential operators on the function was examined, providing new insights within the framework of fractional calculus. Finally, recurrence relations were obtained, offering a deeper understanding of the characteristic properties of the function. These findings provide a foundation for further analytical studies and potential applications in mathematical physics, fractional differential equations, and other areas involving generalized special functions.