On (p,q)-Sumudu and (p,q)-Laplace Transforms of the Basic Analogue of Aleph-Function

Abstract

In this paper, we introduce the definitions of Sumudu and Laplace transforms of first and second kind in quantum calculus by using functions of several variables. On account of the general nature of the $(p,q)$-analogue of Aleph-function, a large number of new and known results for these transforms were obtained. Also, we obtain some interesting relationships and identities for these transforms. We also derive some correlations among Aleph function and the above-mentioned integral transforms in quantum calculus.

1. Introduction

There are numerous integral transforms to solve differential equations. The Laplace transform is the most popular and widely used in several branches of engineering and applied mathematics among these transforms. In the sequence of the Laplace, Fourier, Mellin, and Hankel transforms, in 1993 Watugala [1,2] introduced a new integral transform, named the Sumudu transform, and further applied it to the solution of ordinary differential equation in control engineering problems. This transform has a property that it has larger frequency domain than Laplace transform. That is why they attracted the attention of so many researchers [3,4,5]. Two of the most frequently used formulas in the area of integral transforms are the classical Laplace and Sumudu transform and their corresponding q-analogues. They were successively applied in the theory of differential/difference equations, quantum mechanics, special functions, and calculating integrals [6].

The theory of q-analysis and the concepts of difference equations dates back to eighteenth century by Euler and developed by Gauss and Ramnaujan. The number of studies in q-analysis have been grown rapidly in many ares of mathematics such as number theory, orthogonal polynomials, in ordinary, and partial differential equations and also in physical disciplines such as quantum mechanics and relativity [7,8,9].

In the past century, many authors have generalized H-function. In a recent paper, Sudland et al. [10] introduced a generalization of Saxena’s I-function [11], which is also a generalization of Fox’s H-function. This function is known as Aleph function. In their paper, Saxena and Pogany [12] studied fractional integration formulae for the Aleph functions. Fractional driftless Fokker–Planck equation were studied by Sudland et al. [13], a special case of the Aleph function. This subject has tremendous applications in the field of applied mathematics [14,15,16,17,18,19]. Motivated by these applications, various researchers are working on these operators to evaluate q-analogue of various integral transforms and special functions.

The Aleph function [12,20] is defined as follows.

$$\begin{array}{ccc}\hfill \aleph \left(z\right)& =& {\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z\left|\begin{array}{c}{(a_j,A_j)}_{1,n};{\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i;r}\\ {(b_j,B_j)}_{1,m};{\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i;r}\end{array}\right.\right]\hfill \\ & =& \frac{1}{2\pi \omega }\underset{L}{\int }{\mathsf{\Omega }}_{p_i,q_i,{\tau }_i;r}^{m,n}\left(s\right)z^{-s}ds\hfill \end{array}$$

(1)

where $z\ne 0,\omega =\sqrt{-1}$ and

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_{p_i,q_i,{\tau }_i;r}^{m,n}\left(s\right)=\frac{{\displaystyle \prod _{j=1}^m}\mathsf{\Gamma }(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}\mathsf{\Gamma }(1-a_j-A_{j}s)}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}\mathsf{\Gamma }(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}\mathsf{\Gamma }(a_{ji}+A_{ji}s)\right]}.\end{array}$$

The q-gamma function was first introduced by Thomae and later by Jackson. The q-analogue of gamma function which is defined by Jackson [21] is given by

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_q\left(x\right)=\frac{{(q;q)}_{\infty }{(1-q)}^{1-x}}{{(q^x;q)}_{\infty }},0<q<1.\end{array}$$

(2)

Jackson gave the general definition which is given below

$$\begin{array}{c}\hfill {\int }_a^{b}f\left(t\right)d_{q}t={\int }_0^{b}f\left(t\right)d_{q}t-{\int }_0^{a}f\left(t\right)d_{q}t\end{array}$$

(3)

where

$$\begin{array}{c}\hfill {\int }_0^{a}f\left(t\right)d_{q}t=a(1-q){\displaystyle \sum _{n=0}^{\infty }}f\left(a{q}^n\right)q^n\end{array}$$

(4)

Jackson also defined an integral, i.e.,

$$\begin{array}{c}\hfill {\int }_0^{\infty }f\left(t\right)d_{q}t=(1-q){\displaystyle \sum _{n=-\infty }^{\infty }}f\left(q^n\right)q^n\end{array}$$

(5)

q-analogues of gamma and beta functions are defined as follows (see [22]),

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_q\left(\alpha \right)={\int }_0^{\frac{1}{1-q}}{x}^{\alpha -1}{E}_q\left(q(1-q)x\right)d_{q}x\hspace{1em}\hspace{1em}(\alpha >0).\end{array}$$

(6)

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_q\left(\alpha \right)=K(A;\alpha ){\int }_0^{\frac{\infty }{A(1-q)}}{x}^{\alpha -1}{e}_q(-(1-q)x)d_{q}x\hspace{1em}\hspace{1em}(\alpha >0).\end{array}$$

(7)

$$\begin{array}{c}\hfill B_q(t;s)={\int }_0^1{x}^{t-1}{(1-qx)}_q^{s-1}{d}_{q}x\hspace{1em}\hspace{1em}(t,s>0).\end{array}$$

(8)

where

$$\begin{array}{c}\hfill K(A;t)=A^{t-1}\frac{{(-q/A;q)}_{\infty }}{{(-q^t/A;q)}_{\infty }}\frac{{(-A;q)}_{\infty }}{{(-A{q}^{t-1};q)}_{\infty }}\hspace{1em}(t\in \mathbb{R})\end{array}$$

(9)

P. Njionou Sadjang [23] introduced the so-called the shifted factorial as follows.

$$\begin{array}{c}\hfill {(x\ominus a)}_{p,q}^n=(x-a)(px-aq)(p^{2}x-a{q}^2)\cdots (x{p}^{n-1}-a{q}^{n-1})\\ \hfill {(x\oplus a)}_{p,q}^n=(x+a)(px+aq)(p^{2}x+a{q}^2)\cdots (x{p}^{n-1}+a{q}^{n-1})\end{array}$$

These definitions are extended as follows.

$$\begin{array}{c}\hfill {(a\ominus b)}_{p,q}^n={\displaystyle \prod _{k=0}^{\infty }}(a{p}^k-b{q}^k)\\ \hfill {(a\oplus b)}_{p,q}^n={\displaystyle \prod _{k=0}^{\infty }}(a{p}^k+b{q}^k)\end{array}$$

Let x be a complex number, the $(p,q$)-Gamma function is defined by P. Njionou Sadjang [24]

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_{p,q}\left(x\right)=\frac{{(p\ominus q)}_{p,q}^{\infty }}{{(p^x\ominus q^x)}_{p,q}^{\infty }}{(p-q)}^{1-x},0<q<p\end{array}$$

If we put $p=1$, then ${\mathsf{\Gamma }}_{p,q}$ reduces to ${\mathsf{\Gamma }}_q$.

The ($p,q$)-Gamma function fulfills the following fundamental relation.

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_{p,q}(x+1)={\left[x\right]}_{p,q}{\mathsf{\Gamma }}_{p,q}\left(x\right)\end{array}$$

If x is a nonnegative integer, it follows from above that

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_{p,q}(x+1)={\left[x\right]}_{p,q}!\end{array}$$

It can be also easily seen from the definition that

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_{p,q}(x+1)=\frac{{(p\ominus q)}_{p,q}^x}{{(p-q)}_{p,q}^x}\end{array}$$

P. Njionou Sadjang [24] also defined the $(p,q)$-Beta function as

$$\begin{array}{c}\hfill B_{p,q}(x,y)=\frac{{\mathsf{\Gamma }}_{p,q}\left(x\right){\mathsf{\Gamma }}_{p,q}\left(y\right)}{{\mathsf{\Gamma }}_{p,q}(x+y)}\end{array}$$

The ($p,q$)-derivative of the function $f\left(x\right)$ is defined as follows [23,24],

$$\begin{array}{c}\hfill D_{p,q}f\left(x\right)=\frac{f\left(px\right)-f\left(qx\right)}{(p-q)x},x\ne 0\end{array}$$

where $D_{p,q}f\left(0\right)=f^{\prime }\left(0\right),$ provided that $f\left(x\right)$ is differentiable at $x=0$

The $(p,q)$-numbers ${\left[n\right]}_{p,q}$ and $(p,q)$ factorials ${\left[n\right]}_{p,q}!$ are defined [23] as:

$$\begin{array}{c}\hfill {\left[n\right]}_{p,q}=\frac{p^n-q^n}{(p-q)}\end{array}$$

and

$$\begin{array}{c}\hfill {\left[n\right]}_{p,q}!={\left[1\right]}_{p,q}{\left[2\right]}_{p,q}\cdots {\left[n\right]}_{p,q}\end{array}$$

respectively.

Also it happens that $D_{p,q}\left(x^n\right)={\left[n\right]}_{p,q}{x}^{n-1}$.

The ($p,q$)-analogue of the exponential function are given [25] as follows.

$$e_{p,q}\left(z\right)={\displaystyle \sum _{n=0}^{\infty }}\frac{p^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}}{{\left[n\right]}_{p,q}!}{z}^n$$

$$E_{p,q}\left(z\right)={\displaystyle \sum _{n=0}^{\infty }}\frac{q^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}}{{\left[n\right]}_{p,q}!}{z}^n$$

For $0<q<p$, the ($p,q$)-Gamma function of first kind is defined in [25] as follows,

$${\mathsf{\Gamma }}_{p,q}\left(z\right)=p^{\frac{z(z-1)}{2}}{\int }_0^{\infty }{t}^{z-1}{E}_{p,q}(-qt)d_{p,q}t.\hspace{1em}Re\left(z\right)>0.$$

For $0<q<p$, the ($p,q$)-Gamma function of second kind is defined in [25] as follows,

$${\gamma }_{p,q}\left(z\right)=q^{\frac{z(z-1)}{2}}{\int }_0^{\infty }{t}^{z-1}{e}_{p,q}(-pt)d_{p,q}t.\hspace{1em}Re\left(z\right)>0.$$

Remark 1.

$D_{p,q}\left(x\right)$ reduces to Hahn Derivative $d_{q}f\left(x\right)$ iff $p\to 1$.

Remark 2.

${\left[n\right]}_{p,q}={\left[n\right]}_q$ (Hahn Basic Number) iff $p\to 1$. where ${\left[n\right]}_q=\frac{1-q^n}{1-q},q\ne 1$.

Note 1.

$$\begin{array}{c}\hfill D_{p,q}^n\left(x^{\mu }\right)=\frac{{\mathsf{\Gamma }}_{p,q}(\mu +1)}{{\mathsf{\Gamma }}_{p,q}(\mu -n+1)}{x}^{\mu -n},Re\left(\mu \right)+1>0\\ \hfill I_{p,q}^n\left(x^{\mu }\right)=\frac{{\mathsf{\Gamma }}_{p,q}(\mu +1)}{{\mathsf{\Gamma }}_{p,q}(\mu +n+1)}{x}^{\mu +n},Re\left(\mu \right)+1>0\end{array}$$

Classical Laplace transform:

Suppose $f\left(t\right)$ is a real-valued function defined over the interval $(0,\infty )$. The Laplace transform of $f\left(t\right)$ is defined by

$$\begin{array}{c}\hfill F\left(s\right)=L\left[f\left(t\right)\right]={\int }_0^{\infty }{e}^{-st}f\left(t\right)dt,\hspace{1em}Re\left(s\right)>0.\end{array}$$

(10)

The Laplace transform is said to exist if the integral (10) is convergent for some values of s.

$\mathit{q}$-Laplace transform:

Hahn [26] defined the q-image of classical Laplace transform by means of the following q-integrals as

$$\begin{array}{c}\hfill L_q\left[f\left(s\right)\right]=\frac{1}{(1-q)}{\int }_0^{s^{-1}}{E}_q^{qsx}f\left(x\right)d_{q}x,\hspace{1em}Re\left(s\right)>0.\end{array}$$

(11)

$$\begin{array}{c}\hfill or\hspace{1em}{L}_q\left[f\left(s\right)\right]=\frac{1}{(1-q)}{\int }_0^{\infty }{e}_q^{-sx}f\left(x\right)d_{q}x,\hspace{1em}Re\left(s\right)>0.\end{array}$$

(12)

where $e_q^{-sx}$ and $E_q^{qsx}$ are the q-analogues of exponential function [26].

The Laplace transform of the power function is defined as

$$\begin{array}{c}\hfill L\left[t^{\mu }\right]=\frac{\mathsf{\Gamma }(\mu +1)}{s^{\mu +1}}.\end{array}$$

(13)

The q-Laplace transform of the power function is defined as in [27,28]

$$\begin{array}{c}\hfill L_q\left[t^{\mu }\right]=\frac{{\mathsf{\Gamma }}_q(\mu +1){(1-q)}^{\mu }}{s^{\mu +1}}.\end{array}$$

(14)

For a given function f(t), Sadjang [25] defined $(p,q)$-Laplace transform of the first kind as

$$F_1\left(s\right)=L_{p,q}f\left(t\right)\left(s\right)={\int }_0^{\infty }f\left(t\right)E_{p,q}(-qts)dp,qt,\hspace{1em}s>0.$$

Also for $\alpha >-1$, Sadjang [25] defined the $(p,q)$-Laplace transform of first kind of the power function as

$$L_{p,q}{t}^{\alpha }=\frac{{\mathsf{\Gamma }}_{p,q}(\alpha +1)}{p^{\frac{\alpha (\alpha +1)}{2}}{s}^{\alpha +1}}\hspace{1em}s>0.$$

For a given function f(t), Sadjang [25] defined $(p,q)$-Laplace transform of the second kind as

$$F_2\left(s\right)={\mathbb{L}}_{p,q}f\left(t\right)\left(s\right)={\int }_0^{\infty }f\left(t\right)e_{p,q}(-pts)dp,qt,\hspace{1em}s>0.$$

Also for $\alpha >-1$, Sadjang [25] defined the $(p,q)$-Laplace transform of second kind of the power function as

$${\mathbb{L}}_{p,q}{t}^{\alpha }=\frac{{\gamma }_{p,q}(\alpha +1)}{q^{\frac{\alpha (\alpha +1)}{2}}{s}^{\alpha +1}}\hspace{1em}s>0.$$

Classical Sumudu Transform:

Over the set of function

$$\begin{array}{c}\hfill A=\left\{f\left(t\right)|\exists M,{\tau }_1,{\tau }_2>0,\left|f\left(t\right)\right|<M{e}^{\left|t\right|/{\tau }_j},\hspace{1em}\mathtt{if}\hspace{1em}t\in {(-1)}^j\times [0,\infty )\right\}\end{array}$$

the Sumudu transform is defined by [2]

$$\begin{array}{c}\hfill G\left(u\right)=S\left[f\left(t\right)\right]={\int }_0^{\infty }f\left(ut\right)e^{-t}dt,u\in (-{\tau }_1,{\tau }_2).\end{array}$$

(15)

$\mathit{q}$-Sumudu Transform:

Albayrak et al. [29] defined the q-analogues of the Sumudu transform by means of the following.

$$\begin{array}{c}\hfill S_q[f\left(t\right);s]=\frac{1}{(1-q)}{\int }_0^1{E}_q\left(qt\right)f\left(st\right)d_{q}t,\hspace{1em}s\in (-{\tau }_1,{\tau }_2)\end{array}$$

$$\begin{array}{c}\hfill \mathtt{or}\hspace{1em}{S}_q[f\left(t\right);s]=\frac{1}{(1-q)s}{\int }_0^s{E}_q\left(\frac{q}{s}t\right)f\left(t\right)d_{q}t,\hspace{1em}s\in (-{\tau }_1,{\tau }_2)\end{array}$$

(16)

$$\begin{array}{c}\hfill S_q[x^{\alpha -1};s]=s^{\alpha -1}{(1-q)}^{\alpha -1}{\mathsf{\Gamma }}_q\left(\alpha \right)\end{array}$$

(17)

$$\begin{array}{c}\hfill \mathtt{Also},\hspace{1em}{(1-q)}^{\alpha -1}{\mathsf{\Gamma }}_q\left(\alpha \right)={(q;q)}_{\alpha -1}.\end{array}$$

The ($p,q$)-Sumudu transform of first kind for a given function $f\left(t\right)$ is defined by Sadjang [30] as follows,

$$\begin{array}{ll}{G}_1\left(s\right)& =S_{p,q}f\left(t\right)\left(s\right)=\frac{1}{s}{\int }_0^{\infty }f\left(t\right)E_{p,q}\left(\frac{-qt}{s}\right)dp,qt\\ & ={\int }_0^{\infty }f\left(st\right)E_{p,q}(-qt)dp,qt,\hspace{1em}s>0.\end{array}$$

Sadjang [30] also defined the $(p,q)$-Sumudu transform of first kind of the power function as

$$S_{p,q}{t}^{\alpha }=\frac{{\mathsf{\Gamma }}_{p,q}(\alpha +1)}{p^{\frac{\alpha (\alpha +1)}{2}}}{s}^{\alpha }\hspace{1em}s>0.$$

The ($p,q$)-Sumudu transform of second kind for a given function $f\left(t\right)$ is defined by Sadjang [30] as follows,

$$\begin{array}{ll}{G}_2\left(s\right)& ={\mathbb{S}}_{p,q}f\left(t\right)\left(s\right)=\frac{1}{s}{\int }_0^{\infty }f\left(t\right)e_{p,q}\left(\frac{-pt}{s}\right)dp,qt\\ & ={\int }_0^{\infty }f\left(st\right)e_{p,q}(-qt)dp,qt,\hspace{1em}s>0.\end{array}$$

Sadjang [30] also defined the $(p,q)$-Sumudu transform of first kind of the power function as

$${\mathbb{S}}_{p,q}{t}^{\alpha }=\frac{{\gamma }_{p,q}(\alpha +1)}{q^{\frac{\alpha (\alpha +1)}{2}}}{s}^{\alpha }\hspace{1em}s>0.$$

2. The $(\mathit{p},\mathit{q})$-Analogue of Aleph-Function

Dutta and Arora [31] gave the definition of q-analogue of Aleph-function as follows,

$$\begin{array}{c}\hfill \aleph (z;q)={\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;q\left|\begin{array}{c}{(a_j,A_j)}_{1,n};{\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i;r}\\ {(b_j,B_j)}_{1,m};{\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i;r}\end{array}\right.\right]\\ \hfill =\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}G\left(q^{b_j+B_{j}s}\right){\displaystyle \prod _{j=1}^n}G\left(q^{1-a_j-A_{j}s}\right)\pi z^{-s}{d}_{q}s}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}G\left(q^{1-b_{ji}-B_{ji}s}\right){\displaystyle \prod _{j=n+1}^{p_i}}G\left(q^{a_{ji}+A_{ji}s}\right)\right]G\left(q^s\right)G\left(q^{1-s}\right)sin\pi s},\end{array}$$

(18)

where $z\ne 0,\omega =\sqrt{-1}$ and

$$\begin{array}{c}\hfill G\left(q^{\alpha }\right)={\displaystyle \prod _{n=0}^{\infty }}{(1-q^{\alpha +n})}^{-1}=\frac{1}{{(q^{\alpha };q)}_{\infty }}.\end{array}$$

(19)

And the convergence conditions are defined in [31].

Altaf et al. [17] defined q-analogue of Aleph-function by applying q-Gamma function as follows,

$$\begin{array}{c}\hfill \aleph (z;q)={\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;q\left|\begin{array}{c}{(a_j,A_j)}_{1,n};{\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i;r}\\ {(b_j,B_j)}_{1,m};{\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i;r}\end{array}\right.\right]\end{array}$$

$$\begin{array}{c}\hfill =\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_q(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_q(1-a_j-A_{j}s)z^{-s}{d}_{q}s}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_q(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_q(a_{ji}+A_{ji}s)\right]{\mathsf{\Gamma }}_q\left(s\right){\mathsf{\Gamma }}_q(1-s)sin\pi s}\end{array}$$

(20)

The convergence conditions remains same as defined in [31].

Remark 3.

Put ${\tau }_i$ = 1 in (20), we will get the special case for the basic analogue of the I-function [32,33]:

$$\begin{array}{c}\hfill I_{p_i,q_i;r}^{m,n}\left[z;q\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {(a_{ji},A_{ji})}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {(b_{ji},B_{ji})}_{m+1,q_i}\end{array}\right.\right]\end{array}$$

$$\begin{array}{c}\hfill =\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_q(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_q(1-a_j-A_{j}s)z^{-s}{d}_{q}s}{{\sum }_{i=1}^r\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_q(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_q(a_{ji}+A_{ji}s)\right]{\mathsf{\Gamma }}_q\left(s\right){\mathsf{\Gamma }}_q(1-s)sin\pi s}.\end{array}$$

(21)

Taking $r=1$ in (21) we have known results [34] as

$$\begin{array}{c}\hfill H_{P,Q}^{m,n}\left[z;q\left|\begin{array}{c}{(a_j,A_j)}_{1,P}\\ {(b_j,B_j)}_{1,Q}\end{array}\right.\right]\end{array}$$

$$\begin{array}{c}\hfill =\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_q(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_q(1-a_j-A_{j}s)z^{-s}{d}_{q}s}{{\displaystyle \prod _{j=m+1}^Q}{\mathsf{\Gamma }}_q(1-b_j-B_{j}s){\displaystyle \prod _{j=n+1}^P}{\mathsf{\Gamma }}_q(a_j+A_{j}s){\mathsf{\Gamma }}_q\left(s\right){\mathsf{\Gamma }}_q(1-s)sin\pi s}.\end{array}$$

(22)

The existence conditions will remain same as in above for Equation (20).

Alok et al. [35] defined ($p,q$)-analogue of Aleph-function as follows,

$$\begin{array}{c}\hfill {\aleph }_{p_i,q_i;{\tau }_i;r}^{m,n}\left[\left(z;q\left|\begin{array}{c}{(a_j,A_j)}_{1,n};{\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m};{\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right)\right]\end{array}$$

$$\begin{array}{c}\hfill =\frac{1}{2\pi i}\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_{p,q}(a_{ji}+A_{ji}s){\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s\right]}ds\end{array}$$

where L is contour of integration running from $-i\infty$ to $+i\infty$ in such a manner so that all poles of ${\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s);1\le j\le m$ are to right of the path and those of ${\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s);1\le j\le n$, are to left. The integral converges as defined in [35].

Special Cases:

Taking ${\tau }_i=1$, in the above equation we get the ($p,q$)-analogue of I-function defined by Altaf et al. [36] as follows.

$$\begin{array}{c}\hfill I_{p_i,q_i;r}^{m,n}\left[\left(z;(p,q)\left|\begin{array}{c}{(a_j,A_j)}_{1,n};{(a_{ji},A_{ji})}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m};{(b_{ji},B_{ji})}_{m+1,q_i}\end{array}\right.\right)\right]\end{array}$$

$$\begin{array}{c}\hfill =\frac{1}{2\pi i}\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s)\pi z^{-s}}{{\sum }_{i=1}^r\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_{p,q}(a_{ji}+A_{ji}s){\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s\right]}ds\hspace{1em}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\end{array}$$

Taking $r=1$, we will get the ($p,q$)-analogue of Fox’s H-function defined by Altaf et al. [36] as follows.

$$\begin{array}{c}\hfill H_{P,Q}^{m,n}\left[\left(z;(p,q)\left|\begin{array}{c}{(a_j,A_j)}_{1,P}\\ {(b_j,B_j)}_{1,Q}\end{array}\right.\right)\right]\end{array}$$

$$\begin{array}{c}\hfill =\frac{1}{2\pi i}\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s)\pi z^{-s}ds}{{\displaystyle \prod _{j=m+1}^Q}{\mathsf{\Gamma }}_{p,q}(1-b_j-B_{j}s){\displaystyle \prod _{j=n+1}^P}{\mathsf{\Gamma }}_{p,q}(a_j+A_{j}s){\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s}\hspace{1em}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\end{array}$$

If we take $A_j=B_j=1$, in the above equation, we will get the ($p,q$)-analogue of Meijer’s G-Function defined by Swati Pathak et al. [37] as follows,

$$\begin{array}{c}\hfill G_{P,Q}^{m,n}\left[\left(z;(p,q)\left|\begin{array}{c}{a}_1,a_2,\cdots ,a_P\\ b_1,b_2,\cdots ,b_Q\end{array}\right.\right)\right]\end{array}$$

$$\begin{array}{c}\hfill =\frac{1}{2\pi i}\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-s)\pi z^{-s}ds}{{\displaystyle \prod _{j=m+1}^Q}{\mathsf{\Gamma }}_{p,q}(1-b_j-s){\displaystyle \prod _{j=n+1}^P}{\mathsf{\Gamma }}_{p,q}(a_j+s){\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s}\end{array}$$

Put $p=1$, the above results of ($p,q$)-analogue change into well known results of basic analogues of Aleph-function, I-function, H-function and G-function [17,38].

3. $(\mathit{p},\mathit{q})$-Sumudu Transforms of First and Second Kind of the $(\mathit{p},\mathit{q})$ Analogue of ℵ-Function

In this section we derive $(p,q)$-Sumudu transforms of first and second kind of the $(p,q)$-analogue of Aleph-function with the help of $(p,q)$-Gamma function and Mellin–Barnes-type integral [39]. We use here notation

$$\begin{array}{c}\hfill {\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}[z;(p,q)|.]\hspace{1em}\mathrm{for}\hspace{1em}{\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

3.1. $(p,q)$-Sumudu Transform of First and Second Kind for ${\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}[z;(p,q)|.]:$

Theorem 1.

Let $z,\rho \in \mathbb{C},u\in (-{\tau }_1,{\tau }_2)$ and $Re\left(s\right)>0$ and $|\frac{p}{q}|>1$. Further, let the constants $A_j$,$B_j$,$A_{ji}$,$B_{ji}$ be real and positive, $a_j$,$b_j$,$a_{ji}$,$b_{ji}$ are complex numbers $(i=1,2,\cdots ,p_i;j=1,2,\cdots ,q_i)$, ${\tau }_i>0$ for $i=1,2,\cdots ,r$. The conditions as given in above for $(p,q)$-Aleph function also holds. Then, the $(p,q)$-Sumudu transform of first kind of the ℵ-function exists and the following relation holds,

$$\begin{array}{c}\hfill S_{p,q}\left[t^{\rho -1}{\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right]\right]\end{array}$$

$$\begin{array}{c}\hfill ={\left[p^{\frac{-\rho }{2}}u\right]}^{\rho -1}{\aleph }_{p_i,q_i+1,{\tau }_i;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ (\rho ,0){(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

(23)

Proof. 

To prove (23), we first express the $(p,q)$-analogue of ℵ-function occurring on the left hand side of (23) in terms of Mellin–Barnes contour integral and apply $(p,q)$-Sumudu transform integral (say I).

$$\begin{array}{c}\hfill I={\int }_0^{\infty }{\left(ut\right)}^{\rho -1}{\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right]\hspace{1em}{E}_{p,q}(-qt)d_{p,q}t.\end{array}$$

$$\begin{array}{cc}=& {\int }_0^{\infty }{\left(ut\right)}^{\rho -1}\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_{p,q}(a_{ji}+A_{ji}s)\right]{\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s}\hfill \\ \times & \left\{E_{p,q}(-qt)d_{p,q}t{d}_{p,q}s\right\}\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill I& =& u^{\rho -1}[\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_{p,q}(a_{ji}+A_{ji}s)\right]{\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s}\hfill \\ \hfill & \times & {\int }_0^{\infty }{\left(t\right)}^{\rho -1}{E}_{p,q}(-qt)d_{p,q}t]d_{p,q}s.\hfill \end{array}$$

For the second integral using $(p,q)$-Gamma function

$$\begin{array}{c}\hfill S_{p,q}\left(x^{\alpha }\right)=\frac{{\mathsf{\Gamma }}_{p,q}(\alpha +1)s^{\alpha }}{p^{\frac{\alpha (\alpha +1)}{2}}}\hspace{1em}(\alpha >0),\end{array}$$

we arrive at

$$\begin{array}{c}\hfill I=p^{\frac{-\rho (\rho -1)}{2}}{u}^{\rho -1}\left[\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s)\pi z^{-s}{\mathsf{\Gamma }}_{p,q}(\rho -0.s)d_{p,q}s}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_{p,q}(a_{ji}+A_{ji}s)\right]{\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s}\right]\end{array}$$

$$\begin{array}{c}\hfill ={\left[u{p}^{\frac{-\rho }{2}}\right]}^{\rho -1}{\aleph }_{p_i,q_i+1,{\tau }_i;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ (\rho ,0),{(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

By interpreting the Mellin–Barnes counter integral thus obtained in terms of the $(p,q)$-analogue of ℵ-function, we obtain the result (23). This completes the proof of Theorem 1. □

Theorem 2.

Let $z,\rho \in \mathbb{C},u\in (-{\tau }_1,{\tau }_2)$ and $Re\left(s\right)>0$ and $|\frac{p}{q}|>1$. Further, let the constants $A_j$,$B_j$,$A_{ji}$,$B_{ji}$ are real and positive, $a_j$,$b_j$,$a_{ji}$,$b_{ji}$ are complex numbers $(i=1,2,\cdots ,p_i;j=1,2,\cdots ,q_i)$, ${\tau }_i>0$ for $i=1,2,\cdots ,r$. The conditions as given in above for $(p,q)$-Aleph function also holds. Then, the $(p,q)$-Sumudu transform of second type of the ℵ-function exists and the following relation holds,

$$\begin{array}{c}\hfill {\mathbb{S}}_{p,q}\left[t^{\rho -1}{\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right]\right]\end{array}$$

$$\begin{array}{c}\hfill ={\left[q^{\frac{-\rho }{2}}u\right]}^{\rho -1}{\aleph }_{p_i,q_i+1,{\tau }_i;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ (\rho ,0){(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

(25)

Proof. 

In order to prove (25), we first express the $(p,q)$-analogue of ℵ-function occurring on the left hand side of (25) in terms of Mellin–Barnes contour integral and apply $(p,q)$-Sumudu transform integral (say I).

$$\begin{array}{c}\hfill I={\int }_0^{\infty }{\left(ut\right)}^{\rho -1}{\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right]\hspace{1em}{e}_{p,q}(-pt)d_{p,q}t.\end{array}$$

$$\begin{array}{cc}=& {\int }_0^{\infty }{\left(ut\right)}^{\rho -1}\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\gamma }_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\gamma }_{p,q}(1-a_j-A_{j}s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\gamma }_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\gamma }_{p,q}(a_{ji}+A_{ji}s)\right]{\gamma }_{p,q}\left(s\right){\gamma }_{p,q}(1-s)sin\pi s}\hfill \\ \times & \left\{e_{p,q}(-pt)d_{p,q}t{d}_{p,q}s\right\}\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill I& =& u^{\rho -1}[\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\gamma }_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\gamma }_{p,q}(1-a_j-A_{j}s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\gamma }_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\gamma }_{p,q}(a_{ji}+A_{ji}s)\right]{\gamma }_{p,q}\left(s\right){\gamma }_{p,q}(1-s)sin\pi s}\hfill \\ & \times & {\int }_0^{\infty }{\left(t\right)}^{\rho -1}{E}_{p,q}(-pt)d_{p,q}t]d_{p,q}s.\hfill \end{array}$$

For the second integral using $(p,q)$-gamma function

$$\begin{array}{c}\hfill {\mathbb{S}}_{p,q}\left(x^{\alpha }\right)=\frac{{\gamma }_{p,q}(\alpha +1)s^{\alpha }}{q^{\frac{\alpha (\alpha +1)}{2}}}\hspace{1em}Re\left(\alpha \right)>0,\end{array}$$

we arrive at

$$\begin{array}{c}\hfill I=q^{\frac{-\rho (\rho -1)}{2}}{u}^{\rho -1}\left[\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\gamma }_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\gamma }_{p,q}(1-a_j-A_{j}s)\pi z^{-s}{\gamma }_{p,q}(\rho -0.s)d_{p,q}s}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\gamma }_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\gamma }_{p,q}(a_{ji}+A_{ji}s)\right]{\gamma }_{p,q}\left(s\right){\gamma }_{p,q}(1-s)sin\pi s}\right]\end{array}$$

$$\begin{array}{c}\hfill ={\left[u{q}^{\frac{-\rho }{2}}\right]}^{\rho -1}{\aleph }_{p_i,q_i+1,{\tau }_i;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ (\rho ,0),{(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

By interpreting the Mellin–Barnes counter integral thus obtained in terms of the $(p,q)$-analogue of ℵ-function, we obtain the result (25). This completes the proof of Theorem 2. □

Special Cases:

Now, if we put $p=1$ in theorem 1 then it reduces to q-Sumudu transform of Aleph-function [38],

and if we put $p=1$ and $q=1$ in theorem 1 then it reduces to Sumudu transform of Aleph-function [40].

If we put ${\tau }_i$= 1 ($i=1,2,\cdots ,r$) in (23) and take (A) into account, then we arrive at the following result in terms of the $(p,q)$-analogue of I-function [36] as follows.

Corollary 1.

$$\begin{array}{c}\hfill S_{p,q}\left[t^{\rho -1}{I}_{p_i,q_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {(a_{ji},A_{ji})}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {(b_{ji},B_{ji})}_{m+1,q_i}\end{array}\right.\right]\right]\end{array}$$

$$\begin{array}{c}\hfill ={\left[u{q}^{\frac{-\rho }{2}}\right]}^{\rho -1}{I}_{p_i,q_i+1;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {(a_{ji},A_{ji})}_{n+1,p_i}\\ (\rho ,0),{(b_j,B_j)}_{1,m}& \cdots & {(b_{ji},B_{ji})}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

(27)

The existence conditions for (27) are the same as given in Theorem 1.

If we put ${\tau }_i$=1, $i=1,2,\cdots ,r$, $p_i=P,q_i=Q$ and set $r=1$ in (23) and take (B) into account, then we arrive at the following result in terms of $(p,q)$-analogue of H-function [36] as follows.

Corollary 2.

$$\begin{array}{c}\hfill S_{p,q}\left[t^{\rho -1}{H}_{P,Q}^{m,n}\left[z;(p,q)\left|\begin{array}{c}(a_P,A_P)\\ (b_Q,B_Q)\end{array}\right.\right]\right]={\left[u(1-q)\right]}^{\rho -1}{H}_{P+1,Q}^{m,n+1}\left[z;(p,q)\left|\begin{array}{c}(a_P,A_P)\\ (\rho ,0),(b_Q,B_Q)\end{array}\right.\right]\end{array}$$

(28)

Now if we put $p=1$ in theorem 2 then it reduces to q-Sumudu transform of Aleph-function [38],

and if we put $p=1$ and $q=1$ in theorem 2 then it reduces to Sumudu transform of Aleph-function [40].

If we put ${\tau }_i$= 1 ($i=1,2,\cdots ,r$) in (23) and take (A) into account, then we arrive at the following result in terms of the $(p,q)$-analogue of I-function [36] as follows.

Corollary 3.

$$\begin{array}{c}\hfill {\mathbb{S}}_{p,q}\left[t^{\rho -1}{I}_{p_i,q_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {(a_{ji},A_{ji})}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {(b_{ji},B_{ji})}_{m+1,q_i}\end{array}\right.\right]\right]\end{array}$$

$$\begin{array}{c}\hfill ={\left[q^{\frac{-\rho }{2}}u\right]}^{\rho -1}{I}_{p_i,q_i+1;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {(a_{ji},A_{ji})}_{n+1,p_i}\\ (\rho ,0){(b_j,B_j)}_{1,m}& \cdots & {(b_{ji},B_{ji})}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

(29)

The existence conditions for (29) are the same as given in Theorem 2.

If we put ${\tau }_i$=1, $i=1,2,\cdots ,r$, $p_i=P,q_i=Q$ and set $r=1$ in (23) and take (B) into account, then we arrive at the following result in terms of $(p,q)$-analogue of H-function [36] as follows.

Corollary 4.

$$\begin{array}{c}\hfill {\mathbb{S}}_{p,q}\left[t^{\rho -1}{H}_{P,Q}^{m,n}\left[z;(p,q)\left|\begin{array}{c}(a_P,A_P)\\ (b_Q,B_Q)\end{array}\right.\right]\right]={\left[u(1-q)\right]}^{\rho -1}{H}_{P+1,Q}^{m,n+1}\left[z;(p,q)\left|\begin{array}{c}(a_P,A_P)\\ (\rho ,0),(b_Q,B_Q)\end{array}\right.\right]\end{array}$$

(30)

Next, we find the $(p,q)$-Laplace transform of first and second kind for $(p,q)$-analogue of ℵ-function.

3.2. $(p,q)$-Laplace Transform of First and Second Kind for ${\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}[z;(p,q)|.]:$

Theorem 3.

Let z,$\rho \in \mathbb{C}$ and $Re\left(s\right)>0$, $Re\left(u\right)>0$ and $|\frac{p}{q}|>1$. Further, let the constants $A_j$,$B_j$,$A_{ji}$,$B_{ji}$ are real and positive, $a_j$,$b_j$,$a_{ji}$,$b_{ji}$ are complex numbers $(i=1,2,\cdots ,p_i;j=1,2,\cdots ,q_i)$, ${\tau }_i>0\hspace{1em}\mathtt{f}\mathtt{o}\mathtt{r}\hspace{1em}i=1,2,\cdots ,r$. The conditions as given in above for $(p,q)$-Aleph function also holds. Then, the $(p,q)$-Laplace transform of first kind of the ℵ-function exists and the following relation holds,

$$\begin{array}{c}\hfill L_{p,q}\left[t^{\rho -1}{\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right]\right]\end{array}$$

$$\begin{array}{c}\hfill =u^{-\rho }{p}^{\frac{\rho (1-\rho )}{2}}{\aleph }_{p_i,q_i+1,{\tau }_i;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ (\rho ,0),{(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

(31)

Proof. 

To prove (31), we first express the $(p,q)$-analogue of ℵ-function occurring on the left hand side of (31) in terms of Mellin–Barnes contour integral and apply $(p,q)$-Laplace transform integral (say I).

$$\begin{array}{c}\hfill I={\int }_0^{\infty }{t}^{\rho -1}{\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right]\hspace{1em}{E}_{p,q}(-qtu)d_{p,q}t\end{array}$$

$$\begin{array}{ccc}\hfill & =& {\int }_0^{\infty }{t}^{\rho -1}\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_{p,q}(a_{ji}+A_{ji}s)\right]{\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s}\hfill \\ \hfill & \times & \left\{E_{p,q}(-qtu)d_{p,q}t{d}_{p,q}s\right\}\hfill \end{array}$$

$$\begin{array}{cc}=& \frac{1}{2\pi \omega }\underset{L}{\int }[\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_{p,q}(a_{ji}+A_{ji}s)\right]{\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s}\hfill \\ \times & {\int }_0^{\infty }{\left(t\right)}^{\rho -1}{E}_{p,q}(-qtu)d_{p,q}t]d_{p,q}s.\hfill \end{array}$$

(32)

We know that the $(p,q)$-Laplace transform of first kind of certain elementary function is given by

$$\begin{array}{c}\hfill L_{p,q}(x^n;s)={\int }_0^{\infty }{E}_{p,q}(-sqx)x^n{d}_{p,q}x=\frac{{\mathsf{\Gamma }}_{p,q}(n+1)}{p^{\frac{n(n+1)}{2}{s}^{n+1}}},\\ \hfill Re\left(s\right)>0,Re(n+1)>0\hspace{1em}(n\in \mathbb{N}\cup \left\{0\right\})\hspace{1em}and\hspace{1em}|\frac{q}{p}|<1.\end{array}$$

(33)

Then, by using (33) in (32), we arrive at the following.

$$\begin{array}{c}\hfill I=\left[\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s){\mathsf{\Gamma }}_{p,q}(\rho -0.s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_{p,q}(a_{ji}+A_{ji}s){\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s\right]p^{\frac{\rho (\rho -1)}{2}}{u}^{\rho }}\right]d_{p,q}s\end{array}$$

$$\begin{array}{c}\hfill =u^{-\rho }{\left(p\right)}^{\frac{\rho (1-\rho )}{2}}\times \end{array}$$

$$\begin{array}{c}\hfill \frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\mathsf{\Gamma }}_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\mathsf{\Gamma }}_{p,q}(1-a_j-A_{j}s){\mathsf{\Gamma }}_{p,q}(\rho -0.s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\mathsf{\Gamma }}_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\mathsf{\Gamma }}_{p,q}(a_{ji}+A_{ji}s){\mathsf{\Gamma }}_{p,q}\left(s\right){\mathsf{\Gamma }}_{p,q}(1-s)sin\pi s\right]}{d}_{p,q}s\end{array}$$

$$\begin{array}{c}\hfill =u^{-\rho }{\left(p\right)}^{\frac{\rho (1-\rho )}{2}}{\aleph }_{p_i,q_i+1,{\tau }_i;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ (\rho ,0),{(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

By interpreting the Mellin–Barnes counter integral, thus obtained in terms of the $(p,q)$-analogue of ℵ-function, we obtain the result (31). This completes the proof of Theorem 3. □

Theorem 4.

Let z,$\rho \in \mathbb{C}$ and $Re\left(s\right)>0,Re\left(u\right)>0$ and $|\frac{p}{q}|>1$. Further, let the constants $A_j$,$B_j$,$A_{ji}$,$B_{ji}$ are real and positive, $a_j$,$b_j$,$a_{ji}$,$b_{ji}$ are complex numbers $(i=1,2,\cdots ,p_i;j=1,2,\cdots ,q_i)$, ${\tau }_i>0\hspace{1em}\mathtt{f}\mathtt{o}\mathtt{r}\hspace{1em}i=1,2,\cdots ,r$. The conditions as given in above for $(p,q)$-Aleph function also holds. Then, the $(p,q)$-Laplace transform of second kind of the ℵ-function exists and the following relation holds,

$$\begin{array}{c}\hfill {\mathbb{L}}_{p,q}\left[t^{\rho -1}{\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right]\right]\end{array}$$

$$\begin{array}{c}\hfill =u^{-\rho }{q}^{\frac{\rho (1-\rho )}{2}}{\aleph }_{p_i,q_i+1,{\tau }_i;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ (\rho ,0),{(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

(34)

Proof. 

To prove (34), we first express the $(p,q)$-analogue of ℵ-function occurring on the left hand side of (34) in terms of Mellin–Barnes contour integral and apply $(p,q)$-Laplace transform integral (say I).

$$\begin{array}{c}\hfill I={\int }_0^{\infty }{t}^{\rho -1}{\aleph }_{p_i,q_i,{\tau }_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right]\hspace{1em}{e}_{p,q}(-ptu)d_{p,q}t\end{array}$$

$$\begin{array}{cc}=& {\int }_0^{\infty }{t}^{\rho -1}\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\gamma }_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\gamma }_{p,q}(1-a_j-A_{j}s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\gamma }_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\gamma }_{p,q}(a_{ji}+A_{ji}s)\right]{\gamma }_{p,q}\left(s\right){\gamma }_{p,q}(1-s)sin\pi s}\hfill \\ \times & \left\{e_{p,q}(-ptu)d_{p,q}t{d}_{p,q}s\right\}\hfill \end{array}$$

$$\begin{array}{cc}=& \frac{1}{2\pi \omega }\underset{L}{\int }[\frac{{\displaystyle \prod _{j=1}^m}{\gamma }_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\gamma }_{p,q}(1-a_j-A_{j}s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\gamma }_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\gamma }_{p,q}(a_{ji}+A_{ji}s)\right]{\gamma }_{p,q}\left(s\right){\gamma }_{p,q}(1-s)sin\pi s}\hfill \\ \times & {\int }_0^{\infty }{\left(t\right)}^{\rho -1}{e}_{p,q}(-ptu)d_{p,q}t]d_{p,q}s.\hfill \end{array}$$

(35)

We know that the $(p,q)$-Laplace transform of second type of certain elementary function is given by

$$\begin{array}{c}\hfill {\mathbb{L}}_{p,q}(x^n;s)={\int }_0^{\infty }{e}_{p,q}(-spx)x^n{d}_{p,q}x=\frac{{\gamma }_{p,q}(n+1)}{q^{\frac{n(n+1)}{2}{s}^{n+1}}},\\ \hfill Re\left(s\right)>0,Re(n+1)>0\hspace{1em}(n\in \mathbb{N}\cup \left\{0\right\})\hspace{1em}and\hspace{1em}|\frac{q}{p}|<1.\end{array}$$

(36)

Then, by using (36) in (35), we arrive at the following.

$$\begin{array}{c}\hfill I=\left[\frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\gamma }_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\gamma }_{p,q}(1-a_j-A_{j}s){\gamma }_{p,q}(\rho -0.s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\gamma }_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\gamma }_{p,q}(a_{ji}+A_{ji}s){\gamma }_{p,q}\left(s\right){\gamma }_{p,q}(1-s)sin\pi s\right]q^{\frac{\rho (\rho -1)}{2}}{u}^{\rho }}\right]d_{p,q}s\end{array}$$

$$\begin{array}{c}\hfill =u^{-\rho }{\left(q\right)}^{\frac{\rho (1-\rho )}{2}}\times \end{array}$$

$$\begin{array}{c}\hfill \frac{1}{2\pi \omega }\underset{L}{\int }\frac{{\displaystyle \prod _{j=1}^m}{\gamma }_{p,q}(b_j+B_{j}s){\displaystyle \prod _{j=1}^n}{\gamma }_{p,q}(1-a_j-A_{j}s){\gamma }_{p,q}(\rho -0.s)\pi z^{-s}}{{\sum }_{i=1}^r{\tau }_i\left[{\displaystyle \prod _{j=m+1}^{q_i}}{\gamma }_{p,q}(1-b_{ji}-B_{ji}s){\displaystyle \prod _{j=n+1}^{p_i}}{\gamma }_{p,q}(a_{ji}+A_{ji}s){\gamma }_{p,q}\left(s\right){\gamma }_{p,q}(1-s)sin\pi s\right]}{d}_{p,q}s\end{array}$$

$$\begin{array}{c}\hfill =u^{-\rho }{\left(q\right)}^{\frac{\rho (1-\rho )}{2}}{\aleph }_{p_i,q_i+1,{\tau }_i;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {\left[{\tau }_i(a_{ji},A_{ji})\right]}_{n+1,p_i}\\ (\rho ,0),{(b_j,B_j)}_{1,m}& \cdots & {\left[{\tau }_i(b_{ji},B_{ji})\right]}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

By interpreting the Mellin–Barnes counter integral thus obtained in terms of the $(p,q)$-analogue of ℵ-function, we obtain the result (34). This completes the proof of Theorem 4. □

Special Cases:

Now if we put $p=1$ in theorem 3 then it reduces to q-Laplace transform of Aleph-function [38],

and if we put $p=1$ and $q=1$ in theorem 3 then it reduces to Laplace transform of Aleph-function [40].

If we put ${\tau }_i$= 1 ($i=1,2,\cdots ,r$) in (31) and take (A) into account, then we arrive at the following result in terms of the $(p,q)$-analogue of I-function [36] as follows.

Corollary 5.

$$\begin{array}{c}\hfill L_{p,q}\left[t^{\rho -1}{I}_{p_i,q_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {(a_{ji},A_{ji})}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {(b_{ji},B_{ji})}_{m+1,q_i}\end{array}\right.\right]\right]\end{array}$$

$$\begin{array}{c}\hfill =u^{-\rho }{q}^{\frac{\rho (1-\rho )}{2}}{I}_{p_i,q_i+1;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {(a_{ji},A_{ji})}_{n+1,p_i}\\ (\rho ,0),{(b_j,B_j)}_{1,m}& \cdots & {(b_{ji},B_{ji})}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

(37)

The existence conditions for (37) are the same as given in Theorem 3.

If we put ${\tau }_i=1,i=1,2,\cdots ,r$, $p_i=P,q_i=Q$ and set $r=1$ in (31) and take (B) into account, then we arrive at the following result in terms of $(p,q)$-analogue of H-function [36].

Corollary 6.

$$\begin{array}{c}\hfill L_{p,q}\left[t^{\rho -1}{H}_{P,Q}^{m,n}\left[\left(z;(p,q)\left|\begin{array}{c}(a_P,A_P)\\ (b_Q,B_Q)\end{array}\right.\right)\right]\right]\end{array}$$

$$\begin{array}{c}\hfill =u^{-\rho }{q}^{\frac{\rho (1-\rho )}{2}}{H}_{P+1,Q}^{m,n+1}\left[z;(p,q)\left|\begin{array}{c}(a_P,A_P)\\ (\rho ,0),(b_Q,B_Q)\end{array}\right.\right]\end{array}$$

(38)

Now if we put $p=1$ in theorem 4 then it reduces to q-Laplace transform of Aleph-function [38],

and if we put $p=1$ and $q=1$ in theorem 4 then it reduces to Laplace transform of Aleph-function [40].

If we put ${\tau }_i$= 1 ($i=1,2,\cdots ,r$) in (34) and take (A) into account, then we arrive at the following result in terms of the $(p,q)$-analogue of I-function [36] as follows,

Corollary 7.

$$\begin{array}{c}\hfill {\mathbb{L}}_{p,q}\left[t^{\rho -1}{I}_{p_i,q_i;r}^{m,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {(a_{ji},A_{ji})}_{n+1,p_i}\\ {(b_j,B_j)}_{1,m}& \cdots & {(b_{ji},B_{ji})}_{m+1,q_i}\end{array}\right.\right]\right]\end{array}$$

$$\begin{array}{c}\hfill =u^{-\rho }{p}^{\frac{\rho (1-\rho )}{2}}{I}_{p_i,q_i+1;r}^{m+1,n}\left[z;(p,q)\left|\begin{array}{ccc}{(a_j,A_j)}_{1,n}& \cdots & {(a_{ji},A_{ji})}_{n+1,p_i}\\ (\rho ,0),{(b_j,B_j)}_{1,m}& \cdots & {(b_{ji},B_{ji})}_{m+1,q_i}\end{array}\right.\right].\end{array}$$

(39)

The existence conditions for (39) are the same as given in Theorem 4.

If we put ${\tau }_i=1,i=1,2,\cdots ,r$, $p_i=P,q_i=Q$ and set $r=1$ in (34) and take (B) into account, then we arrive at the following result in terms of $(p,q)$-analogue of H-function [36].

Corollary 8.

$$\begin{array}{c}\hfill {\mathbb{L}}_{p,q}\left[t^{\rho -1}{H}_{P,Q}^{m,n}\left[\left(z;(p,q)\left|\begin{array}{c}(a_P,A_P)\\ (b_Q,B_Q)\end{array}\right.\right)\right]\right]\end{array}$$

$$\begin{array}{c}\hfill =u^{-\rho }{p}^{\frac{\rho (1-\rho )}{2}}{H}_{P+1,Q}^{m,n+1}\left[z;(p,q)\left|\begin{array}{c}(a_P,A_P)\\ (\rho ,0),(b_Q,B_Q)\end{array}\right.\right]\end{array}$$

(40)

4. Results and Discussions

In this work, we focus on the definition of Laplace and Sumudu transforms in quantum calculus. Certain properties for these transforms were obtained. We have also introduced the $(p,q)$-analogue of Aleph function and derive some interesting results. Therefore, the main results can play an important role in the theory of integral transforms and special functions. We conclude our investigation by remarking that several new techniques can be obtained from the main results by appropriate choices of hypergeometric functions.