On the Riemann–Liouville fractional q-calculus operator involving q-Mittag–Leffler function

ABSTRACT

The aim of this paper is to establish four theorems under the fractional q-calculus operators that provide an image formula for the q-analogue of generalized Mittag–Leffler function. Due to the general nature of the q-calculus operators and the generalized q-Mittag–Leffler function, a number of results involving special functions that can be achieved only by making appropriate values for the parameters.

KEYWORDS:

• Fractional q-calculus operators
• generalized q-Mittag–Leffler function
• q-gamma function
• q-beta function

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 33D60
• 33D90
• secondary 26A33

1. Introduction

From the perspective of course of the utility of these results in the evaluation of q-integrals, and the solution of q-differential and q-integral equations, it is necessary to determine the fractional q-derivatives (and fractional q-integrals) of special functions of one and much more variables. The concept of q-analogue of generalized Mittag–Leffler (M-L) functions and the fractional q-calculus of one and much more variables has a broad variety of applications in diverse areas in physical, mathematical and technical sciences, namely, partition theory, number theory, lie theory, mixture analysis, quantum theory, etc. Influenced by these lines of study, a number of scholars have used the fractional q-calculus operators (see Al-Salam, 1966; Kalla et al., 2005; Purohit & Kalla, 2009; Purohit & Selvakumaran, 2015), q-integral inequalities (see Agarwal et al., 2014, 2015; Baleanu & Agarwal, 2014; Choi & Agarwal, 2014; Khan et al., 2022; Nosheen et al., 2023; Wang et al., 2022) and fractional quantum calculus and applications (see, for details, Al-Omari et al., 2018; Purohit & Ucar, 2018; Tariboon et al., 2015) in the theory of special functions in several variables.

In the theory of q-series (see Gasper & Rahman, 1990), $n\in \mathbb{N},$ and for a real or complex a, the q-shifted factorial is defined by

$\begin{array}{l}{(a;q)}_0=1,{(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k),{(a;q)}_{\mathrm{\infty }}=\prod _{k=0}^{\mathrm{\infty }}(1-a{q}^k).\end{array}$

Gasper and Rahman (1990) have given the definitions of q-gamma function and q-analogue of the beta function as

(1.1) $\begin{array}{ll}{\mathrm{\Gamma }}_q\left(\alpha \right)=& \frac{{(q;q)}_{\mathrm{\infty }}}{{(q^{\alpha };q)}_{\mathrm{\infty }}}{(1-q)}^{1-\alpha }=\frac{{(\alpha ;q)}_{\alpha -1}}{{(1-q)}^{\alpha -1}},\\ & (\alpha \in \mathbb{C},\left|q\right|<1),\end{array}$(1.1)

and

(1.2) $\begin{array}{ll}{\mathfrak{B}}_q(x,y)=& {\int }_0^1{z}^{x-1}\frac{{(zq;q)}_{\mathrm{\infty }}}{(z{q}^y;q)\mathrm{\infty }}{d}_{q}z=\frac{{\mathrm{\Gamma }}_q\left(x\right){\mathrm{\Gamma }}_q\left(y\right)}{{\mathrm{\Gamma }}_q(x+y)},\\ & (\mathrm{\Re }(x)>0,\mathrm{\Re }(y)>0).\end{array}$(1.2)

The q-analogue of power function $(z-\tau {)}^{\alpha }$ is defined as

(1.3) $\begin{array}{ll}{(z-\tau )}_q^{\alpha }=& z^{\alpha }{(\tau /z;q)}_{\alpha }=z^{\alpha }\frac{{(\tau /z;q)}_{\mathrm{\infty }}}{{(q^{\alpha }\tau /z;q)}_{\mathrm{\infty }}},\\ & (z\ne 0,0<\left|q\right|<1).\end{array}$(1.3)

Also, Rajkovic et al. (2007) defined a q-derivative of a function f(x) by

(1.4) $\left(D_{q}f\right)(x)=\frac{f(x)-f(qx)}{(1-q)x},(x\ne 0,q\ne 1),$(1.4)

and

(1.5) $\underset{q\to 1}{lim}{D}_{q}f(x)=\frac{df(x)}{dx}.$(1.5)

$\left(D_{\frac{1}{q}}f\right)(x)=q^n(D_q^{n}f)\left(\frac{x}{q^n}\right),$

$D_q(f(x)g(x))=g(x)D_{q}f(x)+f(qx)D_{q}g(x),$

$D_q(f(x)/g(x))=(g(x)D_{q}f(x)-f(x)D_{q}g(x))/g(x)g(qx).$

Recently, q-analogue of generalized M-L function was introduced by Nadeem et al. (2020) for $\mathrm{\Re }(\ell )>\mathrm{\Re }(\omega )>0$ and $\left|q\right|<1$ in the following manner:

(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6)

where $B_q(.)$ is the q-analogue of beta function.

We list the relationships as special cases of q-analogue of the generalized M-L function with other special functions as shown below:

(1)	On setting $\ell =1$ in (1.6(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6) ), we obtain (1.7) $E_{\epsilon ,\delta }^{(\omega ;1)}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(q^{\omega };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}=E_{\epsilon ,\delta }^{\omega }(z;q),$(1.7) which is introduced by Sharma and Jain (2014).
(2)	Again, on making ω = 1 in (1.6(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6) ), we obtain (1.8) $E_{\epsilon ,\delta }^{(1;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}=e_{\epsilon ,\delta }(z;q),$(1.8) the function $e_{\epsilon ,\delta }(z;q)$ can be termed as q-analogue of the M-L function which is introduced by Mansour (2009).
(3)	On letting $\omega =\epsilon =\delta =1$ in (1.6(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6) ), we obtain (1.9) $\begin{array}{ll}{E}_{1,1}^{(1;\ell )}(z;q)=& \sum _{n=0}^{\mathrm{\infty }}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}{z}^n=\frac{{(q^{\ell }n;q)}_{\mathrm{\infty }}}{{(q;q)}_{\mathrm{\infty }}}\\ =& {}_1{\varphi }_0(q^{\ell };-;q,z),\end{array}$(1.9) where the function ${}_1{\varphi }_0(q^{\ell };-;q,z)=(1-z{)}^{-\ell }$ can be termed as q-binomial function.

In fractional calculus, the classical M-L function is now regarded as the queen function by some mathematicians. The M-L function has drawn the attention of numerous researchers in the field of fractional calculus and its applications because it can solve some issues expressed in terms of fractional order differential, difference and integral equations.

Motivated by these avenues of applications, a massive amount of research in the theory of M-L functions has been studied in the literature. For a detailed account of the various generalizations, properties and applications of the M-L function, readers may refer to the literature (see Amsalu & Suthar, 2018; Din & Abidin, 2022; Nisar et al., 2020; Suthar et al., 2019, 2019, 2020; Uçar et al., 2021; Yavuz, 2022; Yavuz & Abdeljawad, 2020). Our goal in this study is to prove four theorems under the fractional q-calculus operators, which give an image formula for the extended M-L function’s q-analogue. Before the conclusion, special cases of the main results are also presented.

2. Fractional q-calculus approach

In this section, first, we would like to provide some definitions of these q-fractional operators.

As a well-known, left-sided Riemann–Liouville (R-L) q-fractional operator introduced by Agarwal (1969) when a = 0 and by Rajkovic et al. (2009) for $a\ne 0$ as follows:

(2.1) $\left(I_{q,a+}^{\vartheta }f\right)(x)=\frac{x^{\vartheta -1}}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_a^x{(qz/x;q)}_{\vartheta -1}f(z)d_{q}z.$(2.1)

Further, Mansour (2016) defines a right-sided R-L q-fractional operator as

(2.2) $\left(I_{q,b-}^{\vartheta }f\right)(x)=\frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_{qx}^b{z}^{\vartheta -1}{(qx/z;q)}_{\vartheta -1}f(z)d_{q}z.$(2.2)

From Mansour (2016), for ϑ > 0 and $\lceil \vartheta \rceil =m$, the left- and right-sided R-L fractional q-derivatives of order ϑ are defined by

(2.3) $\left(D_{q,a+}^{\vartheta }f\right)(x)=\left(D_q^m{I}_{q,a+}^{m-\vartheta }f\right)(x),$(2.3)

and

(2.4) $\left(D_{q,b-}^{\vartheta }f\right)(x)={\left(\frac{-1}{q}\right)}^m\left(D_{q^{-1}}^m{I}_{q,b-}^{m-\vartheta }f\right)(x),$(2.4)

the left- and right-sided Caputo fractional q-derivatives of order ϑ are defined by

(2.5) $\begin{array}{ll}\left({}^c{D}_{q,a+}^{\vartheta }f\right)(x)=& \left(I_{q,a+}^{m-\vartheta }{D}_q^{m}f\right)(x),\\ \left({}^c{D}_{q,b-}^{\vartheta }f\right)(x)=& {\left(\frac{-1}{q}\right)}^m\left(I_{q,b-}^{m-\vartheta }{D}_{q^{-1}}^{m}f\right)(x).\end{array}$(2.5)

This section would establish the following fascinating outcomes in the form of theorems. Here, we present the generalized q-M-L function in view of the fractional q-integrals and fractional q-derivative representations.

Theorem 2.1.

Let $\epsilon ,\delta ,\omega \in \mathbb{C}$, $\mathrm{\Re }(\epsilon )>0,\mathrm{\Re }(\delta )>0,\text{break}\mathrm{\Re }(\ell )>\mathrm{\Re }(\omega )>0,a\in \mathrm{\Re }$, $\vartheta >0,$ $\left|q\right|<1$, and $I_{q,0+}^{\vartheta }$ be the left side of R-L q-fractional integral operator (2.1(2.1) $\left(I_{q,a+}^{\vartheta }f\right)(x)=\frac{x^{\vartheta -1}}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_a^x{(qz/x;q)}_{\vartheta -1}f(z)d_{q}z.$(2.1) ). Then, there holds the following formula true:

(2.6) $\left(I_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta +\delta -1}{E}_{\epsilon ,\delta +\vartheta }^{(\omega ;\ell )}(a{x}^{\epsilon };q).$(2.6)

Proof.

Let Ω₁ be the left-hand side of (2.6(2.6) $\left(I_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta +\delta -1}{E}_{\epsilon ,\delta +\vartheta }^{(\omega ;\ell )}(a{x}^{\epsilon };q).$(2.6) ), and using the generalized q-M-L function (1.6(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6) ) and the left-sided R-L q-fractional operator (2.1(2.1) $\left(I_{q,a+}^{\vartheta }f\right)(x)=\frac{x^{\vartheta -1}}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_a^x{(qz/x;q)}_{\vartheta -1}f(z)d_{q}z.$(2.1) ) on the left-hand side of (2.6(2.6) $\left(I_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta +\delta -1}{E}_{\epsilon ,\delta +\vartheta }^{(\omega ;\ell )}(a{x}^{\epsilon };q).$(2.6) ), we have

$\begin{array}{ll}{\mathrm{\Omega }}_1=& \frac{x^{\vartheta -1}}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_0^x{(qz/x;q)}_{\vartheta -1}{z}^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{\epsilon };q)d_{q}z\\ =& \frac{x^{\vartheta -1}}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_0^x{(qz/x;q)}_{\vartheta -1}{z}^{\delta -1}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\\ & \frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{a^n{z}^{\epsilon n}}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{d}_{q}z\\ =& \sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\frac{x^{\vartheta -1}}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\\ & {\int }_0^x{(qz/x;q)}_{\vartheta -1}{z}^{\delta +\epsilon n-1}{d}_{q}z.\end{array}$

Now, using Equation (1.3)(1.3) $\begin{array}{ll}{(z-\tau )}_q^{\alpha }=& z^{\alpha }{(\tau /z;q)}_{\alpha }=z^{\alpha }\frac{{(\tau /z;q)}_{\mathrm{\infty }}}{{(q^{\alpha }\tau /z;q)}_{\mathrm{\infty }}},\\ & (z\ne 0,0<\left|q\right|<1).\end{array}$(1.3) in the above expression reduces to

(2.7) $\begin{array}{ll}{\mathrm{\Omega }}_1=& \frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \times x^{\vartheta -1}{\int }_0^x\frac{{(qz/x;q)}_{\mathrm{\infty }}}{{(q^{\vartheta }z/x;q)}_{\mathrm{\infty }}}{z}^{\delta +\epsilon n-1}{d}_{q}z.\end{array}$(2.7)

Substituting $z=\varphi x$, then $\varphi =z/x$ in (2.7(2.7) $\begin{array}{ll}{\mathrm{\Omega }}_1=& \frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \times x^{\vartheta -1}{\int }_0^x\frac{{(qz/x;q)}_{\mathrm{\infty }}}{{(q^{\vartheta }z/x;q)}_{\mathrm{\infty }}}{z}^{\delta +\epsilon n-1}{d}_{q}z.\end{array}$(2.7) ), we get

$\begin{array}{ll}{\mathrm{\Omega }}_1=& \frac{x^{\vartheta +\delta -1}}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a{x}^{\epsilon }\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & {\int }_0^1\frac{{(q\varphi ;q)}_{\mathrm{\infty }}}{{(q^{\vartheta }\varphi ;q)}_{\mathrm{\infty }}}{\varphi }^{\delta +\epsilon n-1}{d}_q\varphi .\end{array}$

Using Equation (1.2)(1.2) $\begin{array}{ll}{\mathfrak{B}}_q(x,y)=& {\int }_0^1{z}^{x-1}\frac{{(zq;q)}_{\mathrm{\infty }}}{(z{q}^y;q)\mathrm{\infty }}{d}_{q}z=\frac{{\mathrm{\Gamma }}_q\left(x\right){\mathrm{\Gamma }}_q\left(y\right)}{{\mathrm{\Gamma }}_q(x+y)},\\ & (\mathrm{\Re }(x)>0,\mathrm{\Re }(y)>0).\end{array}$(1.2) , we obtain

(2.8) $\begin{array}{ll}{\mathrm{\Omega }}_1=& \frac{x^{\vartheta +\delta -1}}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a{x}^{\epsilon }\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \frac{{\mathrm{\Gamma }}_q(\delta +\epsilon n){\mathrm{\Gamma }}_q\left(\vartheta \right)}{{\mathrm{\Gamma }}_q(\delta +\vartheta +\epsilon n)}.\end{array}$(2.8)

Interpreting the right-hand side of (2.8(2.8) $\begin{array}{ll}{\mathrm{\Omega }}_1=& \frac{x^{\vartheta +\delta -1}}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a{x}^{\epsilon }\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \frac{{\mathrm{\Gamma }}_q(\delta +\epsilon n){\mathrm{\Gamma }}_q\left(\vartheta \right)}{{\mathrm{\Gamma }}_q(\delta +\vartheta +\epsilon n)}.\end{array}$(2.8) ), with regard to description (1.6(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6) ), we conclude at outcome (2.6(2.6) $\left(I_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta +\delta -1}{E}_{\epsilon ,\delta +\vartheta }^{(\omega ;\ell )}(a{x}^{\epsilon };q).$(2.6) ).

Theorem 2.2.

Let $\epsilon ,\delta ,\omega \in \mathbb{C}$, $\mathrm{\Re }(\epsilon )>0,\mathrm{\Re }(\delta )>0,\text{break}\mathrm{\Re }(\ell )>\mathrm{\Re }(\omega )>0,a\in \mathrm{\Re }$, $\vartheta >0,$ $\left|q\right|<1$ be the right side of R-L q-fractional integral operator (2.2(2.2) $\left(I_{q,b-}^{\vartheta }f\right)(x)=\frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_{qx}^b{z}^{\vartheta -1}{(qx/z;q)}_{\vartheta -1}f(z)d_{q}z.$(2.2) ). Then, it holds the subsequent formula:

(2.9) $\left(I_{q,-}^{\vartheta }\left\{z^{-\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q}{E}_{\epsilon ,\delta +\vartheta }^{(\omega ;\ell )}(a{x}^{-\epsilon };q).$(2.9)

Proof.

Let Ω₂ be the left-hand side of (2.9(2.9) $\left(I_{q,-}^{\vartheta }\left\{z^{-\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q}{E}_{\epsilon ,\delta +\vartheta }^{(\omega ;\ell )}(a{x}^{-\epsilon };q).$(2.9) ), and using Equations (1.6)(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6) and (2.2(2.2) $\left(I_{q,b-}^{\vartheta }f\right)(x)=\frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_{qx}^b{z}^{\vartheta -1}{(qx/z;q)}_{\vartheta -1}f(z)d_{q}z.$(2.2) ) on the left-hand side of (2.9(2.9) $\left(I_{q,-}^{\vartheta }\left\{z^{-\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q}{E}_{\epsilon ,\delta +\vartheta }^{(\omega ;\ell )}(a{x}^{-\epsilon };q).$(2.9) ), we have

$\begin{array}{ll}{\mathrm{\Omega }}_2=& \frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_{qx}^{\mathrm{\infty }}{z}^{\vartheta -1}{(qx/z;q)}_{\vartheta -1}{z}^{-\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{-\epsilon };q)d_{q}z\\ =& \sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\\ & {\int }_{qx}^{\mathrm{\infty }}{(qx/z;q)}_{\vartheta -1}{z}^{-\delta -\epsilon n-1}{d}_{q}z.\end{array}$

Now, using Equation (1.3)(1.3) $\begin{array}{ll}{(z-\tau )}_q^{\alpha }=& z^{\alpha }{(\tau /z;q)}_{\alpha }=z^{\alpha }\frac{{(\tau /z;q)}_{\mathrm{\infty }}}{{(q^{\alpha }\tau /z;q)}_{\mathrm{\infty }}},\\ & (z\ne 0,0<\left|q\right|<1).\end{array}$(1.3) in the above expression reduces to

(2.10) $\begin{array}{ll}{\mathrm{\Omega }}_2=& \frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \times {\int }_{qx}^{\mathrm{\infty }}\frac{{(qx/z;q)}_{\mathrm{\infty }}}{{(q^{\vartheta -1}qx/z;q)}_{\mathrm{\infty }}}{z}^{-\delta -\epsilon n-1}{d}_{q}z.\end{array}$(2.10)

Substituting $z=qx/\theta$, then $\theta =qx/z$ in (2.10(2.10) $\begin{array}{ll}{\mathrm{\Omega }}_2=& \frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \times {\int }_{qx}^{\mathrm{\infty }}\frac{{(qx/z;q)}_{\mathrm{\infty }}}{{(q^{\vartheta -1}qx/z;q)}_{\mathrm{\infty }}}{z}^{-\delta -\epsilon n-1}{d}_{q}z.\end{array}$(2.10) ), we get

(2.11) $\begin{array}{ll}{\mathrm{\Omega }}_2=& \frac{x^{-\delta }}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a{x}^{-\epsilon }\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \times q^{-\delta -\epsilon n-1}{\int }_0^1\frac{{(\theta ;q)}_{\mathrm{\infty }}}{{(q^{\vartheta -1}\theta ;q)}_{\mathrm{\infty }}}{\theta }^{\delta +\epsilon n-1}{d}_q\theta .\end{array}$(2.11)

Replacing $\theta =\lambda q$ in (2.11(2.11) $\begin{array}{ll}{\mathrm{\Omega }}_2=& \frac{x^{-\delta }}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a{x}^{-\epsilon }\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \times q^{-\delta -\epsilon n-1}{\int }_0^1\frac{{(\theta ;q)}_{\mathrm{\infty }}}{{(q^{\vartheta -1}\theta ;q)}_{\mathrm{\infty }}}{\theta }^{\delta +\epsilon n-1}{d}_q\theta .\end{array}$(2.11) ), we get

(2.12) $\begin{array}{ll}{\mathrm{\Omega }}_2=& \frac{x^{-\delta }}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a{x}^{-\epsilon }\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{q}^{-1}\\ & \times {\int }_0^1\frac{{(\lambda q;q)}_{\mathrm{\infty }}}{{(\lambda q^{\vartheta };q)}_{\mathrm{\infty }}}(\lambda {)}^{\delta +\epsilon n-1}{d}_q(\lambda ).\end{array}$(2.12)

Using Equations (1.2)(1.2) $\begin{array}{ll}{\mathfrak{B}}_q(x,y)=& {\int }_0^1{z}^{x-1}\frac{{(zq;q)}_{\mathrm{\infty }}}{(z{q}^y;q)\mathrm{\infty }}{d}_{q}z=\frac{{\mathrm{\Gamma }}_q\left(x\right){\mathrm{\Gamma }}_q\left(y\right)}{{\mathrm{\Gamma }}_q(x+y)},\\ & (\mathrm{\Re }(x)>0,\mathrm{\Re }(y)>0).\end{array}$(1.2) in (2.12(2.12) $\begin{array}{ll}{\mathrm{\Omega }}_2=& \frac{x^{-\delta }}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a{x}^{-\epsilon }\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{q}^{-1}\\ & \times {\int }_0^1\frac{{(\lambda q;q)}_{\mathrm{\infty }}}{{(\lambda q^{\vartheta };q)}_{\mathrm{\infty }}}(\lambda {)}^{\delta +\epsilon n-1}{d}_q(\lambda ).\end{array}$(2.12) ), we get

(2.13) ${\mathrm{\Omega }}_2=\frac{x^{-\delta }}{q}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a{x}^{-\epsilon }\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta +\vartheta )}.$(2.13)

Interpreting the right-hand side of (2.13(2.13) ${\mathrm{\Omega }}_2=\frac{x^{-\delta }}{q}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a{x}^{-\epsilon }\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta +\vartheta )}.$(2.13) ), with regard to description (1.6(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6) ), we conclude at result (2.9(2.9) $\left(I_{q,-}^{\vartheta }\left\{z^{-\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q}{E}_{\epsilon ,\delta +\vartheta }^{(\omega ;\ell )}(a{x}^{-\epsilon };q).$(2.9) ).

Theorem 2.3.

Let $\epsilon ,\delta ,\omega \in \mathbb{C}$, $\mathrm{\Re }(\epsilon )>0,\mathrm{\Re }(\delta )>0,\text{break}\mathrm{\Re }(\ell )>\mathrm{\Re }(\omega )>0,a\in \mathrm{\Re }$, $\vartheta >0,$ $\left|q\right|<1$ and $D_{q,0+}^{\vartheta }$ be the left-sided operator of R-L q-fractional integral (2.3(2.3) $\left(D_{q,a+}^{\vartheta }f\right)(x)=\left(D_q^m{I}_{q,a+}^{m-\vartheta }f\right)(x),$(2.3) ). Then, there holds the following formula true:

(2.14) $\left(D_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta -\delta -1}{E}_{\epsilon ,\delta -\vartheta }^{(\omega ;\ell )}(a{x}^{\epsilon };q).$(2.14)

Proof.

On the left-hand side of (2.14(2.14) $\left(D_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta -\delta -1}{E}_{\epsilon ,\delta -\vartheta }^{(\omega ;\ell )}(a{x}^{\epsilon };q).$(2.14) ), let Ω₃, and using (1.6(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6) ) and (2.3(2.3) $\left(D_{q,a+}^{\vartheta }f\right)(x)=\left(D_q^m{I}_{q,a+}^{m-\vartheta }f\right)(x),$(2.3) ), we have

$\begin{array}{ll}{\mathrm{\Omega }}_3=& D_q^m\left(I_{q,0+}^{m-\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{\epsilon };q)\right\}\right)(x)\\ =& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\\ & \times \frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{D}_q^m{x}^{m-\vartheta -1}\\ & \times {\int }_0^x{z}^{\delta +\epsilon n-1}{(qz/x;q)}_{m-\vartheta -1}{d}_{q}z.\end{array}$

Now, using Equation (1.3)(1.3) $\begin{array}{ll}{(z-\tau )}_q^{\alpha }=& z^{\alpha }{(\tau /z;q)}_{\alpha }=z^{\alpha }\frac{{(\tau /z;q)}_{\mathrm{\infty }}}{{(q^{\alpha }\tau /z;q)}_{\mathrm{\infty }}},\\ & (z\ne 0,0<\left|q\right|<1).\end{array}$(1.3) in the above expression reduces to

(2.15) $\begin{array}{ll}{\mathrm{\Omega }}_3=& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\\ & \times \frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{D}_q^m{x}^{m-\vartheta -1}\\ & \times {\int }_0^x{z}^{\delta +\epsilon n-1}\frac{{(qz/x;q)}_{\mathrm{\infty }}}{{(q^{m-\vartheta }z/x;q)}_{\mathrm{\infty }}}{d}_{q}z.\end{array}$(2.15)

Substituting $z=\varphi x$, then $\varphi =z/x$ in (2.15(2.15) $\begin{array}{ll}{\mathrm{\Omega }}_3=& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\\ & \times \frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{D}_q^m{x}^{m-\vartheta -1}\\ & \times {\int }_0^x{z}^{\delta +\epsilon n-1}\frac{{(qz/x;q)}_{\mathrm{\infty }}}{{(q^{m-\vartheta }z/x;q)}_{\mathrm{\infty }}}{d}_{q}z.\end{array}$(2.15) ), we get

$\begin{array}{ll}{\mathrm{\Omega }}_3=& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\\ & \times \frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{D}_q^m{x}^{m-\vartheta +\delta +\epsilon n-1}\\ & \times {\int }_0^1\frac{{(\varphi q;q)}_{\mathrm{\infty }}}{{(\varphi q^{m-\vartheta };q)}_{\mathrm{\infty }}}{\varphi }^{\delta +\epsilon n-1}{d}_q\varphi .\end{array}$

Using Equation (1.2)(1.2) $\begin{array}{ll}{\mathfrak{B}}_q(x,y)=& {\int }_0^1{z}^{x-1}\frac{{(zq;q)}_{\mathrm{\infty }}}{(z{q}^y;q)\mathrm{\infty }}{d}_{q}z=\frac{{\mathrm{\Gamma }}_q\left(x\right){\mathrm{\Gamma }}_q\left(y\right)}{{\mathrm{\Gamma }}_q(x+y)},\\ & (\mathrm{\Re }(x)>0,\mathrm{\Re }(y)>0).\end{array}$(1.2) on the above expression, we obtain

(2.16) $\begin{array}{ll}{\mathrm{\Omega }}_3=& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\\ & \times \frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{D}_q^m{x}^{m-\vartheta +\delta +\epsilon n-1}{\mathfrak{B}}_q(\delta +\epsilon n,m-\vartheta ).\end{array}$(2.16)

Implementing Equation (1.2)(1.2) $\begin{array}{ll}{\mathfrak{B}}_q(x,y)=& {\int }_0^1{z}^{x-1}\frac{{(zq;q)}_{\mathrm{\infty }}}{(z{q}^y;q)\mathrm{\infty }}{d}_{q}z=\frac{{\mathrm{\Gamma }}_q\left(x\right){\mathrm{\Gamma }}_q\left(y\right)}{{\mathrm{\Gamma }}_q(x+y)},\\ & (\mathrm{\Re }(x)>0,\mathrm{\Re }(y)>0).\end{array}$(1.2) and making a simple calculation, we arrive at result (2.14(2.14) $\left(D_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta -\delta -1}{E}_{\epsilon ,\delta -\vartheta }^{(\omega ;\ell )}(a{x}^{\epsilon };q).$(2.14) ).

Theorem 2.4.

Let $\epsilon ,\delta ,\omega \in \mathbb{C}$, $\mathrm{\Re }(\epsilon )>0,\mathrm{\Re }(\delta )>0,\text{break}\mathrm{\Re }(\ell )>\mathrm{\Re }(\omega )>0,a\in \mathrm{\Re }$, $\delta -\vartheta +\left\{\vartheta \right\}>1,$ $\left|q\right|<1$ and $D_{q,-}^{\vartheta }$ be the right-sided operator of R-L q-fractional integral (2.2(2.2) $\left(I_{q,b-}^{\vartheta }f\right)(x)=\frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_{qx}^b{z}^{\vartheta -1}{(qx/z;q)}_{\vartheta -1}f(z)d_{q}z.$(2.2) ). Then, there holds the subsequent formula:

(2.17) $\left(D_{q,-}^{\vartheta }\left\{z^{\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q^{m+1}}{E}_{\epsilon ,\delta -\vartheta }^{(\omega ;\ell )}(a{x}^{-\epsilon };q).$(2.17)

Proof.

Let Ω₄ be the left-hand side of (2.17(2.17) $\left(D_{q,-}^{\vartheta }\left\{z^{\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q^{m+1}}{E}_{\epsilon ,\delta -\vartheta }^{(\omega ;\ell )}(a{x}^{-\epsilon };q).$(2.17) ), and using Equations (1.6)(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6) and (2.2(2.2) $\left(I_{q,b-}^{\vartheta }f\right)(x)=\frac{1}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}{\int }_{qx}^b{z}^{\vartheta -1}{(qx/z;q)}_{\vartheta -1}f(z)d_{q}z.$(2.2) ), we have

$\begin{array}{ll}{\mathrm{\Omega }}_4=& {\left(\frac{-1}{q}\right)}^m{D}_{q^{-1}}^m\left(I_{q,-}^{m-\vartheta }\left\{z^{\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{-\epsilon };q)\right\}\right)(x)\\ =& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\\ & \times \frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{\left(\frac{-1}{q}\right)}^m{D}_{q^{-1}}^m\\ & \times {\int }_{qx}^{\mathrm{\infty }}{(qx/z;q)}_{m-\vartheta -1}{z}^{m-\delta -\epsilon n-1}{d}_{q}z.\end{array}$

Now, using Equation (1.3)(1.3) $\begin{array}{ll}{(z-\tau )}_q^{\alpha }=& z^{\alpha }{(\tau /z;q)}_{\alpha }=z^{\alpha }\frac{{(\tau /z;q)}_{\mathrm{\infty }}}{{(q^{\alpha }\tau /z;q)}_{\mathrm{\infty }}},\\ & (z\ne 0,0<\left|q\right|<1).\end{array}$(1.3) in the above expression reduces to

(2.18) $\begin{array}{ll}{\mathrm{\Omega }}_4=& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\\ & \times \frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{\left(\frac{-1}{q}\right)}^m{D}_{q^{-1}}^m\\ & \times {\int }_{qx}^{\mathrm{\infty }}\frac{{(qx/z;q)}_{\mathrm{\infty }}}{{(q^{m-\vartheta }x/z;q)}_{\mathrm{\infty }}}{z}^{m-\delta -\epsilon n-1}{d}_{q}z.\end{array}$(2.18)

Substituting $z=qx/\zeta$, then $\zeta =qx/z$ in (2.18(2.18) $\begin{array}{ll}{\mathrm{\Omega }}_4=& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\\ & \times \frac{a^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{\left(\frac{-1}{q}\right)}^m{D}_{q^{-1}}^m\\ & \times {\int }_{qx}^{\mathrm{\infty }}\frac{{(qx/z;q)}_{\mathrm{\infty }}}{{(q^{m-\vartheta }x/z;q)}_{\mathrm{\infty }}}{z}^{m-\delta -\epsilon n-1}{d}_{q}z.\end{array}$(2.18) ), we get

(2.19) $\begin{array}{ll}{\mathrm{\Omega }}_4=& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\\ & \times \frac{{\left(a\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{\left(\frac{-1}{q}\right)}^m{D}_{q^{-1}}^m{x}^{m-\delta -\epsilon n}{q}^{m-\delta -\epsilon n-1}\\ & \times {\int }_0^1\frac{{(\zeta ;q)}_{\mathrm{\infty }}}{{(q^{m-\vartheta -1}\zeta ;q)}_{\mathrm{\infty }}}{\zeta }^{-m+\delta +\epsilon n-1}{d}_q\zeta .\end{array}$(2.19)

Replacing $\zeta =\lambda q$ in (2.19(2.19) $\begin{array}{ll}{\mathrm{\Omega }}_4=& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\\ & \times \frac{{\left(a\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{\left(\frac{-1}{q}\right)}^m{D}_{q^{-1}}^m{x}^{m-\delta -\epsilon n}{q}^{m-\delta -\epsilon n-1}\\ & \times {\int }_0^1\frac{{(\zeta ;q)}_{\mathrm{\infty }}}{{(q^{m-\vartheta -1}\zeta ;q)}_{\mathrm{\infty }}}{\zeta }^{-m+\delta +\epsilon n-1}{d}_q\zeta .\end{array}$(2.19) ), we get

(2.20) $\begin{array}{ll}{\mathrm{\Omega }}_4=& \frac{1}{{\mathrm{\Gamma }}_q(m-\vartheta )}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\\ & \times \frac{{\left(a\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{\left(\frac{-1}{q}\right)}^m\\ & \times D_{q^{-1}}^m{x}^{m-\delta -\epsilon n}{q}^{-1}{\int }_0^1{\lambda }^{\delta -m+\epsilon n-1}\frac{{(\lambda q;q)}_{\mathrm{\infty }}}{{(q^{m-\vartheta }\lambda ;q)}_{\mathrm{\infty }}}{d}_q(\lambda ).\end{array}$(2.20)

Using Equation (1.2)(1.2) $\begin{array}{ll}{\mathfrak{B}}_q(x,y)=& {\int }_0^1{z}^{x-1}\frac{{(zq;q)}_{\mathrm{\infty }}}{(z{q}^y;q)\mathrm{\infty }}{d}_{q}z=\frac{{\mathrm{\Gamma }}_q\left(x\right){\mathrm{\Gamma }}_q\left(y\right)}{{\mathrm{\Gamma }}_q(x+y)},\\ & (\mathrm{\Re }(x)>0,\mathrm{\Re }(y)>0).\end{array}$(1.2) in (2.12(2.12) $\begin{array}{ll}{\mathrm{\Omega }}_2=& \frac{x^{-\delta }}{{\mathrm{\Gamma }}_q\left(\vartheta \right)}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a{x}^{-\epsilon }\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}{q}^{-1}\\ & \times {\int }_0^1\frac{{(\lambda q;q)}_{\mathrm{\infty }}}{{(\lambda q^{\vartheta };q)}_{\mathrm{\infty }}}(\lambda {)}^{\delta +\epsilon n-1}{d}_q(\lambda ).\end{array}$(2.12) ), we get

(2.21) $\begin{array}{ll}{\mathrm{\Omega }}_4=& \frac{1}{q}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \times {\left(\frac{-1}{q}\right)}^m{D}_{q^{-1}}^m{x}^{m-\delta -\epsilon n}\frac{{\mathrm{\Gamma }}_q(\delta -m+\epsilon n)}{{\mathrm{\Gamma }}_q(\delta -\vartheta +\epsilon n)}.\end{array}$(2.21)

Also using Equation (1.4(1.4) $\left(D_{q}f\right)(x)=\frac{f(x)-f(qx)}{(1-q)x},(x\ne 0,q\ne 1),$(1.4) ), we get

(2.22) $\begin{array}{ll}{\mathrm{\Omega }}_4=& \frac{1}{q}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \times {\left(\frac{-1}{q}\right)}^m{q}^m{D}_q^m\left(\frac{x^{m-\delta -\epsilon n}}{q^m}\right)\frac{{\mathrm{\Gamma }}_q(\delta -m+\epsilon n)}{{\mathrm{\Gamma }}_q(\delta -\vartheta +\epsilon n)}.\end{array}$(2.22)

On simplification of the right-hand side of (2.22(2.22) $\begin{array}{ll}{\mathrm{\Omega }}_4=& \frac{1}{q}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{{\left(a\right)}^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}\\ & \times {\left(\frac{-1}{q}\right)}^m{q}^m{D}_q^m\left(\frac{x^{m-\delta -\epsilon n}}{q^m}\right)\frac{{\mathrm{\Gamma }}_q(\delta -m+\epsilon n)}{{\mathrm{\Gamma }}_q(\delta -\vartheta +\epsilon n)}.\end{array}$(2.22) ), with regard to description (1.6(1.6) $E_{\epsilon ,\delta }^{(\omega ;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_q(\omega +n,\ell -\omega )}{{\mathfrak{B}}_q(\omega ,\ell -\omega )}\frac{{(q^{\ell };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )},$(1.6) ), we conclude at the result (2.17(2.17) $\left(D_{q,-}^{\vartheta }\left\{z^{\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;\ell )}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q^{m+1}}{E}_{\epsilon ,\delta -\vartheta }^{(\omega ;\ell )}(a{x}^{-\epsilon };q).$(2.17) ).

3. Special cases

With the aid of the results developed in the prior segment, this section would establish the following fascinating outcomes in the form of corollaries. As these findings are direct implications of Equations (1.7)(1.7) $E_{\epsilon ,\delta }^{(\omega ;1)}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(q^{\omega };q)}_n}{{(q;q)}_n}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}=E_{\epsilon ,\delta }^{\omega }(z;q),$(1.7) and (1.8(1.8) $E_{\epsilon ,\delta }^{(1;\ell )}(z;q)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{\mathrm{\Gamma }}_q(\epsilon n+\delta )}=e_{\epsilon ,\delta }(z;q),$(1.8) ) and theorems 2.1–2.4, they are provided without evidence here.

(i) If we put $\ell =1$, in theorems 2.1–2.4, we obtain the following interesting results in the form of corollaries 3.1–3.4, respectively.

Corollary 3.1.

(3.1) $\left(I_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;1)}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta +\delta -1}{E}_{\epsilon ,\delta +\vartheta }^{\omega }(a{x}^{\epsilon };q).$(3.1)

Corollary 3.2.

(3.2) $\left(I_{q,-}^{\vartheta }\left\{z^{-\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;1)}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q}{E}_{\epsilon ,\delta +\vartheta }^{\omega }(a{x}^{-\epsilon };q).$(3.2)

Corollary 3.3.

(3.3) $\left(D_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(\omega ;1)}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta -\delta -1}{E}_{\epsilon ,\delta -\vartheta }^{\omega }(a{x}^{\epsilon };q).$(3.3)

Corollary 3.4.

(3.4) $\left(D_{q,-}^{\vartheta }\left\{z^{\vartheta -\delta }{E}_{\epsilon ,\delta }^{(\omega ;1)}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q^{m+1}}{E}_{\epsilon ,\delta -\vartheta }^{\omega }(a{x}^{-\epsilon };q).$(3.4)

(ii) If we put ω = 1, in theorems 2.1–2.4, we obtain the following interesting results in the form of corollaries 3.5–3.8, respectively.

Corollary 3.5.

(3.5) $\left(I_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(1;\ell )}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta +\delta -1}{e}_{\epsilon ,\delta +\vartheta }(a{x}^{\epsilon };q).$(3.5)

Corollary 3.6.

(3.6) $\left(I_{q,-}^{\vartheta }\left\{z^{-\vartheta -\delta }{E}_{\epsilon ,\delta }^{(1;\ell )}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q}{e}_{\epsilon ,\delta +\vartheta }(a{x}^{-\epsilon };q).$(3.6)

Corollary 3.7.

(3.7) $\left(D_{q,0+}^{\vartheta }\left\{z^{\delta -1}{E}_{\epsilon ,\delta }^{(1;\ell )}(a{z}^{\epsilon };q)\right\}\right)(x)=x^{\vartheta -\delta -1}{e}_{\epsilon ,\delta -\vartheta }(a{x}^{\epsilon };q).$(3.7)

Corollary 3.8.

(3.8) $\left(D_{q,-}^{\vartheta }\left\{z^{\vartheta -\delta }{E}_{\epsilon ,\delta }^{(1;\ell )}(a{z}^{-\epsilon };q)\right\}\right)(x)=\frac{x^{-\delta }}{q^{m+1}}{e}_{\epsilon ,\delta -\vartheta }(a{x}^{-\epsilon };q).$(3.8)

4. Conclusion

The results presented in this article contribute to the study of the q-fractional calculus, specifically the q-analogue of the generalized functions of the M-L. The findings seen in this paper tend to be recent and likely to have useful implications for a wide variety of mathematical, statistical and physical science problems.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

No data were used to support this study.

Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/27684830.2023.2292549

Additional information

Notes on contributors

Mulugeta Dawud Ali

Mulugeta Dawud Ali is an Assistant Professor in the Department of Mathematics, College of Natural Science at Wollo University’s KIOT Campus. He is now doing his PhD at Wollo University. My research interests include q-Fractional Calculus, q-Special functions, q-Integral transformations, q-Hypergeometric Series, functional spaces, and Mathematical Physics. I am very familiar with mathematical application software like MATLAB, Mathematica, and Latex.

D.L. Suthar

Dr. D.L. Suthar is an Associate Professor at Wollo University in Dessie, Ethiopia’s Amhara Region. I have 15 years of teaching experience and 18 years of research experience, including a Ph.D. program. Among my research interests are special functions, fractional calculus, integral transformations, basic hypergeometric series, geometric function theory, and mathematical physics. ICM (Brazil, Russia) and ICIAM (Spain, Japan) offered me International Travel Grants to attend International Conferences. I have 185 research articles published in national and international journals, as well as three books for engineering students.