Saigo fractional q-integral involving the generalized q-hypergeometric series and a general class of q-polynomials

Abstract

The fractional q-calculus has attracted the interest of a large number of academics over the last four decades or so, due mainly to a wide range of applications that cover natural sciences to social sciences. Many fractional q-calculus operators, particularly those involving various q-special functions, have been deeply studied and widely applied. In this paper, we aim to establish certain image formulas of Saigo fractional q-integral operators involving the product of generalized q-hypergeometric series and a general class of q-polynomials that are primarily expressed in terms of generalized q-hypergeometric series in a systematic manner. We demonstrate their use by studying q-Konhouser biorthogonal polynomials and q-Jacobi polynomials. Additionally, some fascinating special cases of our main findings are taken into consideration, and pertinent connections between some of the findings presented here and those from earlier studies are also made.

Keywords:

• GCq-Ps
• q-hypergeometric series
• q-integrals
• q-Jacobi polynomials
• q-Konhouser biorthogonal
• Saigo’s fractional

2010Mathematics Subject Classification:

• 26A33
• 33D15
• 33D60
• 33D90

1. Introduction and preliminaries

Fractional calculus, also known as calculus of variations, deals with the generalization of traditional calculus operations to non-integer orders of differentiation and integration. Instead of just dealing with integer orders like 1, 2, 3, and so on, fractional calculus allows operations like differentiating and integrating functions by non-integer amounts, such as 0.5, −1.5, and so forth. This concept is useful in modeling various physical and engineering phenomena with memory effects, such as diffusion, viscoelasticity, and more.

Fractal calculus is a mathematical framework that extends traditional calculus to deal with functions defined on fractals, which are complex geometric objects with self-similar patterns at different scales. Fractals often have non-integer dimensions, and fractal calculus aims to provide tools for analyzing and differentiating functions that exhibit fractal-like behavior. Fractal calculus and its geometry explanation (He, 2018) have been becoming hot topics in both mathematics and engineering for non differential solutions. Fractal theory is the theoretical basis for the fractal spacetime (He, 2014), and analysis MEMS in the fractal space and established the corresponding fractal model (Tian et al., 2021) as well. Many researchers have already found the intrinsic relationship between the fractional dimensions and the fractional order (Wu and Liang, 2017). This paper will focus itself on the fractal calculus, a relatively new branch of mathematics with easy understanding and ready applications in the field of fractal solitary waves (Wang, 2022). The two-scale fractal calculus is used to describe transport problems in a porous medium, such as the problem of oil extraction and heat transfer of heat pipes. The porous medium is viewed as a fractal space, so non-linear vibrations in the porous medium can be modeled by fractal vibration theory (He et al., 2021).

q-Calculus, also known as quantum calculus or Jackson’s calculus, is a generalization of traditional calculus that involves a parameter q. This parameter q is a complex number or a formal variable, and when q approaches 1, q-calculus converges to ordinary calculus. q-Calculus extends differentiation and integration operations by introducing q-derivatives and q-integrals, which satisfy different rules compared to traditional calculus. It has applications in physics, particularly in quantum mechanics, statistical mechanics, and combinatorics.

Fractal calculus focuses on functions defined on fractals, whereas fractional calculus deals with generalizing calculus to non-integer orders, and q-calculus involves a parameter q that leads to specialized rules for differentiation and integration. The computation of fractional q-derivatives (and fractional q-integrals) of special functions of one or more variables is significant because these results can be used to evaluate q-integrals and solve q-integral and q-differential equations. The theory of fractional q-calculus and q-hypergeometric functions of one and more variables has several applications in applied mathematics, Engineering and Physical Sciences, such as Lie theory, Number theory, Computational complexity, Partition theory, Quantum field theory, and so on. Due to their use in domains including combinatorics, orthogonal polynomials, calculus of variations, basic hypergeometric functions, the theory of relativity and mechanics, quantum difference operators are significant in mathematics (Gasper and Rahman, 1990). Numerous basic quantum calculus ideas are covered in the book by Kac and Cheung (2002). Researchers have recently paid a lot of attention to q-calculus, and (Annaby and Mansour, 2012; Cao et al., 2023; Rajković et al., 2007; Zhou et al., 2022) and other references cited therein contain a number of new findings.

The fractional calculus has been acknowledged as a tool for the explanation of several phenomena in kinematics, biology, chemistry, finance, etc. throughout the previous two centuries. Moreover, q-calculus was confirmed as a method for handling discrete variations of continuous scientific problems (see (Kac and Cheung, 2002) for more information). It was only a matter of time before those ideas came together.

Inspired by these possibilities, a number of researchers have used fractional q-calculus operators in the theory of special functions of one or more variables. Recently, Kumar et al. (2023) and Kumar et al. (2022) determined several fractional q-integral formulas for the q-analogues of I-function and Aleph function of one and two variables. Vyas et al. (2019), Vyas et al. (2020) and Vyas et al. (2021) analyzed the q-analogues of the numerous special functions and subsequent implementation. It has been derived for basic q-generating series, q-trigonometric functions and q-exponential function by Al-Omari et al. (2021) utilized for different forms of q-Bessel functions and some power series of special type. Purohit et al. (2021), Purohit et al. (2021) and Purohit et al. (2023) derived the unified class of spiral-like functions, analytic function including different type of fractional operators in quantum calculus. Here, all definitions and notations are taken from Gasper and Rahman’s book (Gasper and Rahman, 1990).

In the theory of q-series, the q-shifted factorial is defined as follows for a real or complex a and $\left|q\right|$ $<$ 1: (1.1) $${\left(\gamma ;q\right)}_0=1; q\ne 1, {\left(\gamma ;q\right)}_n=\prod _{k=0}^{n-1}\left(1-\gamma q^k\right), \left(n\in \mathbb{N}\right).$$(1.1)

${(\gamma ;q)}_{\infty }$ then as the infinite product diverges when $\gamma \ne 0$ and $\left|q\right|\ge 1,$ thus we will assume that $\left|q\right|\ge 1$ anytime ${(\gamma ;q)}_{\infty }$ appears in a formula. Moreover, we have the following for any complex number a: (1.2) $${\left(\gamma ;q\right)}_a=\frac{{\left(\gamma ;q\right)}_{\infty }}{{\left(\gamma q^a;q\right)}_{\infty }},\hspace{1em}a\in \mathbb{C}, 0<\left|q\right|<1.$$(1.2) where q^a’s principal value is obtained. The definition of the power function ${(z-\gamma )}^{\vartheta },$ q-analogues is (1.3) $${\left(z-\gamma \right)}_q^{\gamma }=z^{\vartheta }{\left(\gamma /z;q\right)}_{\vartheta }=z^{\vartheta }\prod _{j=0}^{\infty }\left[\frac{1-\left(\gamma /z\right)q^j}{1-\left(\gamma /z\right)q^{j+\vartheta }}\right]=z^{\vartheta }\frac{{\left(\gamma /z;q\right)}_{\infty }}{{\left(q^{\vartheta }\gamma /z;q\right)}_{\infty }},\hspace{1em}0<\left|q\right|<1, \left(z\ne 0\right).$$(1.3)

The mathematical description of the q-gamma function is (1.4) $${\Gamma }_q\left(z\right)=\frac{{\left(q;q\right)}_{\infty }}{{\left(q^z;q\right)}_{\infty }}{\left(1-q\right)}^{1-z}=\frac{{\left(q;q\right)}_{z-1}}{{\left(1-q\right)}^{z-1}},(z\ne 0,-1,-2,\dots ).,$$(1.4)

The q-derivative of an analytic function f(z) is defined as follows. (1.5) $$\left(D_{q}f\right)\left(z\right)=\frac{f\left(z\right)-f\left(qz\right)}{\left(1-q\right)z},\hspace{1em}\left(z\ne 0,\hspace{1em}q\ne 1\right),$$(1.5) and $$\underset{q\to 1}{\lim }{D}_{q}f\left(z\right)=\frac{df\left(z\right)}{dz}.$$

We have (1.6) $$D_q^n(z^{\mu })=\frac{{\Gamma }_q\left(\mu +1\right)}{{\Gamma }_q\left(\mu -n+1\right)} z^{\mu -n},\hspace{1em}(\mathfrak{R}\left(\mu \right)+1>0).$$(1.6)

The q-integrals of a function f(z) are defined as follows. (1.7) $${\int }_0^{a}f\left(z\right)d_{q}z=a\left(1-q\right)\sum _{m=0}^{\infty }{q}^{m}f\left(a{q}^m\right),$$(1.7) and (1.8) $${\int }_a^{\infty }f\left(z\right)d_{q}z=a\left(1-q\right)\sum _{m=1}^{\infty }{q}^{-m}f\left(a{q}^{-m}\right).$$(1.8)

The generalized q-hypergeometric series is given by (1.9) $${}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,z\right]=\sum _{m=0}^{\infty }\frac{{\left(a_1,\dots ,a_r;q\right)}_m}{{\left(b_1,\dots ,b_s;q\right)}_n}\frac{z^n}{{(q;q)}_m}{\left\{{\left(-1\right)}^m{q}^{\left(m(m-1)/2\right)}\right\}}^{(1+s-r)},$$(1.9) with ${(a_1,\dots ,a_r;q)}_m={(a_1;q)}_m\dots {(a_r;q)}_m.$ If $0<\left|q\right|<1,$ the series (1.9) converses for all $\left|z\right|<1$ if $r=s+1,$ and for any z if $r\le s.$ Also if $\left|q\right|>1,$ the series converses absolutely $\left|z\right|<\left|(b_1\dots b_{s}q)/(a_1\dots a_r)\right|.$

The family of basic (or q-) polynomials $f_{n,N}(z;q)$ (cf. Srivastava and Agarwal (1989)) is described in terms of a bounded complex sequence ${\left\{S_{n,q}\right\}}_{n=0}^{\infty }$ as (1.10) $$f_{n,N}\left(z;q\right)=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}{z}^j,\hspace{1em}(n=0,1,2,\dots ),$$(1.10) where N is a positive integer. The q-polynomials family $f_{n,N}(z;q)$ yields a number of well-known q-polynomials as its special cases by appropriately specializing the coefficient $S_{n,q}.$ The q-Laguerre polynomials, q-Hermite polynomials, q-Jacobi polynomials, q-Konhauser polynomials, q-Wall polynomials, and several others are among these.

The definition of Saigo’s fractional q-integral operators has recently been provided by Purohit and Yadav (2010), with the restriction that one of the parameters,ε, must be a non-negative integer. Under that limitation, it was impossible to provide a definition of fractional derivatives. Garg and Chanchlani (2011) provide the definitions of the Saigo’s fractional q-integral operator as follows to get over these issues. For $\vartheta \in \mathbb{C}, \mathfrak{R}(\vartheta )>0, \zeta$ and $\epsilon$ being real or complex, the generalized fractional q-integral operators ${\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }(.)$ and ${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }(.)$ defined in the following manner: (1.11) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=\frac{z^{-\zeta -1}}{{\Gamma }_q(\vartheta )}{\int }_0^z{(tq/z;q)}_{\vartheta -1}\\ \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q^{\vartheta };q\right)}_n{\left(q;q\right)}_n}{\left(-1\right)}^n{q}^{\left(\epsilon -\zeta \right)n}{q}^{-\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right)}{\left(\frac{t}{z}-1\right)}_q^{n}f\left(t\right)d_{q}t,\end{array}$$(1.11) and (1.12) $${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=\frac{q^{-\vartheta \left(\vartheta +1\right)/2-\zeta }}{{\Gamma }_q(\vartheta )}{\int }_z^{\infty }{(z/t;q)}_{\vartheta -1} t^{-\zeta -1} \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q^{\vartheta };q\right)}_n{\left(q;q\right)}_n}{\left(-1\right)}^n{q}^{\left(\epsilon -\zeta \right)n}{q}^{-\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right)}{\left(\frac{z}{qt}-1\right)}_q^{n}f\left(t{q}^{1-\vartheta }\right)d_{q}t,$$(1.12) where $0<\left|q\right|<1,$ a real valued function f(z) on $(0,\infty ).$

Definitions given by (1.11(1.11) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=\frac{z^{-\zeta -1}}{{\Gamma }_q(\vartheta )}{\int }_0^z{(tq/z;q)}_{\vartheta -1}\\ \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q^{\vartheta };q\right)}_n{\left(q;q\right)}_n}{\left(-1\right)}^n{q}^{\left(\epsilon -\zeta \right)n}{q}^{-\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right)}{\left(\frac{t}{z}-1\right)}_q^{n}f\left(t\right)d_{q}t,\end{array}$$(1.11) ) and (1.12(1.12) $${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=\frac{q^{-\vartheta \left(\vartheta +1\right)/2-\zeta }}{{\Gamma }_q(\vartheta )}{\int }_z^{\infty }{(z/t;q)}_{\vartheta -1} t^{-\zeta -1} \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q^{\vartheta };q\right)}_n{\left(q;q\right)}_n}{\left(-1\right)}^n{q}^{\left(\epsilon -\zeta \right)n}{q}^{-\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right)}{\left(\frac{z}{qt}-1\right)}_q^{n}f\left(t{q}^{1-\vartheta }\right)d_{q}t,$$(1.12) ), in view of (1.7(1.7) $${\int }_0^{a}f\left(z\right)d_{q}z=a\left(1-q\right)\sum _{m=0}^{\infty }{q}^{m}f\left(a{q}^m\right),$$(1.7) ) and (1.8(1.8) $${\int }_a^{\infty }f\left(z\right)d_{q}z=a\left(1-q\right)\sum _{m=1}^{\infty }{q}^{-m}f\left(a{q}^{-m}\right).$$(1.8) ) can be written as (1.13) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=z^{-\zeta }{\left(1-q\right)}^{\vartheta } \\ \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q;q\right)}_n}{q}^{\left(\epsilon -\zeta +1\right)n}\sum _{k=0}^{\infty }{q}^k\frac{{\left(q^{\vartheta +n};q\right)}_k}{{\left(q;q\right)}_k}f\left(z q^{k+n}\right),\end{array}$$(1.13) and (1.14) $$\begin{array}{c}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=z^{-\zeta }{q}^{-\vartheta \left(\vartheta +1\right)/2}{\left(1-q\right)}^{\vartheta } \\ \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q;q\right)}_n} q^{\epsilon n}\sum _{k=0}^{\infty }{q}^{\zeta k}\frac{{\left(q^{\vartheta +n};q\right)}_k}{{\left(q;q\right)}_k}f\left(z q^{-\vartheta -k-n}\right).\end{array}$$(1.14)

For $0<\left|q\right|<1;$ $\mathfrak{R}(\vartheta )>0,$ ζ and $\epsilon$ being real or complex. Images of the power function under fractional q-integrals ${\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }$ and ${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }$ are given by (1.15) $${\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left(z^{\mu }\right)=\frac{{\Gamma }_q\left(\mu +1\right){\Gamma }_q\left(\mu -\zeta +\epsilon +1\right)}{{\Gamma }_q\left(\mu -\zeta +1\right){\Gamma }_q\left(\mu +\vartheta +\epsilon +1\right)}{z}^{\mu -\zeta },$$(1.15) provided $\mathfrak{R}(\mu +1)>0$ and $\mathfrak{R}(\mu -\zeta +\epsilon +1)>0.$

and (1.16) $${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left(z^{\mu }\right)=\frac{{\Gamma }_q\left(\zeta -\mu \right){\Gamma }_q\left(\epsilon -\mu \right)}{{\Gamma }_q\left(-\mu \right){\Gamma }_q\left(\zeta +\vartheta -\mu +\epsilon \right)}{z}^{\mu -\zeta }{q}^{-\vartheta \mu -\vartheta \left(\vartheta +1\right)/2},$$(1.16) provided $\mathfrak{R}(\zeta -\mu )>0$ and $\mathfrak{R}(\epsilon -\mu )>0.$

To our opinion, there is far too little literature on fractional q-integrability, q-differentiability. As a result of the foregoing research, the goal of this study is to obtain the Saigo fractional q-calculus formula pertaining to product of the generalized q-hypergeometric series and a the general class of basic (or q-) polynomials (GCq-P). The name general class of basic polynomials, itself indicates the importance of the results, because we can derive a number of fractional q-calculus formulae for various classical orthogonal q-polynomials.

2. Fractional q-calculus approach

2.1. Left-sided fractional q-integration of the product of q-polynomial and generalized q-hypergeometric series

In this section, we consider Saigo fractional q-integral operators (1.11(1.11) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=\frac{z^{-\zeta -1}}{{\Gamma }_q(\vartheta )}{\int }_0^z{(tq/z;q)}_{\vartheta -1}\\ \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q^{\vartheta };q\right)}_n{\left(q;q\right)}_n}{\left(-1\right)}^n{q}^{\left(\epsilon -\zeta \right)n}{q}^{-\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right)}{\left(\frac{t}{z}-1\right)}_q^{n}f\left(t\right)d_{q}t,\end{array}$$(1.11) ) involving the product of the GCq-P (1.10(1.10) $$f_{n,N}\left(z;q\right)=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}{z}^j,\hspace{1em}(n=0,1,2,\dots ),$$(1.10) ) and the generalized q-hypergeometric function (1.9(1.9) $${}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,z\right]=\sum _{m=0}^{\infty }\frac{{\left(a_1,\dots ,a_r;q\right)}_m}{{\left(b_1,\dots ,b_s;q\right)}_n}\frac{z^n}{{(q;q)}_m}{\left\{{\left(-1\right)}^m{q}^{\left(m(m-1)/2\right)}\right\}}^{(1+s-r)},$$(1.9) ) as the kernels and derive the following theorem:

Theorem 1.

Let $\vartheta ,\zeta ,\epsilon ,\rho ,\delta \in \mathbb{C},$ $0<\left|q\right|<1$; $\mathfrak{R}(\vartheta )>0,\mathfrak{R}(\delta +j+1)>0$ and $\mathfrak{R}(\delta +j-\zeta +\epsilon +1)>0$, then the Saigo fractional q-integral ${\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }$ of the product of a GCq-P $f_{n,N}(.)$ and the generalized q-hypergeometric function ${}_r{\phi }_s(.)$ is given by (2.1) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{\delta }{f}_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho z\right]\right\}\\ =z^{\delta -\zeta } \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q} \frac{{\Gamma }_q\left(\delta +j+1\right){\Gamma }_q\left(\delta +j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(\delta +j-\zeta +1\right){\Gamma }_q\left(\delta +j+\vartheta +\epsilon +1\right)}{z}^j\\ \times { }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\delta +j+1}, q^{\delta +j-\zeta +\epsilon +1}\\ b_1,\dots ,b_s, q^{\delta +j-\zeta +1}, q^{\delta +j+\vartheta +\epsilon +1}\end{array}; q, \rho z\right].\end{array}$$(2.1)

Proof.

We continue to prove (2.1(2.1) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{\delta }{f}_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho z\right]\right\}\\ =z^{\delta -\zeta } \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q} \frac{{\Gamma }_q\left(\delta +j+1\right){\Gamma }_q\left(\delta +j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(\delta +j-\zeta +1\right){\Gamma }_q\left(\delta +j+\vartheta +\epsilon +1\right)}{z}^j\\ \times { }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\delta +j+1}, q^{\delta +j-\zeta +\epsilon +1}\\ b_1,\dots ,b_s, q^{\delta +j-\zeta +1}, q^{\delta +j+\vartheta +\epsilon +1}\end{array}; q, \rho z\right].\end{array}$$(2.1) ) by first presenting a GCq-P occurring on its left-hand side as a series given by (1.10(1.10) $$f_{n,N}\left(z;q\right)=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}{z}^j,\hspace{1em}(n=0,1,2,\dots ),$$(1.10) ), then replacing the generalized q-hypergeomtric series with the help of (1.9(1.9) $${}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,z\right]=\sum _{m=0}^{\infty }\frac{{\left(a_1,\dots ,a_r;q\right)}_m}{{\left(b_1,\dots ,b_s;q\right)}_n}\frac{z^n}{{(q;q)}_m}{\left\{{\left(-1\right)}^m{q}^{\left(m(m-1)/2\right)}\right\}}^{(1+s-r)},$$(1.9) ), interchanging the order of Saigo fractional q-integral and summation, we get the subsequent form (say ${\mathrm{I}}_1$): (2.2) $$\begin{array}{c}{I}_1=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\sum _{m=0}^{\infty }\frac{{\left(a_1,a_2,\dots ,a_r;q\right)}_m}{{\left(b_1,b_2,\dots ,b_s;q\right)}_m}{\rho }^m{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}\\ \times \hspace{0.17em}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{\frac{z^{\delta +j+m}}{{\left(q;q\right)}_m}\right\}.\end{array}$$(2.2)

Now, using the image formula (1.15(1.15) $${\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left(z^{\mu }\right)=\frac{{\Gamma }_q\left(\mu +1\right){\Gamma }_q\left(\mu -\zeta +\epsilon +1\right)}{{\Gamma }_q\left(\mu -\zeta +1\right){\Gamma }_q\left(\mu +\vartheta +\epsilon +1\right)}{z}^{\mu -\zeta },$$(1.15) ), which is true under the circumstances specified in Theorem 1, we get (2.3) $$\begin{array}{c}{I}_1=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\sum _{m=0}^{\infty }\frac{{\left(a_1,a_2,\dots ,a_r;q\right)}_m{\rho }^m}{{\left(b_1,b_2,\dots ,b_s;q\right)}_m}{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}\\ \times \frac{{\Gamma }_q\left(\delta +j+m+1\right){\Gamma }_q\left(\delta +j+m-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(\delta +j+m-\zeta +1\right){\Gamma }_q\left(\delta +j+m+\vartheta +\epsilon +1\right)}{z}^{\delta +j+m-\zeta }.\end{array}$$(2.3)

Using the Eq. (1.4)(1.4) $${\Gamma }_q\left(z\right)=\frac{{\left(q;q\right)}_{\infty }}{{\left(q^z;q\right)}_{\infty }}{\left(1-q\right)}^{1-z}=\frac{{\left(q;q\right)}_{z-1}}{{\left(1-q\right)}^{z-1}},(z\ne 0,-1,-2,\dots ).,$$(1.4) and simplifying, we obtain (2.4) $$\begin{array}{c}{I}_1=z^{\delta -\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\frac{{\Gamma }_q\left(\delta +j+1\right){\Gamma }_q\left(\delta +j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(\delta +j-\zeta +1\right){\Gamma }_q\left(\delta +j+\vartheta +\epsilon +1\right)}{z}^j\\ \times \sum _{m=0}^{\infty }\frac{{\left(a_1,a_2,\dots ,a_r;q\right)}_m}{{\left(b_1,b_2,\dots ,b_s;q\right)}_m}{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}\\ \frac{{\left(q^{\delta +j+1},q\right)}_m{\left(q^{\delta +j-\zeta +\epsilon +1},q\right)}_m}{{\left(q^{\delta +j-\zeta +1};q\right)}_m{\left(q^{\delta +j+\vartheta +\epsilon +1};q\right)}_m}{\left(\rho z\right)}^m.\end{array}$$(2.4)

Upon substituting the right-hand side of Eq. (4.6)(4.6) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{E_q^{-1/z}{f}_{n,N}\left(z;q\right)\right\}=q^{-\vartheta \left(\vartheta -1\right)/2}{z}^{-\zeta -1}\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q\\ S_{n,q}\frac{{\left(z{q}^{-\vartheta }\right)}^j}{\left(1-q\right)}\times {\frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right) }}_{\mathrm{ }\mathrm{ }}\\ { }_2{\phi }_2\left[\begin{array}{c}{q}^{\zeta -j+1},q^{\epsilon -j+1}\\ q^{-j+1},q^{\vartheta +\zeta -j+\epsilon +1}\end{array} ;q ,-q^{1+\vartheta }/z\right],\end{array}$$(4.6) , in view of using definition (1.9(1.9) $${}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,z\right]=\sum _{m=0}^{\infty }\frac{{\left(a_1,\dots ,a_r;q\right)}_m}{{\left(b_1,\dots ,b_s;q\right)}_n}\frac{z^n}{{(q;q)}_m}{\left\{{\left(-1\right)}^m{q}^{\left(m(m-1)/2\right)}\right\}}^{(1+s-r)},$$(1.9) ), we arrive at the result (2.1(2.1) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{\delta }{f}_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho z\right]\right\}\\ =z^{\delta -\zeta } \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q} \frac{{\Gamma }_q\left(\delta +j+1\right){\Gamma }_q\left(\delta +j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(\delta +j-\zeta +1\right){\Gamma }_q\left(\delta +j+\vartheta +\epsilon +1\right)}{z}^j\\ \times { }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\delta +j+1}, q^{\delta +j-\zeta +\epsilon +1}\\ b_1,\dots ,b_s, q^{\delta +j-\zeta +1}, q^{\delta +j+\vartheta +\epsilon +1}\end{array}; q, \rho z\right].\end{array}$$(2.1) ).□

Remark.

If we take n = 0, $j=0,$ then $f_{0,N}(z;q)\to 1$ in the Theorems 1, we get the known result presented by Garg and Chanchlani (2011), page 147, Eq. (3.1)(3.1) $$N=1, \omega \ge 1, S_{n,q}=\frac{{\Gamma }_q\left(\omega n+\mu +1\right){\left(-1\right)}^j{q}^{j\left(j-1\right)}}{{\left(q;q\right)}_n{\Gamma }_q\left(\omega j+\mu +1\right)},$$(3.1) .

We now show several special cases of Theorem 1 as follows:

If we set ζ = 0, define operator in Eq. (1.11)(1.11) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=\frac{z^{-\zeta -1}}{{\Gamma }_q(\vartheta )}{\int }_0^z{(tq/z;q)}_{\vartheta -1}\\ \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q^{\vartheta };q\right)}_n{\left(q;q\right)}_n}{\left(-1\right)}^n{q}^{\left(\epsilon -\zeta \right)n}{q}^{-\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right)}{\left(\frac{t}{z}-1\right)}_q^{n}f\left(t\right)d_{q}t,\end{array}$$(1.11) , and using the following identity (2.5) $${\mathcal{I}}_q^{\vartheta ,0,\epsilon }f(z)=I_q^{\epsilon ,\vartheta }f(z),$$(2.5) where the $I_q^{\epsilon ,\vartheta }(.)$ is the Kober fractional q-integral operator, appears in the right-hand side of (2.5(2.5) $${\mathcal{I}}_q^{\vartheta ,0,\epsilon }f(z)=I_q^{\epsilon ,\vartheta }f(z),$$(2.5) ) due to (Agarwal, 1969) defined as: (2.6) $$I_q^{\epsilon ,\vartheta }\left\{f\left(z\right)\right\}=\frac{z^{-\epsilon -\vartheta }}{{\Gamma }_q\left(\vartheta \right)}{\int }_0^z{\left(z-tq\right)}_{\mu -1}{t}^{\epsilon }f\left(t\right)d_q\left(t\right),(\mathfrak{R}\left(\mu \right)>0;\left|q\right|<1,\epsilon \in \mathfrak{R}).$$(2.6)

Corollary 2.

Let $\vartheta ,\epsilon ,\rho ,\delta \in \mathbb{C},$ $0<\left|q\right|<1$; $\mathfrak{R}(\vartheta )>0,\mathfrak{R}(\delta +j+\epsilon +1)>0$, then the Kober fractional q-integral $I_q^{\epsilon ,\vartheta }$ of the product of GCq-P $f_{n,N}(.)$ and generalized q-hypergeometric function ${}_r{\phi }_s(.)$ is given by (2.7) $$\begin{array}{c}{I}_q^{\epsilon ,\vartheta }\left\{z^{\delta }{f}_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho z\right]\right\}=z^{\delta } \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q} \\ \times \frac{{\Gamma }_q\left(\delta +j+\epsilon +1\right) z^j}{{\Gamma }_q\left(\delta +j+\vartheta +\epsilon +1\right)}{ }_{r+1}{\phi }_{s+1}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\delta +j+\epsilon +1}\\ b_1,\dots ,b_s, q^{\delta +j+\vartheta +\epsilon +1}\end{array}; q, \rho z\right].\end{array}$$(2.7)

Remark.

If we take n = 0, $j=0,$ then $f_{0,N}(z;q)\to 1$ in the Corollary 2, we get the result given by Yadav and Purohit (2006), p. 441, Eq. (25).

If we set $\zeta =-\vartheta$ in the operators (1.11(1.11) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=\frac{z^{-\zeta -1}}{{\Gamma }_q(\vartheta )}{\int }_0^z{(tq/z;q)}_{\vartheta -1}\\ \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q^{\vartheta };q\right)}_n{\left(q;q\right)}_n}{\left(-1\right)}^n{q}^{\left(\epsilon -\zeta \right)n}{q}^{-\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right)}{\left(\frac{t}{z}-1\right)}_q^{n}f\left(t\right)d_{q}t,\end{array}$$(1.11) ), and using the following identity (2.8) $${\mathcal{I}}_q^{\vartheta ,0,\epsilon }f(z)=I_q^{\vartheta }f(z),$$(2.8) where the Riemann-Liouville fractional q-integral operator, appearing in the right-hand side is due to (Agarwal, 1969) defined as: (2.9) $$I_q^{\vartheta }\left\{f\left(z\right)\right\}=\frac{1}{{\Gamma }_q\left(\vartheta \right)}{\int }_0^z{\left(z-tq\right)}_{\mu -1}f\left(t\right)d_q\left(t\right), (\mathfrak{R}\left(\mu \right)>0;\left|q\right|<1).$$(2.9)

Corollary 3.

Let $\vartheta ,\rho ,\delta \in \mathbb{C},$ $0<\left|q\right|<1$; $\mathfrak{R}(\vartheta )>0,\mathfrak{R}(\delta +j+1)>0$, then the Riemann-Liouville fractional q-integral $I_q^{\vartheta }$ of the product of GCq-P $f_{n,N}(.)$ and generalized q-hypergeometric function ${}_r{\phi }_s(.)$ is given by (2.10) $$\begin{array}{c}{I}_q^{\vartheta }\left\{z^{\delta }{f}_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho z\right]\right\}\\ =z^{\delta +\vartheta } \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q} \frac{{\Gamma }_q\left(\delta +j+1\right) z^j}{{\Gamma }_q\left(\delta +j-\zeta +1\right)}{ }_{r+1}{\phi }_{s+1}\\ \left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\delta +j+1}\\ b_1,\dots ,b_s, q^{\delta +j+\vartheta +1}\end{array} ;q, \rho z\right].\end{array}$$(2.10)

Remark.

If we take n = 0, $j=0,$ then $f_{0,N}(z;q)\to 1$ in the Corollary 3, we get the known result presented by Yadav and Purohit (2004), p. 595, Eqs. (2.8)(2.8) $${\mathcal{I}}_q^{\vartheta ,0,\epsilon }f(z)=I_q^{\vartheta }f(z),$$(2.8) .

2.2. Right-sided fractional q-integration of the product of q-polynomial and generalized q-hypergeometric series

In this section, we consider Saigo fractional q-integral operators (1.12(1.12) $${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=\frac{q^{-\vartheta \left(\vartheta +1\right)/2-\zeta }}{{\Gamma }_q(\vartheta )}{\int }_z^{\infty }{(z/t;q)}_{\vartheta -1} t^{-\zeta -1} \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q^{\vartheta };q\right)}_n{\left(q;q\right)}_n}{\left(-1\right)}^n{q}^{\left(\epsilon -\zeta \right)n}{q}^{-\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right)}{\left(\frac{z}{qt}-1\right)}_q^{n}f\left(t{q}^{1-\vartheta }\right)d_{q}t,$$(1.12) ) involving the product of the GCq-P (1.10(1.10) $$f_{n,N}\left(z;q\right)=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}{z}^j,\hspace{1em}(n=0,1,2,\dots ),$$(1.10) ) and the generalized q-hypergeometric function (1.9(1.9) $${}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,z\right]=\sum _{m=0}^{\infty }\frac{{\left(a_1,\dots ,a_r;q\right)}_m}{{\left(b_1,\dots ,b_s;q\right)}_n}\frac{z^n}{{(q;q)}_m}{\left\{{\left(-1\right)}^m{q}^{\left(m(m-1)/2\right)}\right\}}^{(1+s-r)},$$(1.9) ) as the kernels and derive the following theorem:

Theorem 4.

Let $\vartheta ,\zeta ,\epsilon ,\rho ,\delta \in \mathbb{C},$ $0<\left|q\right|<1$; $\mathfrak{R}(\vartheta )>0,\mathfrak{R}(\zeta +\delta -j)>0, \mathfrak{R}(\epsilon +\delta -j)>0$ and $\delta -j\ne 0$, then the Saigo fractional q-integral ${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }$ of the product of GCq-P $f_{n,N}(.)$ and generalized q-hypergeometric series ${}_r{\phi }_s(.)$ is given by (2.11) $$\begin{array}{c}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{-\delta }{f}_{n,N}{\left(z;q\right)}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}\\ =q^{\vartheta \delta -\vartheta \left(\vartheta +1\right)/2} z^{-\delta -\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q\\ S_{n,q}\frac{{\Gamma }_q\left(\zeta +\delta -j\right){\Gamma }_q\left(\epsilon +\delta -j\right)}{{\Gamma }_q\left(\delta -j\right){\Gamma }_q\left(\vartheta +\zeta +\delta -j+\epsilon \right)}{\left(z{q}^{-\vartheta }\right)}^j\\ {\times }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\zeta +\delta -j}, q^{\epsilon +\delta -j}\\ b_1,\dots ,b_s, q^{\delta -j}, q^{\vartheta +\zeta +\delta -j+\epsilon }\end{array}; q, \rho q^{\vartheta }/z\right].\end{array}$$(2.11)

Proof.

Let ${\mathrm{I}}_2$ be the left-hand side of (2.11(2.11) $$\begin{array}{c}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{-\delta }{f}_{n,N}{\left(z;q\right)}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}\\ =q^{\vartheta \delta -\vartheta \left(\vartheta +1\right)/2} z^{-\delta -\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q\\ S_{n,q}\frac{{\Gamma }_q\left(\zeta +\delta -j\right){\Gamma }_q\left(\epsilon +\delta -j\right)}{{\Gamma }_q\left(\delta -j\right){\Gamma }_q\left(\vartheta +\zeta +\delta -j+\epsilon \right)}{\left(z{q}^{-\vartheta }\right)}^j\\ {\times }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\zeta +\delta -j}, q^{\epsilon +\delta -j}\\ b_1,\dots ,b_s, q^{\delta -j}, q^{\vartheta +\zeta +\delta -j+\epsilon }\end{array}; q, \rho q^{\vartheta }/z\right].\end{array}$$(2.11) ), using Eqs. (1.9(1.1) $${\left(\gamma ;q\right)}_0=1; q\ne 1, {\left(\gamma ;q\right)}_n=\prod _{k=0}^{n-1}\left(1-\gamma q^k\right), \left(n\in \mathbb{N}\right).$$(1.1) , 1.10)(1.1) $${\left(\gamma ;q\right)}_0=1; q\ne 1, {\left(\gamma ;q\right)}_n=\prod _{k=0}^{n-1}\left(1-\gamma q^k\right), \left(n\in \mathbb{N}\right).$$(1.1) , By interchanging the order of fractional q-integral and summation, we have $$I_2=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\sum _{m=0}^{\infty }\frac{{\left(a_1,a_2,\dots ,a_r;q\right)}_m}{{\left(b_1,b_2,\dots ,b_s;q\right)}_m}{\rho }^m{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{\frac{x^{j-\delta -m}}{{\left(q;q\right)}_m}\right\},$$

 By using (1.16(1.16) $${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left(z^{\mu }\right)=\frac{{\Gamma }_q\left(\zeta -\mu \right){\Gamma }_q\left(\epsilon -\mu \right)}{{\Gamma }_q\left(-\mu \right){\Gamma }_q\left(\zeta +\vartheta -\mu +\epsilon \right)}{z}^{\mu -\zeta }{q}^{-\vartheta \mu -\vartheta \left(\vartheta +1\right)/2},$$(1.16) ), it becomes $$\begin{array}{l}{I}_2=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\sum _{m=0}^{\infty }\frac{{\left(a_1,a_2,\dots ,a_r;q\right)}_m{\rho }^m}{{\left(b_1,b_2,\dots ,b_s;q\right)}_m}\\ \hspace{1em}\hspace{1em}{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}\\ \hspace{1em}\hspace{1em}\times \frac{{\Gamma }_q\left(\zeta +\delta -j+m\right){\Gamma }_q\left(\epsilon +\delta -j+m\right)}{{\Gamma }_q\left(\delta -j+m\right){\Gamma }_q\left(\vartheta +\zeta +\delta -j+m+\epsilon \right)}\\ \hspace{1em}\hspace{1em}{z}^{\left(j-\delta -m\right)-\zeta }{q}^{-\vartheta \left(j-\delta -m\right)-\vartheta \left(\vartheta +1\right)/2}\\ =\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\frac{{\Gamma }_q\left(\zeta +\delta -j\right){\Gamma }_q\left(\epsilon +\delta -j\right)}{{\Gamma }_q\left(\delta -j\right){\Gamma }_q\left(\vartheta +\zeta +\delta -j+\epsilon \right)}\\ \hspace{1em}\sum _{m=0}^{\infty }\frac{{\left(a_1,a_2,\dots ,a_r;q\right)}_m{\rho }^m}{{\left(b_1,b_2,\dots ,b_s;q\right)}_m}\\ \hspace{1em}\times \hspace{0.17em}{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}\frac{{\left(q^{\zeta +\delta -j};q\right)}_m{\left(q^{\epsilon +\delta -j};q\right)}_m}{{\left(q^{\delta -j};q\right)}_m{\left(q^{\vartheta +\zeta +\delta -j+\epsilon };q\right)}_m}\\ {\left(\rho /z\right)}^m{z}^{-\delta +j-\zeta }{q}^{-\vartheta \left(-\delta +j-m\right)-\vartheta \left(\vartheta +1\right)/2},\end{array}$$

Interpreting the right-hand side of the above equation, in view of Eq. (1.9)(1.9) $${}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,z\right]=\sum _{m=0}^{\infty }\frac{{\left(a_1,\dots ,a_r;q\right)}_m}{{\left(b_1,\dots ,b_s;q\right)}_n}\frac{z^n}{{(q;q)}_m}{\left\{{\left(-1\right)}^m{q}^{\left(m(m-1)/2\right)}\right\}}^{(1+s-r)},$$(1.9) , we arrive at the result (2.11(2.11) $$\begin{array}{c}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{-\delta }{f}_{n,N}{\left(z;q\right)}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}\\ =q^{\vartheta \delta -\vartheta \left(\vartheta +1\right)/2} z^{-\delta -\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q\\ S_{n,q}\frac{{\Gamma }_q\left(\zeta +\delta -j\right){\Gamma }_q\left(\epsilon +\delta -j\right)}{{\Gamma }_q\left(\delta -j\right){\Gamma }_q\left(\vartheta +\zeta +\delta -j+\epsilon \right)}{\left(z{q}^{-\vartheta }\right)}^j\\ {\times }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\zeta +\delta -j}, q^{\epsilon +\delta -j}\\ b_1,\dots ,b_s, q^{\delta -j}, q^{\vartheta +\zeta +\delta -j+\epsilon }\end{array}; q, \rho q^{\vartheta }/z\right].\end{array}$$(2.11) ). □

We cannot directly put $\delta -j=0$ in (2.11(2.11) $$\begin{array}{c}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{-\delta }{f}_{n,N}{\left(z;q\right)}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}\\ =q^{\vartheta \delta -\vartheta \left(\vartheta +1\right)/2} z^{-\delta -\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q\\ S_{n,q}\frac{{\Gamma }_q\left(\zeta +\delta -j\right){\Gamma }_q\left(\epsilon +\delta -j\right)}{{\Gamma }_q\left(\delta -j\right){\Gamma }_q\left(\vartheta +\zeta +\delta -j+\epsilon \right)}{\left(z{q}^{-\vartheta }\right)}^j\\ {\times }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\zeta +\delta -j}, q^{\epsilon +\delta -j}\\ b_1,\dots ,b_s, q^{\delta -j}, q^{\vartheta +\zeta +\delta -j+\epsilon }\end{array}; q, \rho q^{\vartheta }/z\right].\end{array}$$(2.11) ) as in the case ${\Gamma }_q(\delta -j)$ comes in the fractional part of the right hand side of the equation. However, if we consider the limit of (2.11(2.11) $$\begin{array}{c}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{-\delta }{f}_{n,N}{\left(z;q\right)}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}\\ =q^{\vartheta \delta -\vartheta \left(\vartheta +1\right)/2} z^{-\delta -\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q\\ S_{n,q}\frac{{\Gamma }_q\left(\zeta +\delta -j\right){\Gamma }_q\left(\epsilon +\delta -j\right)}{{\Gamma }_q\left(\delta -j\right){\Gamma }_q\left(\vartheta +\zeta +\delta -j+\epsilon \right)}{\left(z{q}^{-\vartheta }\right)}^j\\ {\times }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\zeta +\delta -j}, q^{\epsilon +\delta -j}\\ b_1,\dots ,b_s, q^{\delta -j}, q^{\vartheta +\zeta +\delta -j+\epsilon }\end{array}; q, \rho q^{\vartheta }/z\right].\end{array}$$(2.11) ) as $\delta -j\to 0,$ we arrive at the outcome that follows, which we obtain here independently for the sake of derivation.

Theorem 5.

Let $\vartheta ,\zeta ,\epsilon ,\rho ,\delta \in \mathbb{C},$ $0<\left|q\right|<1$; $\mathfrak{R}(\vartheta )>0,\mathfrak{R}(\zeta -j+1)>0, \mathfrak{R}(\epsilon -j+1)>0$, then the Saigo fractional q-integral ${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }$ of the product of GCq-P $f_{n,N}(.)$ and generalized q-hypergeometric series ${}_r{\phi }_s(.)$ is given by (2.12) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}=\\ q^{-\vartheta \left(\vartheta +1\right)/2}{\left(-1\right)}^{1+s-r}\rho z^{-\zeta -1}\times \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)\left(1-a_1\right)\left(1-a_2\right) \dots \left(1-a_r\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right)\left(1-b_1\right)\left(1-b_2\right) \dots \left(1-b_s\right) }\frac{{\left(z q^{-\vartheta }\right)}^j}{\left(1-q\right)}\\ {\times \hspace{0.17em}}_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_{1}q,\dots ,a_{r}q, q^{\zeta -j+1}, q^{\epsilon -j+1}\\ b_{1}q,\dots ,b_{s}q, q^{-j+1}, q^{\vartheta +\zeta -j+\epsilon +1}\end{array} ;q, \rho q^{1+\vartheta +s-r}/z\right].\end{array}$$(2.12)

Proof.

By using (1.9(1.9) $${}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,z\right]=\sum _{m=0}^{\infty }\frac{{\left(a_1,\dots ,a_r;q\right)}_m}{{\left(b_1,\dots ,b_s;q\right)}_n}\frac{z^n}{{(q;q)}_m}{\left\{{\left(-1\right)}^m{q}^{\left(m(m-1)/2\right)}\right\}}^{(1+s-r)},$$(1.9) ) and (1.10(1.10) $$f_{n,N}\left(z;q\right)=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}{z}^j,\hspace{1em}(n=0,1,2,\dots ),$$(1.10) ), the left-hand side of (2.12(2.12) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}=\\ q^{-\vartheta \left(\vartheta +1\right)/2}{\left(-1\right)}^{1+s-r}\rho z^{-\zeta -1}\times \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)\left(1-a_1\right)\left(1-a_2\right) \dots \left(1-a_r\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right)\left(1-b_1\right)\left(1-b_2\right) \dots \left(1-b_s\right) }\frac{{\left(z q^{-\vartheta }\right)}^j}{\left(1-q\right)}\\ {\times \hspace{0.17em}}_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_{1}q,\dots ,a_{r}q, q^{\zeta -j+1}, q^{\epsilon -j+1}\\ b_{1}q,\dots ,b_{s}q, q^{-j+1}, q^{\vartheta +\zeta -j+\epsilon +1}\end{array} ;q, \rho q^{1+\vartheta +s-r}/z\right].\end{array}$$(2.12) ), say ${\mathrm{I}}_3,$ which can be written as: (2.13) $$I_3={\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\sum _{m=0}^{\infty }\frac{{\left(a_1,a_2,\dots ,a_r;q\right)}_m}{{\left(b_1,b_2,\dots ,b_s;q\right)}_m}{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}\frac{{\rho }^n{z}^{j-m}}{{\left(q;q\right)}_m}\right\},$$(2.13)

By interchanging the order of fractional q-integral and summation, we have (2.14) $$I_3=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\sum _{m=0}^{\infty }\frac{{\left(a_1,a_2,\dots ,a_r;q\right)}_m}{{\left(b_1,b_2,\dots ,b_s;q\right)}_m}{\rho }^m{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{j-m}\right\},$$(2.14) which by using the image formula (1.16(1.16) $${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left(z^{\mu }\right)=\frac{{\Gamma }_q\left(\zeta -\mu \right){\Gamma }_q\left(\epsilon -\mu \right)}{{\Gamma }_q\left(-\mu \right){\Gamma }_q\left(\zeta +\vartheta -\mu +\epsilon \right)}{z}^{\mu -\zeta }{q}^{-\vartheta \mu -\vartheta \left(\vartheta +1\right)/2},$$(1.16) ), arrive at (2.15) $$\begin{array}{l}{I}_3=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\sum _{m=0}^{\infty }\frac{{\left(a_1,a_2,\dots ,a_r;q\right)}_m{\rho }^m}{{\left(b_1,b_2,\dots ,b_s;q\right)}_m}\\ \hspace{1em}\hspace{1em}{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}\\ \hspace{1em}\hspace{1em}\times \frac{{\Gamma }_q\left(\zeta -j+m\right){\Gamma }_q\left(\epsilon -j+m\right)}{{\Gamma }_q\left(-j+m\right){\Gamma }_q\left(\vartheta +\zeta -j+m+\epsilon \right)}\\ \hspace{1em}\hspace{1em}{z}^{\left(j-m\right)-\zeta }{q}^{-\vartheta \left(j-m\right)-\vartheta \left(\vartheta +1\right)/2},\end{array}$$(2.15)

Rearranging the parameters, we arrive at $$\begin{array}{l}{I}_3=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\frac{{\Gamma }_q\left(\zeta -j\right){\Gamma }_q\left(\epsilon -j\right)}{{\Gamma }_q\left(-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon \right)}\\ \hspace{1em}\hspace{1em}\sum _{m=0}^{\infty }\frac{{\left(a_1;q\right)}_m{\left(a_2;q\right)}_m\dots {\left(a_r;q\right)}_m}{{\left(b_1;q\right)}_m{\left(b_2;q\right)}_m \dots {\left(b_s;q\right)}_m}\\ \hspace{1em}\hspace{1em}\times {\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}\frac{{\left(q^{\zeta -j};q\right)}_m{\left(q^{\epsilon -j};q\right)}_m}{{\left(q^{-j};q\right)}_m{\left(q^{\vartheta +\zeta -j+\epsilon };q\right)}_m}\\ \hspace{1em}\hspace{1em}{\left(\rho /z\right)}^m{z}^{j-\zeta }{q}^{-\vartheta \left(j-m\right)-\vartheta \left(\vartheta +1\right)/2}.\end{array}$$

Changing the summation index to run between 0 and $\infty ,$ and then utilizing the outcome (2.16) $${\left(a_1;q\right)}_m=\left(1-a_1\right){\left(a_{1}q;q\right)}_{m-1}\iff {\left(a_1;q\right)}_{m+1}=\left(1-a_1\right){\left(a_{1}q;q\right)}_m.$$(2.16)

Additionally, after some reductions in complexity, we obtain (2.17) $$\begin{array}{l}{I}_3=q^{-\vartheta \left(\vartheta +1\right)/2}{\left(-1\right)}^{1+s+r}\rho z^{-\zeta -1}\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \times \frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)\left(1-a_1\right)\left(1-a_2\right) \dots \left(1-a_r\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right)\left(1-b_1\right)\left(1-b_2\right) \dots \left(1-b_s\right) }\\ \frac{1}{\left(1-q\right)}{\left(z{q}^{-\vartheta }\right)}^j\\ \times \sum _{m=0}^{\infty }\frac{{\left(a_{1}q;q\right)}_m{\left(a_{2}q;q\right)}_m\dots {\left(a_{r}q;q\right)}_m}{{\left(b_{1}q;q\right)}_m{\left(b_{2}q;q\right)}_m \dots {\left(b_{s}q;q\right)}_m}{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}\\ \times \frac{{\left(q^{\zeta -j+1};q\right)}_m{\left(q^{\epsilon -j+1};q\right)}_m}{{\left(q^{1-j};q\right)}_m{\left(q^{\vartheta +\zeta -j+\epsilon +1};q\right)}_m}{\left(\rho /z\right)}^m{q}^{m\left(\vartheta +1+s-r\right)},\end{array}$$(2.17)

Interpreting the right-hand side of (2.17(2.17) $$\begin{array}{l}{I}_3=q^{-\vartheta \left(\vartheta +1\right)/2}{\left(-1\right)}^{1+s+r}\rho z^{-\zeta -1}\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \times \frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)\left(1-a_1\right)\left(1-a_2\right) \dots \left(1-a_r\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right)\left(1-b_1\right)\left(1-b_2\right) \dots \left(1-b_s\right) }\\ \frac{1}{\left(1-q\right)}{\left(z{q}^{-\vartheta }\right)}^j\\ \times \sum _{m=0}^{\infty }\frac{{\left(a_{1}q;q\right)}_m{\left(a_{2}q;q\right)}_m\dots {\left(a_{r}q;q\right)}_m}{{\left(b_{1}q;q\right)}_m{\left(b_{2}q;q\right)}_m \dots {\left(b_{s}q;q\right)}_m}{\left[{\left(-1\right)}^m{q}^{m(m-1)/2}\right]}^{1+s-r}\\ \times \frac{{\left(q^{\zeta -j+1};q\right)}_m{\left(q^{\epsilon -j+1};q\right)}_m}{{\left(q^{1-j};q\right)}_m{\left(q^{\vartheta +\zeta -j+\epsilon +1};q\right)}_m}{\left(\rho /z\right)}^m{q}^{m\left(\vartheta +1+s-r\right)},\end{array}$$(2.17) ), in view of Eq. (1.9)(1.9) $${}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,z\right]=\sum _{m=0}^{\infty }\frac{{\left(a_1,\dots ,a_r;q\right)}_m}{{\left(b_1,\dots ,b_s;q\right)}_n}\frac{z^n}{{(q;q)}_m}{\left\{{\left(-1\right)}^m{q}^{\left(m(m-1)/2\right)}\right\}}^{(1+s-r)},$$(1.9) , we obtain the desired result (2.12(2.12) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}=\\ q^{-\vartheta \left(\vartheta +1\right)/2}{\left(-1\right)}^{1+s-r}\rho z^{-\zeta -1}\times \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)\left(1-a_1\right)\left(1-a_2\right) \dots \left(1-a_r\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right)\left(1-b_1\right)\left(1-b_2\right) \dots \left(1-b_s\right) }\frac{{\left(z q^{-\vartheta }\right)}^j}{\left(1-q\right)}\\ {\times \hspace{0.17em}}_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_{1}q,\dots ,a_{r}q, q^{\zeta -j+1}, q^{\epsilon -j+1}\\ b_{1}q,\dots ,b_{s}q, q^{-j+1}, q^{\vartheta +\zeta -j+\epsilon +1}\end{array} ;q, \rho q^{1+\vartheta +s-r}/z\right].\end{array}$$(2.12) ). □

Remark.

If we take n = 0, $j=0,$ then $f_{0,N}(z;q)\to 1$ in the Theorems 2 and 3, we arrive at the known result given by Garg and Chanchlani (2011), p. 147, Eqs. (3.2(3.1) $$N=1, \omega \ge 1, S_{n,q}=\frac{{\Gamma }_q\left(\omega n+\mu +1\right){\left(-1\right)}^j{q}^{j\left(j-1\right)}}{{\left(q;q\right)}_n{\Gamma }_q\left(\omega j+\mu +1\right)},$$(3.1) , 3.3)(3.1) $$N=1, \omega \ge 1, S_{n,q}=\frac{{\Gamma }_q\left(\omega n+\mu +1\right){\left(-1\right)}^j{q}^{j\left(j-1\right)}}{{\left(q;q\right)}_n{\Gamma }_q\left(\omega j+\mu +1\right)},$$(3.1) ).

If we set ζ = 0 in the operators (1.12(1.12) $${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=\frac{q^{-\vartheta \left(\vartheta +1\right)/2-\zeta }}{{\Gamma }_q(\vartheta )}{\int }_z^{\infty }{(z/t;q)}_{\vartheta -1} t^{-\zeta -1} \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q^{\vartheta };q\right)}_n{\left(q;q\right)}_n}{\left(-1\right)}^n{q}^{\left(\epsilon -\zeta \right)n}{q}^{-\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right)}{\left(\frac{z}{qt}-1\right)}_q^{n}f\left(t{q}^{1-\vartheta }\right)d_{q}t,$$(1.12) ), and using the following identity (2.18) $${\mathcal{J}}_q^{\vartheta ,0,\epsilon }f(z)=q^{-\vartheta (\vartheta +1)/2}{K}_q^{\epsilon ,\vartheta }f(z),$$(2.18) where the generalized Weyl fractional q-integral operator, appearing in the right-hand side is due to (Al-Salam, 1966) is defined as (2.19) $$K_q^{\epsilon ,\vartheta }f\left(z\right)=\frac{q^{-\epsilon }{z}^{\epsilon }}{{\Gamma }_q\left(\vartheta \right)}{\int }_z^{\infty }{\left(t-z\right)}_{\vartheta -1}{t}^{-\epsilon -\vartheta }f\left(t{q}^{1-\vartheta }\right)d_q(t),$$(2.19) where $\mathfrak{R}(\vartheta )>0,$ $\epsilon$ is arbitrary.

Corollary 6.

Let $\vartheta ,\epsilon ,\rho ,\delta \in \mathbb{C},$ $0<\left|q\right|<1$; $\mathfrak{R}(\vartheta )>0,\mathfrak{R}(\epsilon +\delta -j)>0,$ and $\delta -j\ne 0$, then the generalized Weyl type fractional q-integral $K_q^{\epsilon ,\vartheta }$ of the product of GCq-P $f_{n,N}(.)$ and generalized q-hypergeometric series ${}_r{\phi }_s(.)$ is given by (2.20) $$\begin{array}{l}{K}_q^{\epsilon ,\vartheta }\left\{z^{-\delta }{f}_{n,N}{\left(z;q\right)}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}=\\ q^{\vartheta \delta } z^{-\delta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \hspace{1em}\hspace{1em}\times \frac{{\Gamma }_q\left(\epsilon +\delta -j\right)}{{\Gamma }_q\left(\vartheta +\delta -j+\epsilon \right)}{\left(z{q}^{-\vartheta }\right)}^j{}_{r+1}{\phi }_{s+1}\\ \hspace{1em}\hspace{1em}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\epsilon +\delta -j}\\ b_1,\dots ,b_s, q^{\vartheta +\delta -j+\epsilon }\end{array}; q, \rho q^{\vartheta }/z\right].\end{array}$$(2.20)

Remark.

If we take n = 0, $j=0,$ then $f_{0,N}(z;q)\to 1$ in the Corollary 6, we arrive at the known result given by Yadav et al. (2008).

Corollary 7.

Let $\vartheta ,\epsilon ,\rho ,\delta \in \mathbb{C},$ $0<\left|q\right|<1$; $\mathfrak{R}(\vartheta )>0,\mathfrak{R}(\epsilon -j+1)>0,$ then the generalized Weyl type fractional q-integral $K_q^{\epsilon ,\vartheta }$ of the product of GCq-P $f_{n,N}(.)$ and generalized q-hypergeometric series ${}_r{\phi }_s(.)$ is given by (2.21) $$\begin{array}{l}{K}_q^{\epsilon ,\vartheta }\left\{f_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}=\\ {\left(-1\right)}^{1+s-r}\rho z^{-1}\times \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \frac{{\Gamma }_q\left(\epsilon -j+1\right)\left(1-a_1\right)\left(1-a_2\right) \dots \left(1-a_r\right)}{{\Gamma }_q\left(\vartheta -j+\epsilon +1\right)\left(1-b_1\right)\left(1-b_2\right) \dots \left(1-b_s\right) }\frac{{\left(z q^{-\vartheta }\right)}^j}{\left(1-q\right)}\\ {\times }_{r+1}{\phi }_{s+1}\left[\begin{array}{c}{a}_{1}q,\dots ,a_{r}q, q^{\epsilon -j+1}\\ b_{1}q,\dots ,b_{s}q, q^{\vartheta -j+\epsilon +1}\end{array}; q, \rho q^{1+\vartheta +s-r}/z\right].\end{array}$$(2.21)

If we set $\zeta =-\vartheta$ in the operators (1.12(1.12) $${\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }f(z)=\frac{q^{-\vartheta \left(\vartheta +1\right)/2-\zeta }}{{\Gamma }_q(\vartheta )}{\int }_z^{\infty }{(z/t;q)}_{\vartheta -1} t^{-\zeta -1} \times \sum _{n=0}^{\infty }\frac{{\left(q^{\vartheta +\zeta };q\right)}_n{\left(q^{-\epsilon };q\right)}_n}{{\left(q^{\vartheta };q\right)}_n{\left(q;q\right)}_n}{\left(-1\right)}^n{q}^{\left(\epsilon -\zeta \right)n}{q}^{-\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right)}{\left(\frac{z}{qt}-1\right)}_q^{n}f\left(t{q}^{1-\vartheta }\right)d_{q}t,$$(1.12) ), using the following identity: (2.22) $${\mathcal{J}}_q^{\vartheta ,-\vartheta ,\epsilon }f(z)=K_q^{\vartheta }f(z),$$(2.22) where the Weyl fractional q-integral operator, appearing in the right-hand side is due to (Al-Salam, 1966) is defined as (2.23) $$K_q^{\vartheta }f\left(z\right)=\frac{q^{-\vartheta \left(\vartheta -1\right)/2}}{{\Gamma }_q\left(\vartheta \right)}{\int }_z^{\infty }{(t-z)}_{\vartheta -1}f\left(t q^{1-\vartheta }\right)d_q(t), \mathfrak{R}(\vartheta )>0.$$(2.23)

Corollary 8.

Let $\vartheta ,\rho ,\delta \in \mathbb{C},$ $0<\left|q\right|<1$; $\mathfrak{R}(\vartheta )>0,\mathfrak{R}(\delta -\vartheta -j)>0,$ and $\delta -j\ne 0$, then the Weyl type fractional q-integral $K_q^{\vartheta }$ of the product of GCq-P $f_{n,N}(.)$ and generalized q-hypergeometric series ${}_r{\phi }_s(.)$ is given by $$\begin{array}{l}{K}_q^{\vartheta }\left\{z^{-\delta }{f}_{n,N}{\left(z;q\right)}_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}\\ =q^{\vartheta \delta -\vartheta \left(\vartheta +1\right)/2} z^{\vartheta -\delta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\end{array}$$ (2.24) $$\begin{array}{l}\times \frac{{\Gamma }_q\left(\delta -\vartheta -j\right)}{{\Gamma }_q\left(\delta -j\right)}{\left(z{q}^{-\vartheta }\right)}^j{}_{r+1}{\phi }_{s+1}\\ \left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\delta -\vartheta -j}\\ b_1,\dots ,b_s, q^{\delta -j}\end{array}; q, \rho q^{\vartheta }/z\right].\end{array}$$(2.24)

Remark.

If we take n = 0, $j=0,$ then $f_{0,N}(z;q)\to 1$ in the Corollary 8, we arrive at the known result given by Yadav and Purohit (2006), P. 239, Eqs (24)(3.2) $$Z_n^{\mu }\left(z;q,\omega \right)=\frac{{\Gamma }_q\left(\omega n+\mu +1\right)}{{\left(q;q\right)}_n}\sum _{j=0}^n{\left[\begin{array}{c}n\hfill \\ j\hfill \end{array}\right]}_q\frac{{(-1)}^j{q}^{j\left(j-1\right)}}{{\Gamma }_q\left(\omega j+\mu +1\right)}{z}^{\omega j},$$(3.2) .

Corollary 9.

Let $\vartheta ,\rho ,\delta \in \mathbb{C},$ $0<\left|q\right|<1$; $\mathfrak{R}(\vartheta )>0,\mathfrak{R}(-\vartheta -j+1)>0,$, then the Weyl type fractional q-integral $K_q^{\vartheta }$ of the product of GCq-P $f_{n,N}(.)$ and generalized q-hypergeometric series ${}_r{\phi }_s(.)$ is given by (2.25) $$\begin{array}{l}{K}_q^{\vartheta }\left\{f_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}=\\ q^{-\vartheta \left(\vartheta +1\right)/2}{\left(-1\right)}^{1+s-r}\rho z^{\vartheta -1}\times \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \frac{{\Gamma }_q\left(-\vartheta -j+1\right)\left(1-a_1\right)\left(1-a_2\right) \dots \left(1-a_r\right)}{{\Gamma }_q\left(1-j\right)\left(1-b_1\right)\left(1-b_2\right) \dots \left(1-b_s\right) }\frac{{\left(z q^{-\vartheta }\right)}^j}{\left(1-q\right)}\\ {\times }_{r+1}{\phi }_{s+1}\left[\begin{array}{c}{a}_{1}q,\dots ,a_{r}q, q^{-\vartheta -j+1}\\ b_{1}q,\dots ,b_{s}q, q^{-j+1}\end{array}; q, \rho q^{1+\vartheta +s-r}/z\right].\end{array}$$(2.25)

3. Application of the main results

In the previous section, we deduced the Saigo fractional q-integral formulae associated with a GCq-P and generalized q-hypergeometric series. Here, we can find some applications given the Saigo fractional q-integral operators of all such classical orthogonal q-polynomials which are special cases of the q-polynomial system.

(i) By setting (3.1) $$N=1, \omega \ge 1, S_{n,q}=\frac{{\Gamma }_q\left(\omega n+\mu +1\right){\left(-1\right)}^j{q}^{j\left(j-1\right)}}{{\left(q;q\right)}_n{\Gamma }_q\left(\omega j+\mu +1\right)},$$(3.1) and replace z by $z^{\omega }$ in Eq. (1.10)(1.10) $$f_{n,N}\left(z;q\right)=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}{z}^j,\hspace{1em}(n=0,1,2,\dots ),$$(1.10) , we obtain q-Konhouser biorthogonal polynomial $Z_n^{\mu }(x;q,\omega )$ define by Yadav and Singh (2004) p. 185, Eq. (2.1)(2.1) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{\delta }{f}_{n,N}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho z\right]\right\}\\ =z^{\delta -\zeta } \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q} \frac{{\Gamma }_q\left(\delta +j+1\right){\Gamma }_q\left(\delta +j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(\delta +j-\zeta +1\right){\Gamma }_q\left(\delta +j+\vartheta +\epsilon +1\right)}{z}^j\\ \times { }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\delta +j+1}, q^{\delta +j-\zeta +\epsilon +1}\\ b_1,\dots ,b_s, q^{\delta +j-\zeta +1}, q^{\delta +j+\vartheta +\epsilon +1}\end{array}; q, \rho z\right].\end{array}$$(2.1) as (3.2) $$Z_n^{\mu }\left(z;q,\omega \right)=\frac{{\Gamma }_q\left(\omega n+\mu +1\right)}{{\left(q;q\right)}_n}\sum _{j=0}^n{\left[\begin{array}{c}n\hfill \\ j\hfill \end{array}\right]}_q\frac{{(-1)}^j{q}^{j\left(j-1\right)}}{{\Gamma }_q\left(\omega j+\mu +1\right)}{z}^{\omega j},$$(3.2) where $\mathfrak{R}(\mu )>-1$ and ω is a positive integer. In view of Eq. (3.2)(3.2) $$Z_n^{\mu }\left(z;q,\omega \right)=\frac{{\Gamma }_q\left(\omega n+\mu +1\right)}{{\left(q;q\right)}_n}\sum _{j=0}^n{\left[\begin{array}{c}n\hfill \\ j\hfill \end{array}\right]}_q\frac{{(-1)}^j{q}^{j\left(j-1\right)}}{{\Gamma }_q\left(\omega j+\mu +1\right)}{z}^{\omega j},$$(3.2) , we obtain the succeeding applications of Theorems 1, 4, and 5 in the form of corollaries as below:

Corollary 10.

Assume that the conditions of Theorem 1 are satisfied, then the following formula holds true: $$\begin{array}{l}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{\delta }{Z}_n^{\mu }\left(x;q,\omega \right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho z\right]\right\}\end{array}$$ (3.3) $$\begin{array}{l}=\frac{{\Gamma }_q\left(\omega n+\mu +1\right)}{{\left(q;q\right)}_n}\times \sum _{j=0}^{\infty }{\left[\begin{array}{c}n\hfill \\ j\hfill \end{array}\right]}_q{(-1)}^j{q}^{j\left(j-1\right)}\\ \frac{{\Gamma }_q\left(\delta +\omega j+1\right){\Gamma }_q\left(\delta +\omega j+1-\zeta +\epsilon \right) z^{\delta -\zeta +\omega j}}{{\Gamma }_q\left(\omega j+\mu +1\right) {\Gamma }_q\left(\delta +\omega j+1-\zeta \right){\Gamma }_q\left(\delta +\omega j+1+\epsilon +\vartheta \right)}\\ \times {}_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\delta +\omega j+1}, q^{\delta +\omega j-\zeta +\epsilon +1}\\ b_1,\dots ,b_s, q^{\delta +\omega j-\zeta +1}, q^{\delta +\omega j+\vartheta +\epsilon +1}\end{array}; q, \rho z\right],\end{array}$$(3.3) provided $\mathfrak{R}(\delta +\omega j+1)>0,$ $\mathfrak{R}(\delta +\omega j-\zeta +\epsilon +1)>0.$

Corollary 11.

Assume that the conditions of Theorem 4 are satisfied, the subsequent formula is valid: (3.4) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{-\delta }{Z}_n^{\mu }\left(x;q,\omega \right){ }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}\\ =q^{\vartheta \delta -\vartheta \left(\vartheta +1\right)/2}\frac{{\Gamma }_q\left(\omega n+\mu +1\right)}{{\left(q;q\right)}_n}\sum _{j=0}^{\infty }{\left[\begin{array}{c}n\hfill \\ j\hfill \end{array}\right]}_q{(-1)}^j{q}^{j\left(j-1\right)}\\ \times z^{\delta -\zeta }\frac{{\Gamma }_q\left(\zeta +\delta -\omega j\right){\Gamma }_q\left(\epsilon +\delta -\omega j\right)}{{\Gamma }_q\left(\omega j+\mu +1\right) {\Gamma }_q\left(\delta -\omega j\right){\Gamma }_q\left(\vartheta +\zeta +\delta -\omega j+\epsilon \right)}{\left(z{q}^{-\vartheta }\right)}^{\omega j}\\ {\times }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\zeta +\delta -\omega j}, q^{\epsilon +\delta -\omega j}\\ b_1,\dots ,b_s, q^{\delta -\omega j}, q^{\vartheta +\zeta +\delta -\omega j+\epsilon }\end{array}; q, \rho q^{\vartheta }/z\right],\end{array}$$(3.4) provided $\mathfrak{R}(\zeta +\delta -\omega j)>0, \mathfrak{R}(\epsilon +\delta -\omega j)>0.$

Corollary 12.

Assume that the conditions of Theorem 5 are met, the resulting formula is correct: (3.5) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{Z_n^{\mu }\left(x;q,\omega \right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}\\ =\frac{{\Gamma }_q\left(\omega n+\mu +1\right)}{{\left(q;q\right)}_n}{q}^{-\vartheta \left(\vartheta +1\right)/2}{\left(-1\right)}^{1+s-r}\rho z^{-\zeta -1}\sum _{j=0}^{\infty }{\left[\begin{array}{c}n\hfill \\ j\hfill \end{array}\right]}_q\frac{{(-1)}^j{q}^{j\left(j-1\right)}}{{\Gamma }_q\left(\omega j+\mu +1\right)}\\ \times \frac{{\Gamma }_q\left(\zeta -\omega j+1\right){\Gamma }_q\left(\epsilon -\omega j+1\right)\left(1-a_1\right)\left(1-a_2\right) \dots \left(1-a_r\right)}{{\Gamma }_q\left(1-\omega j\right){\Gamma }_q\left(\vartheta +\zeta -\omega j+\epsilon +1\right)\left(1-b_1\right)\left(1-b_2\right) \dots \left(1-b_s\right) }\\ \frac{{\left(z q^{-\vartheta }\right)}^{\omega j}}{\left(1-q\right)}{\times }_{r+2}{\phi }_{s+2}\\ \left[\begin{array}{c}{a}_{1}q,\dots ,a_{r}q, q^{\zeta -\omega j+1}, q^{\epsilon -\omega j+1}\\ b_{1}q,\dots ,b_{s}q, q^{-\omega j+1}, q^{\vartheta +\zeta -\omega j+\epsilon +1}\end{array}; q, \rho q^{1+\vartheta +s-r}/z\right],\end{array}$$(3.5) provided $\mathfrak{R}(\zeta -\omega j+1)>0, \mathfrak{R}(\epsilon -\omega j+1)>0.$

(ii) Moreover, by putting (3.6) $$N=\rho =1,S_{n,q}=\frac{{\Gamma }_q\left(n+\tau +1\right){\left(1-q\right)}^n{\Gamma }_q\left(\tau +\varsigma +n+1+j\right){\left(-1\right)}^j{q}^{\left(j\left(j+1\right)/2\right)-nj}}{{\left(q;q\right)}_n{\Gamma }_q\left(\tau +\varsigma +n+1\right){\Gamma }_q\left(\tau +1+j\right)}$$(3.6) in Eq. (1.10)(1.10) $$f_{n,N}\left(z;q\right)=\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}{z}^j,\hspace{1em}(n=0,1,2,\dots ),$$(1.10) , then we obtain q-Jacobi polynomials defined as (3.7) $${\mathrm{P}}_n^{\left(\tau ,\alpha \right)}\left(z;q\right)={\frac{{\left(q^{\tau +1};q\right)}_n}{{\left(q;q\right)}_n}}_2{\phi }_1\left(q^{-n};q^{\tau +\alpha +n+1};q^{\tau +1};q,qz\right).$$(3.7)

In the view of Eq. (3.7)(3.7) $${\mathrm{P}}_n^{\left(\tau ,\alpha \right)}\left(z;q\right)={\frac{{\left(q^{\tau +1};q\right)}_n}{{\left(q;q\right)}_n}}_2{\phi }_1\left(q^{-n};q^{\tau +\alpha +n+1};q^{\tau +1};q,qz\right).$$(3.7) , we obtain the succeeding applications of Theorems 1, 4, and 5 in the form of corollaries as below:

Corollary 13.

Assume that the conditions of Theorem 1 are fulfilled, the subsequent formula is true: (3.8) $$\begin{array}{l}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{\delta } p_n^{\left(\tau ,\alpha \right)}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho z\right]\right\}\\ =z^{\delta -\zeta }\frac{{\left(q^{\tau +1};q\right)}_n}{{\left(q;q\right)}_n}\times \sum _{l=0}^n\frac{{\left(q^{-n};q\right)}_l}{{\left(q;q\right)}_l}\frac{{\left(q^{\tau +\alpha +n+1};q\right)}_l}{{\left(q^{\tau +1};q\right)}_l}{(qz)}^l\\ \frac{{\Gamma }_q\left(\delta +l+1\right){\Gamma }_q\left(\delta +l-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(\delta +l-\zeta +1\right){\Gamma }_q\left(\delta +l+\vartheta +\epsilon +1\right)}\\ \times \hspace{0.17em}{}_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\delta +l+1}, q^{\delta +l-\zeta +\epsilon +1}\\ b_1,\dots ,b_s, q^{\delta +l-\zeta +1}, q^{\delta +l+\vartheta +\epsilon +1}\end{array}; q, \rho z\right],\end{array}$$(3.8) provided $\mathfrak{R}(\delta +l+1)>0,\mathfrak{R}(\delta +l-\zeta +\epsilon +1)>0.$

Corollary 14.

Assume that the conditions of Theorem 4 are observed, the preceding formula is correct: (3.9) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{-\delta }{p}_n^{\left(\tau ,\alpha \right)}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}\\ =q^{\vartheta \delta -\vartheta \left(\vartheta +1\right)/2}\frac{{\left(q^{\tau +1};q\right)}_n}{{\left(q;q\right)}_n}\times \sum _{l=0}^n\frac{{\left(q^{-n};q\right)}_l}{{\left(q;q\right)}_l}\frac{{\left(q^{\tau +\alpha +n+1};q\right)}_l}{{\left(q^{\tau +1};q\right)}_l}\\ z^{\delta -\zeta }\frac{{\Gamma }_q\left(\zeta +\delta -l\right){\Gamma }_q\left(\epsilon +\delta -l\right)}{ {\Gamma }_q\left(\delta -l\right){\Gamma }_q\left(\vartheta +\zeta +\delta -l+\epsilon \right)}{\left(zq{q}^{-\vartheta }\right)}^l\\ {\times }_{r+2}{\phi }_{s+2}\left[\begin{array}{c}{a}_1,\dots ,a_r, q^{\zeta +\delta -l}, q^{\epsilon +\delta -l}\\ b_1,\dots ,b_s, q^{\delta -l}, q^{\vartheta +\zeta +\delta -l+\epsilon }\end{array}; q, \rho q^{\vartheta }/z\right],\end{array}$$(3.9) provided $\mathfrak{R}(\zeta +\delta -l)>0, \mathfrak{R}(\epsilon +\delta -l)>0.$

Corollary 15.

Assume that the conditions of Theorem 5 are met, the resulting formula is correct: (3.10) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{p_n^{\left(\tau ,\alpha \right)}\left(z;q\right) { }_r{\phi }_s\left[\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array};q,\rho /z\right]\right\}\\ =q^{-\vartheta \left(\vartheta +1\right)/2}\frac{{\left(q^{\tau +1};q\right)}_n}{{\left(q;q\right)}_n}\sum _{l=0}^n\frac{{\left(q^{-n};q\right)}_l}{{\left(q;q\right)}_l}\frac{{\left(q^{\tau +\alpha +n+1};q\right)}_l}{{\left(q^{\tau +1};q\right)}_l}{\left(-1\right)}^{1+s-r}\rho z^{-\zeta -1} \\ \times \frac{{\Gamma }_q\left(\zeta -l+1\right){\Gamma }_q\left(\epsilon -l+1\right)\left(1-a_1\right)\left(1-a_2\right) \dots \left(1-a_r\right)}{{\Gamma }_q\left(1-l\right){\Gamma }_q\left(\vartheta +\zeta -l+\epsilon +1\right)\left(1-b_1\right)\left(1-b_2\right) \dots \left(1-b_s\right) }\\ \frac{{\left(zq q^{-\vartheta }\right)}^l}{\left(1-q\right)}{\times }_{r+2}{\phi }_{s+2}\\ \left[\begin{array}{c}{a}_{1}q,\dots ,a_{r}q, q^{\zeta -l+1}, q^{\epsilon -l+1}\\ b_{1}q,\dots ,b_{s}q, q^{-l+1}, q^{\vartheta +\zeta -l+\epsilon +1}\end{array}; q, \rho q^{1+\vartheta +s-r}/z\right],\end{array}$$(3.10) provided $\mathfrak{R}(\zeta -l+1)>0, \mathfrak{R}(\epsilon -l+1)>0.$

4. Special cases

Here, we take further interesting special cases of Theorems.

(i) For $0<\left|q\right|<1,$ q-analogues of exponential function defined in Gasper and Rahman (1990) is given by (4.1) $$e_q^z{=}_1{\varphi }_0\left(\begin{array}{c}0\\ -\end{array}; q, z\right)=\sum _{n=0}^{\infty }\frac{z^n}{{\left(q;q\right)}_n}=,\frac{1}{{\left(z;q\right)}_{\infty }},\left|z\right|<1.$$(4.1)

If we put $r=1, s=0, \delta =0, \rho =1, a_1=0$ in Theorems 1 and 5, the function ${}_r{\varphi }_s$ reduces to $e_q^z$ defined in (4.1(4.1) $$e_q^z{=}_1{\varphi }_0\left(\begin{array}{c}0\\ -\end{array}; q, z\right)=\sum _{n=0}^{\infty }\frac{z^n}{{\left(q;q\right)}_n}=,\frac{1}{{\left(z;q\right)}_{\infty }},\left|z\right|<1.$$(4.1) ), we obtain the following results (4.2) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{e_q^z{f}_{n,N}\left(z;q\right)\right\}=z^{-\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \times \frac{{\Gamma }_q\left(j+1\right){\Gamma }_q\left(j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(j-\zeta +1\right){\Gamma }_q\left(j+\vartheta +\epsilon +1\right)}{z}^j\\ { }_3{\phi }_2\left[\begin{array}{c}0, q^{j+1},q^{j-\zeta +\epsilon +1}\\ q^{j-\zeta +1},q^{j+\vartheta +\epsilon +1}\end{array};q,z\right],\end{array}$$(4.2) provided $\mathfrak{R}(j+1)>0, \mathfrak{R}(j-\zeta +\epsilon +1)>0,\left|z\right|<1,$ and (4.3) $$\begin{array}{c}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right)e_q^{1/z}\right\}=q^{-\vartheta \left(\vartheta -1\right)/2}{z}^{-\zeta -1}\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \times \frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right)}\frac{{\left(z{q}^{-\vartheta }\right)}^j}{\left(1-q\right)} \\ { }_3{\phi }_2\left[\begin{array}{c}0, q^{\zeta -j+1}, q^{\epsilon -j+1}\\ q^{-j+1}, q^{\vartheta +\zeta -j+\epsilon +1}\end{array}; q, q^{\vartheta }/z\right].\end{array}$$(4.3) provided $\mathfrak{R}(\zeta -j+1)>0, \mathfrak{R}(\epsilon -j+1)>0$ and $\left|z\right|>\left|q^{\vartheta }\right|.$

(ii) For $0<\left|q\right|<1,$ q-analogues of exponential functions (Gasper and Rahman, 1990), is given by (4.4) $$E_q^z{=}_0{\varphi }_0\left(\begin{array}{c}-\\ -\end{array} ;q, -z\right)=\sum _{n=0}^{\infty }\frac{q^{n\left(n-1\right)/2}{z}^n}{{\left(q;q\right)}_n}={\left(-z;q\right)}_{\infty }.$$(4.4)

If we take $r=s=0, \delta =0, \rho =-1$ in Theorem 1 and 5, the function ${}_r{\varphi }_s$ reduces to $E_q^z$ defined by (4.4(4.4) $$E_q^z{=}_0{\varphi }_0\left(\begin{array}{c}-\\ -\end{array} ;q, -z\right)=\sum _{n=0}^{\infty }\frac{q^{n\left(n-1\right)/2}{z}^n}{{\left(q;q\right)}_n}={\left(-z;q\right)}_{\infty }.$$(4.4) ), we obtain the following results (4.5) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{E_q^z{f}_{n,N}\left(z;q\right)\right\}=z^{-\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q} z^j\\ \times \frac{{\Gamma }_q\left(j+1\right){\Gamma }_q\left(j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(j-\zeta +1\right){\Gamma }_q\left(j+\vartheta +\epsilon +1\right)} \\ { }_2{\phi }_2\left[\begin{array}{c}{q}^{j+1},q^{j-\zeta +\epsilon +1}\\ q^{j-\zeta +1},q^{j+\vartheta +\epsilon +1}\end{array};q,-z\right],\end{array}$$(4.5) provided $\mathfrak{R}(j+1)>0, \mathfrak{R}(j-\zeta +\epsilon +1)>0,$and (4.6) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{E_q^{-1/z}{f}_{n,N}\left(z;q\right)\right\}=q^{-\vartheta \left(\vartheta -1\right)/2}{z}^{-\zeta -1}\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q\\ S_{n,q}\frac{{\left(z{q}^{-\vartheta }\right)}^j}{\left(1-q\right)}\times {\frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right) }}_{\mathrm{ }\mathrm{ }}\\ { }_2{\phi }_2\left[\begin{array}{c}{q}^{\zeta -j+1},q^{\epsilon -j+1}\\ q^{-j+1},q^{\vartheta +\zeta -j+\epsilon +1}\end{array} ;q ,-q^{1+\vartheta }/z\right],\end{array}$$(4.6) provided $\mathfrak{R}(\zeta -j+1)>0, \mathfrak{R}(\epsilon -j+1)>0.$

(iii) The q-binomial theorem (Gasper and Rahman, 1990) is given by (4.7) $${}_1{\phi }_0\left[\begin{array}{c}\vartheta \\ -\end{array} ;q, z\right]=\frac{{\left(\vartheta z;q\right)}_{\infty }}{{\left(z;q\right)}_{\infty }} , \left|z\right|<1, 0<\left|q\right|<1.$$(4.7)

From (1.3(1.3) $${\left(z-\gamma \right)}_q^{\gamma }=z^{\vartheta }{\left(\gamma /z;q\right)}_{\vartheta }=z^{\vartheta }\prod _{j=0}^{\infty }\left[\frac{1-\left(\gamma /z\right)q^j}{1-\left(\gamma /z\right)q^{j+\vartheta }}\right]=z^{\vartheta }\frac{{\left(\gamma /z;q\right)}_{\infty }}{{\left(q^{\vartheta }\gamma /z;q\right)}_{\infty }},\hspace{1em}0<\left|q\right|<1, \left(z\ne 0\right).$$(1.3) ) and (4.7(4.7) $${}_1{\phi }_0\left[\begin{array}{c}\vartheta \\ -\end{array} ;q, z\right]=\frac{{\left(\vartheta z;q\right)}_{\infty }}{{\left(z;q\right)}_{\infty }} , \left|z\right|<1, 0<\left|q\right|<1.$$(4.7) ), it follows that (4.8) $${\left(z-a\right)}_q^{\vartheta }=z^{\vartheta }{}_1{\phi }_0\left(\begin{array}{c}{q}^{-\vartheta }\\ -\end{array}; q, (a/z)q^{\vartheta }\right).$$(4.8)

If we take $r=1, s=0, a_1=q^{-v},v\in \mathbb{C}$ in Theorems 1 and 4 using (4.8(4.8) $${\left(z-a\right)}_q^{\vartheta }=z^{\vartheta }{}_1{\phi }_0\left(\begin{array}{c}{q}^{-\vartheta }\\ -\end{array}; q, (a/z)q^{\vartheta }\right).$$(4.8) ), we get (4.9) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{\delta }{f}_{n,N}\left(z;q\right){\left(1-\rho z{q}^{-v}\right)}_q^v\right\}=z^{\delta -\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\\ \times \hspace{0.17em}\frac{{\Gamma }_q\left(\delta +j+1\right){\Gamma }_q\left(\delta +j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(\delta +j-\zeta +1\right){\Gamma }_q\left(\delta +j+\vartheta +\epsilon +1\right)}{z}^j\\ { }_3{\phi }_2\left[\begin{array}{c}{q}^{-v},q^{\delta +j+1},q^{\delta +j-\zeta +\epsilon +1}\\ q^{\delta +j-\zeta +1},q^{\delta +j+\vartheta +\epsilon +1}\end{array};q,\rho z\right],\end{array}$$(4.9) provided $\mathfrak{R}(\delta +j+1)>0, \mathfrak{R}(\delta +j-\zeta +\epsilon +1)>0$ and $\left|\rho z\right|<1,$and (4.10) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{z^{-\delta }{f}_{n,N}\left(z;q\right){\left(1-\rho q^{-v}/z\right)}_q^v\right\}=\\ q^{\vartheta \delta -\vartheta \left(\vartheta +1\right)/2}{z}^{-\delta -\zeta } \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q\\ \times S_{n,q}{\left(z{q}^{-\vartheta }\right)}^j\frac{{\Gamma }_q\left(\zeta +\delta -j\right){\Gamma }_q\left(\epsilon +\delta -j\right)}{{\Gamma }_q\left(\delta -j\right){\Gamma }_q\left(\vartheta +\zeta +\delta -j+\epsilon \right)}\\ { }_3{\phi }_2\left[\begin{array}{c}{q}^{-v},q^{\zeta +\delta -j},q^{\epsilon +\delta -j}\\ q^{\delta -j},q^{\vartheta +\zeta +\delta -j+\epsilon }\end{array};q,\rho q^{\vartheta }/z\right],\end{array}$$(4.10) provided $\mathfrak{R}(\zeta +\delta -j)>0$ and $\mathfrak{R}(\epsilon +\delta -j)>0,$ $\left|z\right|>\left|\rho q^{\vartheta }\right|.$

(iv) The little q-Jacobi polynomial is defined in Koekoek et al. (2010) as (4.11) $$p_n\left(z;a,b;q\right){=}_2{\phi }_1\left(\begin{array}{c}{q}^{-n},ab{q}^{n+1}\\ aq\end{array} ;q, qz\right).$$(4.11)

If we take $r=2, s=1, \delta =0, \rho =q, a_1=q^{-n}, a_2=ab{q}^{n+1} \mathit{and} b_1=aq$ in Theorems 1, 5, the function ${}_r{\varphi }_s$ reduces to Little q-Jacobi polynomials defined by (4.11(4.11) $$p_n\left(z;a,b;q\right){=}_2{\phi }_1\left(\begin{array}{c}{q}^{-n},ab{q}^{n+1}\\ aq\end{array} ;q, qz\right).$$(4.11) ), we obtain the following results (4.12) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right)p_n\left(z;a,b;q\right)\right\}=z^{-\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}{z}^j\\ \times \frac{{\Gamma }_q\left(j+1\right){\Gamma }_q\left(j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(j-\zeta +1\right){\Gamma }_q\left(j+\vartheta +\epsilon +1\right)} \\ { }_4{\phi }_3\left[\begin{array}{c}{q}^{-n}, ab{q}^{n+1}, q^{j+1}, q^{j-\zeta +\epsilon +1}\\ aq, q^{j-\zeta +1}, q^{j+\vartheta +\epsilon +1}\end{array};q, qz\right],\end{array}$$(4.12) provided $\mathfrak{R}(j+1)>0, \mathfrak{R}(j-\zeta +\epsilon +1)>0,\left|qz\right|<1,$and $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right)p_n\left(1/z;a,b;q\right)\right\}=q^{1-\vartheta \left(\vartheta -1\right)/2}{z}^{-\zeta -1}\end{array}$$ (4.13) $$\begin{array}{l}\times \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right)}\\ \frac{\left(1-q^{-n}\right)\left(1-ab{q}^{n+1}\right)}{\left(1-aq\right)}\frac{{\left(z{q}^{-\vartheta }\right)}^j}{\left(1-q\right)}\\ {\times }_4{\phi }_3\left[\begin{array}{c}{q}^{-n+1},ab{q}^{n+2},q^{\zeta -j+1},q^{\epsilon -j+1}\\ a{q}^2,q^{-j+1},q^{\vartheta +\zeta -j+\epsilon +1}\end{array};q,q^{1+\vartheta }/z\right],\end{array}$$(4.13) provided $\mathfrak{R}(\zeta -j+1)>0, \mathfrak{R}(\epsilon -j+1)>0 \mathit{and} \left|z\right|<\left|q^{1+\vartheta }\right|.$

(v) The little q-Legendre polynomials in Koekoek et al. (2010). (4.14) $$p_n\left(z;q\right){=}_2{\phi }_1\left(\begin{array}{c}{q}^{-n}, q^{n+1}\\ q\end{array}; q, qz\right).$$(4.14)

If we take $r=2, s=1, \delta =0, a_1=q^{-n}, a_2=q^{n+1} \mathit{and} b_1=q, p=q$ in Theorems 1 and 5 and use (4.14(4.14) $$p_n\left(z;q\right){=}_2{\phi }_1\left(\begin{array}{c}{q}^{-n}, q^{n+1}\\ q\end{array}; q, qz\right).$$(4.14) ) to get the following results (4.15) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right)p_n\left(z;q\right)\right\}=z^{-\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}{z}^j\\ \times \frac{{\Gamma }_q\left(j+1\right){\Gamma }_q\left(j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(j-\zeta +1\right){\Gamma }_q\left(j+\vartheta +\epsilon +1\right)} \\ { }_4{\phi }_3\left[\begin{array}{c}{q}^{-n},a{q}^{n+1},q^{j+1},q^{j-\zeta +\epsilon +1}\\ ,q^{j-\zeta +1},q^{j+\vartheta +\epsilon +1}\end{array};q,qz\right],\end{array}$$(4.15) provided $\mathfrak{R}(j+1)>0, \mathfrak{R}(j-\zeta +\epsilon +1)>0, \left|qz\right|<1,$and (4.16) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(x;q\right)p_n\left(1/z;q\right)\right\}=q^{1-\vartheta \left(\vartheta -1\right)/2}{z}^{-\zeta -1}\\ \times \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right)}\\ \frac{\left(1-q^{-n}\right)\left(1-q^{n+1}\right)}{\left(1-q\right)}\frac{{\left(z{q}^{-\vartheta }\right)}^j}{\left(1-q\right)}\\ {\times }_4{\phi }_3\left[\begin{array}{c}{q}^{-n+1},q^{n+2},q^{\zeta -j+1},q^{\epsilon -j+1}\\ q^2,q^{-j+1},q^{\vartheta +\zeta -j+\epsilon +1}\end{array};q,q^{1+\vartheta }/z\right],\end{array}$$(4.16) provided $\mathfrak{R}(\zeta -j+1)>0, \mathfrak{R}(\epsilon -j+1)>0 \mathit{and} \left|z\right|<\left|q^{1+\vartheta }\right|.$

(vi) From Koekoek et al. (2010), the little q-Laguerre polynomials (4.17) $$p_n\left(z;a,q\right){=}_2{\phi }_1\left(\begin{array}{c}{q}^{-n},0\\ aq\end{array}; q, qz\right).$$(4.17)

If we take $r=2, s=1, \delta =0, \rho =q, a_1=q^{-n}, a_2=0 \mathit{and} b_1=aq$ in Theorems 1 and 5 and use (4.17(4.17) $$p_n\left(z;a,q\right){=}_2{\phi }_1\left(\begin{array}{c}{q}^{-n},0\\ aq\end{array}; q, qz\right).$$(4.17) ) to get the following results (4.18) $$\begin{array}{c}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right)p_n\left(z;a;q\right)\right\}=z^{-\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q} z^j\\ \times \frac{{\Gamma }_q\left(j+1\right){\Gamma }_q\left(j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(j-\zeta +1\right){\Gamma }_q\left(j+\vartheta +\epsilon +1\right)}\\ { }_4{\phi }_3\left[\begin{array}{c}{q}^{-n},0,q^{j+1},q^{j-\zeta +\epsilon +1}\\ aq,q^{j-\zeta +1},q^{j+\vartheta +\epsilon +1}\end{array};q,qz\right],\end{array}$$(4.18) provided $\mathfrak{R}(j+1)>0, \mathfrak{R}(j-\zeta +\epsilon +1)>0, \left|qz\right|<1,$and (4.19) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right)p_n\left(1/z;a;q\right)\right\}=q^{1-\vartheta \left(\vartheta +1\right)/2}{z}^{-\zeta -1}\\ \times \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right)}\\ \frac{\left(1-q^{-n}\right)}{\left(1-aq\right)}\frac{{\left(z{q}^{-\vartheta }\right)}^j}{\left(1-q\right)}\\ {\times }_4{\phi }_3\left[\begin{array}{c}{q}^{-n+1},0,q^{\zeta -j+1},q^{\epsilon -j+1}\\ a{q}^2,q^{-j+1},q^{\vartheta +\zeta -j+\epsilon +1}\end{array};q,q^{1+\vartheta }/z\right],\end{array}$$(4.19) provided $\mathfrak{R}(\zeta -j+1)>0, \mathfrak{R}(\epsilon -j+1)>0 \mathit{and} \left|z\right|<\left|q^{1+\vartheta }\right|.$

(vii) The q-Laguerre polynomial is defined in Koekoek et al. (2010) as (4.20) $$L_n^{\left(a\right)}\left(z;q\right)={\frac{{\left(q^{\vartheta +1};q\right)}_n}{{\left(q;q\right)}_n}}_1{\phi }_1\left(\begin{array}{c}{q}^{-n}\\ q^{\vartheta +1}\end{array};q,-q^{n+\vartheta +1}z\right).$$(4.20)

If we take $r=1, s=1, \delta =0, a_1=q^{-n}, b_1=q^{\vartheta +1} \mathit{and} \rho =-q^{n+a+1}$ in Theorems 1 and 5 and use (4.20(4.20) $$L_n^{\left(a\right)}\left(z;q\right)={\frac{{\left(q^{\vartheta +1};q\right)}_n}{{\left(q;q\right)}_n}}_1{\phi }_1\left(\begin{array}{c}{q}^{-n}\\ q^{\vartheta +1}\end{array};q,-q^{n+\vartheta +1}z\right).$$(4.20) ) to get the following results (4.21) $$\begin{array}{l}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right)L_n^{\left(a\right)}\left(z;q\right)\right\}=z^{-\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q\\ S_{n,q}\frac{{\Gamma }_q\left(j+1\right){\Gamma }_q\left(j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(j-\zeta +1\right){\Gamma }_q\left(j+\vartheta +\epsilon +1\right)}{z}^j\\ \times {\frac{{\left(q;q\right)}_n}{{\left(q^{a+1};q\right)}_n}}_3{\phi }_3\left[\begin{array}{c}{q}^{-n},q^{j+1},q^{j-\zeta +\epsilon +1}\\ q^{a+1},q^{j-\zeta +1},q^{j+\vartheta +\epsilon +1}\end{array};q,-q^{n+\vartheta +1}z\right],\end{array}$$(4.21) provided $\mathfrak{R}(j+1)>0, \mathfrak{R}(j-\zeta +\epsilon +1)>0,$ and (4.22) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right)L_n^{\left(a\right)}\left(1/z;q\right)\right\}=q^{-\vartheta \left(\vartheta +1\right)/2+n+a+1}{z}^{-\zeta -1}\\ \times \frac{{\left(q;q\right)}_n}{{\left(q^{a+1};q\right)}_n}\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right) }\\ \frac{\left(1-q^{-n}\right)}{\left(1-q\right)}\frac{{\left(z{q}^{-\vartheta }\right)}^j}{\left(1-q\right)}\\ {\times }_3{\phi }_3\left[\begin{array}{c}{q}^{-n+1},q^{\zeta -j+1},q^{\epsilon -j+1}\\ q^{\vartheta +2},q^{-j+1},q^{\vartheta +\zeta -j+\epsilon +1}\end{array};q,-q^{n+2\vartheta +2}/z\right],\end{array}$$(4.22) provided $\mathfrak{R}(\zeta -j+1)>0, \mathfrak{R}(\epsilon -j+1)>0.$

(viii) The Stieltjes-Wigert polynomial is defined in Koekoek et al. (2010) as (4.23) $$S_n\left(z;q\right)={\frac{1}{{\left(q;q\right)}_n}}_1{\phi }_1\left(\begin{array}{c}{q}^{-n}\\ 0\end{array};q,-q^{n+1}z\right).$$(4.23)

If we take $r=1, s=1, \delta =0, a_1=q^{-n}, b_1=0 \mathit{and} \rho =-q^{n+1}$ in Theorems 1 and 5 and use (4.23(4.23) $$S_n\left(z;q\right)={\frac{1}{{\left(q;q\right)}_n}}_1{\phi }_1\left(\begin{array}{c}{q}^{-n}\\ 0\end{array};q,-q^{n+1}z\right).$$(4.23) ) to get the following results $$\begin{array}{l}{\mathcal{I}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right)S_n\left(z;q\right)\right\}=z^{-\zeta }\sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q\end{array}$$ (4.24) $$\begin{array}{l}{S}_{n,q}\frac{{\Gamma }_q\left(j+1\right){\Gamma }_q\left(j-\zeta +\epsilon +1\right)}{{\Gamma }_q\left(j-\zeta +1\right){\Gamma }_q\left(j+\vartheta +\epsilon +1\right)}{z}^j\\ \times { }_3{\phi }_3\left[\begin{array}{c}{q}^{-n},q^{0+j+1},q^{0+j-\zeta +\epsilon +1}\\ 0,q^{0+j-\zeta +1},q^{0+j+\vartheta +\epsilon +1}\end{array};q,-q^{n+\vartheta +1}z\right],\end{array}$$(4.24) provided $\mathfrak{R}(j+1)>0, \mathfrak{R}(j-\zeta +\epsilon +1)>0,$ and (4.25) $$\begin{array}{l}{\mathcal{J}}_q^{\vartheta ,\zeta ,\epsilon }\left\{f_{n,N}\left(z;q\right)S_n\left(1/z;q\right)\right\}=q^{-\vartheta \left(\vartheta +1\right)/2+n+1}{z}^{-\zeta -1}\\ \times \sum _{j=0}^{\left[n/N\right]}{\left[\begin{array}{c}n\hfill \\ Nj\hfill \end{array}\right]}_q{S}_{n,q}\frac{{\Gamma }_q\left(\zeta -j+1\right){\Gamma }_q\left(\epsilon -j+1\right)}{{\Gamma }_q\left(1-j\right){\Gamma }_q\left(\vartheta +\zeta -j+\epsilon +1\right) }\\ \frac{\left(1-q^{-n}\right)}{\left(1-q\right)}\frac{{\left(z{q}^{-\vartheta }\right)}^j}{\left(1-q\right)}\\ {\times }_3{\phi }_3\left[\begin{array}{c}{q}^{-n+1},q^{\zeta -j+1},q^{\epsilon -j+1}\\ 0,q^{-j+1},q^{\vartheta +\zeta -j+\epsilon +1}\end{array};q,-q^{n+\vartheta +2}/z\right],\end{array}$$(4.25) provided $\mathfrak{R}(\zeta -j+1)>0, \mathfrak{R}(\epsilon -j+1)>0$.

Although several similar results can be obtained from our theorem, we omit further details.

5. Discussion and conclusions

With relation to the Saigo fractional q-integral operators provided by Garg and Chanchalani, theorems and corollaries are developed in this paper. With the use of image power function formulae, all theorems have been developed along the well-organized path of the product of a GCq-P and q-hypergeometric series. After some appropriate parametric replacement, the results presented in this paper are easily convertible. Additionally, as special cases, the findings of this study also apply to generalized Weyl, Kober, and Riemann-Liouville fractional q-calculus operators. Additionally, our results can be reduced to Jacobi, Legendre, Hermite, Bessel, Gould-Hopper polynomial and their various special cases by appropriately specializing a variety of parameters of the GCq-P. As a result, the conclusions drawn in this article would immediately lead to a vast array of conclusions involving a wide variety of special functions present in problems in science, engineering, mathematical physics, etc.

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There is no conflict of interest regarding the publication of this article.

Data availability statement

No data were used to support this study.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.