Certain bilinear generating relations for q-analogue of I-function

ABSTRACT

The author presents a comprehensive analysis of some bilinear generating relations pertaining the q-polynomial class and the q-analogue of the I-function. The related conclusions for q-polynomials of q-Laguerre, q-Jacobi polynomials, and so on are very simple to derive as particular or limiting examples of the generating functions provided here. It is also presented various generalized integrals related to q-polynomials and I_q-function. Furthermore, we obtain several results as extensions of these evaluations.

KEYWORDS:

• q-generating function
• general class of q-polynomial
• q-analogue of I-function
• q-Jacobi polynomial

Mathematics Subject Classification:

• 33D45
• 33D60
• 33D90

1. Introduction and preliminaries

Extensions of the ordinary calculus are called the q-calculus. Recent applications of the theory of q-calculus operators include q-transform analysis, optimum control problems, solving the q-difference and q-integral equations, and ordinary fractional calculus. Analyzing bilinear generating relations within the context of q-polynomial classes involves examining relationships between different generating functions that arise in combinatorial problems.

Generating functions play a crucial role in enumerative combinatorics. They encode sequences of numbers (often counting the number of objects of a certain type) into a formal power series. Bilinear generating relations often connect different generating functions, providing insights into their combinatorial interpretations and properties. Within the field of generating q-theory, several researchers are investigating applications of these relations in solving combinatorial problems, deriving identities, and understanding the behavior of special functions in various contexts. Among them are Al-Omari et al. (2021), Purohit et al. (2012), Srivastava et al. (2010), Meena et al. (2022), Ahuja & Çetinkaya (2021), Vyas et al. (2021) and Araci (2021), presented a unique method of generating function and examined its characteristic. Proving bilinear generating relations for the q-analogue of the I-function typically involves manipulating q-series expansions, exploiting properties of q-Pochhammer symbols, contour integration techniques, and using combinatorial arguments.

The q-Pochhammer symbol, denoted as $(\mathit{a};\mathit{q}{)}_n$, is another essential q-polynomial related to q-factorials. Bilinear generating relations involving q-Pochhammer symbols often lead to interesting identities and transformations.

The q-shifted factorial (see Gasper & Rahman, 2004) for $\mathit{a},\mathit{q}\in \mathbb{C}$ is defined as

(1.1) ${(\mathrm{\wp };\mathit{q})}_{\mathit{ı}}=\{\begin{array}{l}1;\mathit{ı}=0\\ (1-\mathrm{\wp })(1-\mathrm{\wp }\mathit{q})\dots (1-\mathrm{\wp }{q}^{\mathit{n}-1});\mathit{ı}\in \mathbb{N},\end{array}$(1.1)

as well as its natural continuation

(1.2) $(\mathrm{\wp };\mathit{q}{)}_{\mathit{\vartheta }}=\frac{(\mathrm{\wp };\mathit{q}{)}_{\mathrm{\infty }}}{(\mathrm{\wp }{q}^{\mathit{\vartheta }};\mathit{q}{)}_{\mathrm{\infty }}},\mathit{\vartheta }\in \mathbb{C},\left|\mathit{q}\right|<1.$(1.2)

Since $\mathit{ı}=\mathrm{\infty }$ is a convergent infinite product, the expression (1.1(1.1) ${(\mathrm{\wp };\mathit{q})}_{\mathit{ı}}=\{\begin{array}{l}1;\mathit{ı}=0\\ (1-\mathrm{\wp })(1-\mathrm{\wp }\mathit{q})\dots (1-\mathrm{\wp }{q}^{\mathit{n}-1});\mathit{ı}\in \mathbb{N},\end{array}$(1.1) ) is still valid.

(1.3) $(\mathrm{\wp };\mathit{q}{)}_{\mathrm{\infty }}=\prod _{\mathit{ȷ}=0}^{\mathrm{\infty }}(1-\mathrm{\wp }{q}^{\mathit{ȷ}}).$(1.3)

According to Ernst (2003), the power function $(\mathit{g}\pm \mathit{h}{)}^{\mathit{ı}}$ has the following q-analogue:

(1.4) $\begin{array}{ll}(\mathit{g}\pm \mathit{h}{)}^{ı}& \equiv (\mathit{g}\pm \mathit{h}{)}_{\mathit{ı}}\equiv g^{\mathit{ı}}(\mp \mathit{h}/\mathit{g};\mathit{q}{)}_{\mathit{ı}}\\ & =g^{\mathit{ı}}\sum _{\mathit{ȷ}=0}^{\mathit{ı}}{\left[\begin{array}{l}\mathit{ı}\\ \mathit{ȷ}\end{array}\right]}_q{q}^{\frac{\mathit{ȷ}(\mathit{ȷ}-1)}{2}}(\pm \mathit{h}/\mathit{g}{)}^{\mathit{ȷ}},\end{array}$(1.4)

where the description of the q-binomial coefficient is

(1.5) ${\left[\begin{array}{l}\mathit{ı}\\ \mathit{ȷ}\end{array}\right]}_q=\frac{({\mathit{q}}^{-\mathit{ı}};\mathit{q}{)}_{\mathit{ȷ}}}{(\mathit{q};\mathit{q}{)}_{\mathit{ȷ}}}(-q^{ı}{)}^{\mathit{ȷ}}{q}^{-\mathit{ȷ}(\mathit{ȷ}-1)/2},(\mathit{ȷ}\in \mathbb{N},\mathit{ı}\in \mathbb{R})$(1.5)

It satisfies

(1.6) ${\left[\begin{array}{l}\mathit{ı}\\ \mathit{ȷ}\end{array}\right]}_q=\frac{(\mathit{q};\mathit{q}{)}_{\mathit{ı}}}{(\mathit{q};\mathit{q}{)}_{\mathit{ı}-\mathit{ȷ}}(\mathit{q};\mathit{q}{)}_{\mathit{ȷ}}}.$(1.6)

From Ernst (2003) (p. 502, equation (3.18)), given a restricted sequence of real or complex numbers $Z_{\mathit{ı}}$ and letting $\mathit{f}(\mathit{g})=\sum _{\mathit{ı}=0}^{\mathrm{\infty }}{Z}_{\mathit{ı}}{g}^{\mathit{ı}}$ be a power series in g, we may calculate the following:

(1.7) $\mathit{f}[(\mathit{g}\pm \mathit{h})]=\sum _{\mathit{ı}=0}^{\mathrm{\infty }}{Z}_{\mathit{ı}}{g}^{\mathit{ı}}(\mp \mathit{h}/\mathit{g};\mathit{q}{)}_{\mathit{ı}}.$(1.7)

According to Gasper & Rahman (2004), the q-gamma function is described as

(1.8) ${\mathrm{\Gamma }}_q(\mathit{\varpi })=\frac{(q;q{)}_{\mathrm{\infty }}}{(q^{\mathit{\varpi }};\mathit{q}{)}_{\mathrm{\infty }}}(1-\mathit{q}{)}^{1-\mathit{\varpi }},(\mathit{\varpi }\notin 0,-1,-2,\dots ,\mathit{\varpi }\in \mathbb{C}).$(1.8)

and it satisfies

(1.9) ${\mathrm{\Gamma }}_q(\mathit{\varpi }+1)=\frac{1-{\mathit{q}}^{\mathit{\varpi }}}{1-\mathit{q}}{\mathrm{\Gamma }}_q(\mathit{\varpi }).$(1.9)

According to Srivastava & Agarwal (1989), family of q-polynomials ($\mathit{G}{C}_q\mathit{P}$) $f_{\mathit{n},\mathit{\gamma }}(\mathit{x};\mathit{q})$ described in terms of a bounded complex sequence ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ as

(1.10) $f_{\mathit{n},\mathit{\gamma }}(\mathit{x};\mathit{q})=\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{x}^{\delta },(\mathit{n}\in \mathbb{N}),$(1.10)

where γ is an integer with a positive sign.

The family of q-polynomial $f_{\mathit{n},\mathit{\gamma }}(\mathit{x};\mathit{q})$ provides numerous well-known q-polynomials as their particular cases when the sequence ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is chosen appropriately. These include Wall polynomials, q-Konhauser polynomials, q-Jacobi polynomials, q-Laguerre polynomials, and a number of others.

Now, as indicated in Saxena & Kumar (1995), q-analogue of I-function (called I_q-function) in terms of Mellin–Barnes type basic contour integral is defined as

(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11)

where

(1.12) $\mathit{\psi }(\mathit{s};\mathit{q})=\frac{\{\prod _{\mathit{j}=1}^{\mathrm{\Im }}\mathit{G}(q^{b_j-{\xi }_j\mathit{s}})\}\{\prod _{\mathit{j}=1}^{\mathrm{\wp }}\mathit{G}(q^{1-a_j+{\vartheta }_j\mathit{s}})\mathit{\vartheta }\}\mathit{\pi }}{\sum _{\mathit{i}=1}^{\mathit{r}}\{\prod _{\mathit{j}=\mathrm{\Im }+1}^{{\mathit{Q}}_{\mathit{i}}}\mathit{G}(q^{1-{\mathit{b}}_{\mathit{ji}}+{\xi }_{\mathit{ji}}\mathit{s}})\}\{\prod _{\mathit{j}=\mathrm{\wp }+1}^{{\mathit{P}}_{\mathit{i}}}\mathit{G}(q^{{\mathit{a}}_{\mathit{ji}}-{\vartheta }_{\mathit{ji}}\mathit{s}})\}\mathit{G}(q^{1-\mathit{s}})\mathit{sin\pi s}},$(1.12)

and

(1.13) $\mathit{G}(q^a)={\{\prod _{\mathit{\epsilon }=0}^{\mathrm{\infty }}(1-q^{\mathit{a}+\mathit{\epsilon }})\}}^{-1}=\frac{1}{(q^a;\mathit{q}{)}_{\mathrm{\infty }}}.$(1.13)

Also, $0\le \mathrm{\Im }\le Q_{\mathit{i}},0\le \mathrm{\wp }\le P_{\mathit{i}}$, $\mathit{i}=1,2,\dots \mathit{r},\mathit{r}$ is finite; $\mathit{w}=\sqrt{-1}$ and $a_{\mathit{j}},b_{\mathit{j}},a_{\mathit{ji}},b_{\mathit{ji}}$ are complex numbers; ${\vartheta }_{\mathit{j}},{\xi }_{\mathit{j}},\text{allowbreak}{\vartheta }_{\mathit{ji}},{\xi }_{\mathit{ji}}$ are real and positive.

The contour C extends from $-\mathit{w}\mathrm{\infty }$ to $\mathit{w}\mathrm{\infty }$, with poles of $\mathit{G}(q^{1-a_j+{\vartheta }_j\mathit{s}}),1\le \mathit{j}\le \mathrm{\wp }$ to the left and those $\mathit{G}(q^{b_j-{\xi }_j\mathit{s}}),1\le \mathit{j}\le \mathrm{\Im }$ to the right of C. The integral converges for large values of $\left|\mathit{s}\right|$, if $\mathrm{\Re }\left[\mathit{s}\mathit{log}(\mathit{x})-\mathit{log}\mathit{sin\pi s}\right]<0$ on the contour C, which means if $\left|\mathit{arg}(\mathit{x})\right|<\mathit{\pi }$. It is possible to see that the integration contour C can be substituted by other appropriately indented contours parallel to the imaginary axis.

It is intriguing to note that for r = 1, $P_1=\mathit{P};Q_1=\mathit{Q}$; thus, (1.11(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11) ) provides q-analogue of H-function attributed to Saxena et al. (1983), precisely

(1.14) $\begin{array}{ll}& H_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array} ]=\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}{\psi }^{\star }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.14)

where

(1.15) ${\psi }^{\star }(\mathit{s};\mathit{q})=\frac{\{\prod _{\mathit{j}=1}^{\mathrm{\Im }}\mathit{G}(q^{b_j-{\xi }_j\mathit{s}})\}\{\prod _{\mathit{j}=1}^{\mathrm{\wp }}\mathit{G}(q^{1-a_j+{\vartheta }_j\mathit{s}})\}\mathit{\pi }}{\{\prod _{\mathit{j}=\mathrm{\Im }+1}^{\mathit{Q}}\mathit{G}(q^{1-b_j+{\xi }_j\mathit{s}})\}\{\prod _{\mathit{j}=\mathrm{\wp }+1}^{\mathit{P}}\mathit{G}(q^{a_j-{\vartheta }_j\mathit{s}})\}\mathit{G}(q^{1-\mathit{s}})\mathit{sin\pi s}}$(1.15)

In addition, ${\vartheta }_i^{\prime }\mathit{s}$ and ${\xi }_j^{\prime }\mathit{s}$ are all positive integers and $0\le \mathrm{\Im }\le \mathit{Q},0\le \mathrm{\wp }\le \mathit{P}$. The contour C is a line parallel to $\mathrm{\Re }(\mathit{ws})=0$ with any required indentations placed so that the poles of both $\mathit{G}(q^{b_j-{\xi }_j\mathit{s}}),1\le \mathit{j}\le \mathrm{\Im }$ and $\mathit{G}(q^{1-a_j+{\vartheta }_j\mathit{s}}),1\le \mathit{j}\le \mathrm{\wp }$ are to the left and right of C, respectively. The integral converges for large values of $\left|\mathit{s}\right|$, if $\mathrm{\Re }\left[\mathit{s}\mathit{log}(\mathit{x})-\mathit{log}\mathit{sin\pi s}\right]<0$ on the contour C, i.e. if $\left|\mathit{arg}(\mathit{x})-w_2{w}_1^{-1}\mathit{log}\left|\mathit{x}\right|\right|<\mathit{\pi }$, where $\mathit{log}\mathit{q}=-\mathit{w}=-(w_1+\mathit{i}{w}_2),w_1\mathit{and}{w}_2$ being real and $0<\left|\mathit{q}\right|<1$.

Furthermore, r = 1, $P_1=\mathit{P};Q_1=\mathit{Q}$; ${\vartheta }_{\mathit{j}}={\xi }_j=1$, $\mathit{j}=1,\dots \mathit{P};\mathit{i}=1,\dots \mathit{Q}$, (1.11(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11) ) relates to the q-analogue of Meijer’s G-function according to Saxena et al. (1983), precisely

(1.16) $\begin{array}{l}{G}_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}\left|\begin{array}{l}{a}_1,a_2,\dots ,a_P\\ b_1,b_2,\dots ,b_Q\end{array}\right] \\ & =\frac{1}{2\mathit{\pi i}}{\int }_{\mathit{C}}{\psi }^{\star \star }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.16)

where

(1.17) ${\psi }^{\star \star }(\mathit{s};\mathit{q})=\frac{\{\prod _{\mathit{j}=1}^{\mathrm{\Im }}\mathit{G}(q^{b_j-\mathit{s}})\}\{\prod _{\mathit{j}=1}^{\mathrm{\wp }}\mathit{G}(q^{1-a_j+\mathit{s}})\}\mathit{\pi }}{\{\prod _{\mathit{j}=\mathrm{\Im }+1}^{\mathit{Q}}\mathit{G}(q^{1-b_j+\mathit{s}})\}\{\prod _{\mathit{j}=\mathrm{\wp }+1}^{\mathit{P}}\mathit{G}(q^{a_j-\mathit{s}})\}\mathit{G}(q^{1-\mathit{s}})\mathit{sin\pi s}}.$(1.17)

A full description of Meijer’s G-function, Fox’s H-function, and numerous functions accessible in terms of Fox’s H-function can be seen in research monographs by Mathai & Saxena (1973, 1978). Furthermore, the articles of Yadav & Purohit (2006) and Yadav et al. (2008) also provide the basic functions of one variable defined in terms of the functions $G_q(.)$.

2. The q-generating relations

In this part, using the I_q-function and a family of q-polynomials, we shall derive several bilinear q-generating relations.

Theorem 1.

Let $\mathrm{\Im },\mathrm{\wp },P_i,Q_i$ to be positive integers;$0\le \mathrm{\wp }\le P_i$, $0\le \mathrm{\Im }\le Q_i;$ λ > 0, and let γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, then the subsequent bilinear q-generating relation follows:

$\begin{array}{ll}& \sum _{\mathit{n}=0}^{\mathrm{\infty }}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\mu }\mathit{x};\mathit{q})I_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}\\ & [\mathit{y};\mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{n},\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right]\frac{t^n}{(\mathit{q};\mathit{q}{)}_n} \end{array}$

$=\frac{1}{(1-\mathit{t}{)}^{(\mathit{\kappa })}}\sum _{\mathit{\delta }=0}^{\mathrm{\infty }}\frac{S_{\mathit{\delta },\mathit{q}}}{(\mathit{q};\mathit{q}{)}_{\mathit{\gamma \delta }}}\frac{(\mu x{t}^{\gamma }{)}^{\delta }}{(1-\mathit{t}{q}^{\kappa }{)}^{(\mathit{\gamma \delta })}}$

(2.1) $\begin{array}{ll}& \times I_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}[\frac{y}{(1-\mathit{t}{\mathit{q}}^{\mathit{\kappa }+\mathit{\gamma \delta }}{)}^{(\mathit{\lambda })}}; \\ & \mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{\gamma \delta },\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] ,\end{array}$(2.1)

where µ and κ are arbitrary numbers, $0<\left|\mathit{q}\right|<1$ and $\left|\mathit{t}\right|<1$.

Proof.

Taking the L.H.S. of (2.1(2.1) $\begin{array}{ll}& \times I_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}[\frac{y}{(1-\mathit{t}{\mathit{q}}^{\mathit{\kappa }+\mathit{\gamma \delta }}{)}^{(\mathit{\lambda })}}; \\ & \mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{\gamma \delta },\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] ,\end{array}$(2.1) ) written by $\mathrm{\Theta }$ and applying the contour integral expression (1.11(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11) ) for the q-polynomials $f_{\mathit{n},\mathit{\gamma }}(\mathit{\mu }\mathit{x};\mathit{q})$ and the I_q-function, we get

$\begin{array}{ll}& \mathit{\Theta }=\frac{1}{2\mathit{\pi w}}\sum _{\mathit{n}=0}^{\mathrm{\infty }}\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q\\ & S_{\mathit{\delta },\mathit{q}}(\mathit{\mu }\mathit{x}{)}^{\delta }\{{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})\mathit{G}(q^{\mathit{\kappa }+\mathit{n}+\mathit{\lambda s}})y^s{d}_q\mathit{s}\}\frac{t^n}{(\mathit{q};\mathit{q}{)}_n}.\end{array}$

Reversing the sequence of summations and integration yields

(2.2) $\begin{array}{ll}& \mathit{\Theta }=\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})\sum _{\mathit{n}=0}^{\mathrm{\infty }}\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}\frac{\mathit{G}({\mathit{q}}^{\mathit{\kappa }+\mathit{n}+\mathit{\lambda s}})}{(\mathit{q};\mathit{q}{)}_n}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q\\ & S_{\mathit{\delta },\mathit{q}}(\mathit{\mu }\mathit{x}{)}^{\delta }{t}^n{y}^s{d}_q\mathit{s},\end{array}$(2.2)

where (1.12(1.12) $\mathit{\psi }(\mathit{s};\mathit{q})=\frac{\{\prod _{\mathit{j}=1}^{\mathrm{\Im }}\mathit{G}(q^{b_j-{\xi }_j\mathit{s}})\}\{\prod _{\mathit{j}=1}^{\mathrm{\wp }}\mathit{G}(q^{1-a_j+{\vartheta }_j\mathit{s}})\mathit{\vartheta }\}\mathit{\pi }}{\sum _{\mathit{i}=1}^{\mathit{r}}\{\prod _{\mathit{j}=\mathrm{\Im }+1}^{{\mathit{Q}}_{\mathit{i}}}\mathit{G}(q^{1-{\mathit{b}}_{\mathit{ji}}+{\xi }_{\mathit{ji}}\mathit{s}})\}\{\prod _{\mathit{j}=\mathrm{\wp }+1}^{{\mathit{P}}_{\mathit{i}}}\mathit{G}(q^{{\mathit{a}}_{\mathit{ji}}-{\vartheta }_{\mathit{ji}}\mathit{s}})\}\mathit{G}(q^{1-\mathit{s}})\mathit{sin\pi s}},$(1.12) ) provides $\mathit{\psi }(\mathit{s};\mathit{q})$. Utilizing the q-gamma function relationship, specifically

(2.3) $\mathit{G}(q^{\varpi })=\frac{{\mathrm{\Gamma }}_q(\mathit{\varpi })(1-\mathit{q}{)}^{\mathit{\varpi }-1}}{(\mathit{q};\mathit{q}{)}_{\mathrm{\infty }}},$(2.3)

We find

$\begin{array}{ll}& \mathit{\Theta }=\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})\frac{{\mathrm{\Gamma }}_q(\mathit{\kappa }+\mathit{\lambda s})(1-\mathit{q}{)}^{\mathit{\kappa }+\mathit{\lambda s}-1}}{(\mathit{q};\mathit{q}{)}_{\mathrm{\infty }}}\\ & \sum _{\mathit{n}=0}^{\mathrm{\infty }}\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}\frac{({\mathit{q}}^{\mathit{\kappa }+\mathit{\lambda s}};\mathit{q}{)}_n}{(\mathit{q};\mathit{q}{)}_n}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}(\mathit{\mu }\mathit{x}{)}^{\delta }{t}^n{y}^s{d}_q\mathit{s}.\end{array}$

Utilizing the series reordering a connection once more and altering the order of summations (Srivastava & Manocha, 1984)

(2.4) $\sum _{\mathit{n}=0}^{\mathrm{\infty }}\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}\mathbb{B}(\mathit{\delta },\mathit{n})=\sum _{\mathit{\delta }=0}^{\mathrm{\infty }}\sum _{\mathit{n}=0}^{\mathrm{\infty }}\mathbb{B}(\mathit{\delta },\mathit{n}+\mathit{\gamma \delta }),$(2.4)

we obtain

(2.5) $\begin{array}{ll}& \mathit{\Theta }=\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})\frac{{\mathrm{\Gamma }}_q(\mathit{\kappa }+\mathit{\lambda s})(1-\mathit{q}{)}^{\mathit{\kappa }+\mathit{\lambda s}-1}}{(\mathit{q};\mathit{q}{)}_{\mathrm{\infty }}}\\ & \sum _{\mathit{\delta }=0}^{\mathrm{\infty }}{S}_{\mathit{\delta },\mathit{q}}\frac{(\mu x{t}^{\gamma }{)}^{\delta }}{(\mathit{q};\mathit{q}{)}_{\mathit{\gamma \delta }}}\sum _{\mathit{n}=0}^{\mathrm{\infty }}\frac{({\mathit{q}}^{\mathit{\kappa }+\mathit{\lambda s}};\mathit{q}{)}_{\mathit{n}+\mathit{\gamma \delta }}}{(\mathit{q};\mathit{q}{)}_n}{t}^n{y}^s{d}_q\mathit{s}.\end{array}$(2.5)

Employing the q-binomial theorem (see Gasper & Rahman, 2004) to sum the inner series, specifically

(2.6) ${}_1{\mathrm{\Phi }}_0(\mathit{\varpi };-;\mathit{q};\mathit{x})=\frac{(\varpi x;q{)}_{\mathrm{\infty }}}{(\mathit{x};\mathit{q}{)}_{\mathrm{\infty }}},\left|\mathit{x}\right|<1,0<\left|\mathit{q}\right|<1,$(2.6)

we find that

(2.7) $\begin{array}{ll}& \mathit{\Theta }=\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})\frac{{\mathrm{\Gamma }}_q(\mathit{\kappa }+\mathit{\lambda s})(1-\mathit{q}{)}^{\mathit{\kappa }+\mathit{\lambda s}-1}}{(\mathit{q};\mathit{q}{)}_{\mathrm{\infty }}}\\ & \sum _{\mathit{\delta }=0}^{\mathrm{\infty }}\frac{({\mathit{q}}^{\mathit{\kappa }+\mathit{\lambda s}};\mathit{q}{)}_{\mathit{\gamma \delta }}(\mathit{\mu }\mathit{x}{t}^{\gamma }{)}^{\delta }}{(\mathit{t};\mathit{q}{)}_{\mathit{\kappa }+\mathit{\lambda s}+\mathit{\gamma \delta }}(\mathit{q};\mathit{q}{)}_{\mathit{\gamma \delta }}}{S}_{\mathit{\delta },\mathit{q}}{y}^s{d}_q\mathit{s}.\end{array}$(2.7)

Currently, by reversing the contour integral and summation’s order and employing the q-identities (see Gasper & Rahman, 2004), i.e.

(2.8) $(\mathit{\varpi };\mathit{q}{)}_{\mathit{n}+\mathit{\delta }}=(\mathit{\varpi };\mathit{q}{)}_n(\mathit{\varpi }{q}^n;\mathit{q}{)}_{\delta },$(2.8)

and

(2.9) $(\mathit{\varpi };\mathit{q}{)}_n=\frac{{\mathrm{\Gamma }}_q(\mathit{\varpi }+\mathit{n})(1-\mathit{q}{)}^n}{{\mathrm{\Gamma }}_q(\mathit{\varpi })},(\mathit{n}>0),$(2.9)

we obtain

(2.10) $\begin{array}{ll}& \mathit{\Theta }=\frac{1}{(t;q{)}_{\kappa }}\sum _{\mathit{\delta }=0}^{\mathrm{\infty }}\frac{(\mu x{t}^{\gamma }{)}^{\delta }}{(\mathit{t}{q}^{\kappa };\mathit{q}{)}_{\mathit{\gamma \delta }}(\mathit{q};\mathit{q}{)}_{\mathit{\gamma \delta }}}\\ & S_{\mathit{\delta },\mathit{q}}\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})\frac{{\mathrm{\Gamma }}_q(\mathit{\kappa }+\mathit{\lambda s}+\mathit{\gamma \delta })(1-\mathit{q}{)}^{\mathit{\kappa }+\mathit{\lambda s}+\mathit{\gamma \delta }-1}}{(\mathit{t}{q}^{\mathit{\kappa }+\mathit{\gamma \delta }};\mathit{q}{)}_{\mathit{\lambda s}}(\mathit{q};\mathit{q}{)}_{\mathrm{\infty }}}{y}^s{d}_q\mathit{s}.\end{array}$(2.10)

By analyzing the contour integral of (2.10(2.10) $\begin{array}{ll}& \mathit{\Theta }=\frac{1}{(t;q{)}_{\kappa }}\sum _{\mathit{\delta }=0}^{\mathrm{\infty }}\frac{(\mu x{t}^{\gamma }{)}^{\delta }}{(\mathit{t}{q}^{\kappa };\mathit{q}{)}_{\mathit{\gamma \delta }}(\mathit{q};\mathit{q}{)}_{\mathit{\gamma \delta }}}\\ & S_{\mathit{\delta },\mathit{q}}\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})\frac{{\mathrm{\Gamma }}_q(\mathit{\kappa }+\mathit{\lambda s}+\mathit{\gamma \delta })(1-\mathit{q}{)}^{\mathit{\kappa }+\mathit{\lambda s}+\mathit{\gamma \delta }-1}}{(\mathit{t}{q}^{\mathit{\kappa }+\mathit{\gamma \delta }};\mathit{q}{)}_{\mathit{\lambda s}}(\mathit{q};\mathit{q}{)}_{\mathrm{\infty }}}{y}^s{d}_q\mathit{s}.\end{array}$(2.10) ) in accordance with its description (1.11(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11) ) and the notation (2.3(2.3) $\mathit{G}(q^{\varpi })=\frac{{\mathrm{\Gamma }}_q(\mathit{\varpi })(1-\mathit{q}{)}^{\mathit{\varpi }-1}}{(\mathit{q};\mathit{q}{)}_{\mathrm{\infty }}},$(2.3) ), the intended outcome is obtained. Theorem 1 proof is now complete.

The family of q-polynomials has one value if we determine the bounded sequence $S_{\mathit{\delta },\mathit{q}}=1$ along with µ = 0.

$f_{\mathit{n},\mathit{\gamma }}(\mathit{\mu }\mathit{x};\mathit{q})=1,$

so we arrive at the subsequent theorem in light of the right side of (2.1(2.1) $\begin{array}{ll}& \times I_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}[\frac{y}{(1-\mathit{t}{\mathit{q}}^{\mathit{\kappa }+\mathit{\gamma \delta }}{)}^{(\mathit{\lambda })}}; \\ & \mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{\gamma \delta },\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] ,\end{array}$(2.1) ) for δ = 0.

Theorem 2.

Let λ > 0, κ be an arbitrary number and let $0\le \mathrm{\Im }\le Q_i,0\le \mathrm{\wp }\le P_i$ be satisfied by positive integers $\mathrm{\Im },\mathrm{\wp },P_i,Q_i$. The q-generating connection for the I_q-function is thus given by

$\begin{array}{ll}& \sum _{\mathit{n}=0}^{\mathrm{\infty }}{I}_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}\\ & [\mathit{y};\mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{n},\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] \frac{t^n}{(\mathit{q};\mathit{q}{)}_n}\end{array}$

(2.11) $\begin{array}{ll}& =\frac{1}{(1-\mathit{t}{)}^{\mathit{\kappa }}}{I}_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}\\ & [\frac{y}{(1-t{q}^{\kappa }{)}^{\mathit{\lambda }}};\mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa },\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] ,\end{array}$(2.11)

where $0<|\mathit{q}|<1$ and $|\mathit{t}|<1$.

3. An integral associated with I_q-function

Following theorems and corollaries, integral is shown involving the I_q-function and family of q-polynomials:

Theorem 3.

If $\mathrm{\Re }(\mathit{\sigma })>0$, $\mathrm{\Re }(\mathit{\varrho })>0$ ,$|\mathit{arg}\mathit{\mu }|<\mathit{\pi }$, τ be an arbitrary number; assume $\mathrm{\Im },\mathrm{\wp },P_i,Q_i$ to be positive integers; $0\le \mathrm{\wp }\le P_i$, $0\le \mathrm{\Im }\le Q_i;$ and let γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, we have the following integral

$\begin{array}{ll}& \frac{\mathit{G}(\mathit{q})}{1-\mathit{q}}{\int }_0^1{y}^{\mathit{\sigma }-1}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\tau y};\mathit{q})E_q(\mathit{qy})I_{P_i,Q_i}^{\mathrm{\Im },\mathrm{\wp }}\\ & [\mathit{\mu }{y}^{-\mathit{\varrho }};\mathit{q}\left|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] d_q\mathit{y}\end{array}$

(3.1) $\begin{array}{ll}=& \sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{I}_{P_i,Q_i+1}^{\mathrm{\Im }+1,\mathrm{\wp }}\\ & [\mathit{\mu };\mathit{q}\left|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (\mathit{\sigma }+\mathit{\delta },\mathit{\varrho }),(b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] .\end{array}$(3.1)

Proof.

In view of (1.11(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11) ) and on interchanging the order of integration which is permitted in view of the aforesaid conditions, considering the left-hand side (3.1(3.1) $\begin{array}{ll}=& \sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{I}_{P_i,Q_i+1}^{\mathrm{\Im }+1,\mathrm{\wp }}\\ & [\mathit{\mu };\mathit{q}\left|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (\mathit{\sigma }+\mathit{\delta },\mathit{\varrho }),(b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] .\end{array}$(3.1) ) by $\mathrm{\Lambda }$, transforms in to the integral

$\begin{array}{ll}& \mathit{\Lambda }=\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})\\ & \{\frac{\mathit{G}(\mathit{q})}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{\sigma }-1}\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}(\mathit{\tau }\mathit{y}{)}^{\delta }{E}_q(\mathit{qy})d_q\mathit{y}\}\\ & {\mu }^s{y}^{-\mathit{\varrho s}}{d}_q\mathit{s},\end{array}$

$\begin{array}{ll}& =\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}(\mathit{\tau }{)}^{\delta }\frac{1}{2\mathit{\pi w}}\\ & {\int }_{\mathit{C}}\frac{\{\prod _{\mathit{j}=1}^{\mathrm{\Im }}\mathit{G}(q^{b_j-{\xi }_j\mathit{s}})\}\{\prod _{\mathit{j}=1}^{\mathrm{\wp }}\mathit{G}(q^{1-a_j+{\vartheta }_j\mathit{s}})\mathit{\vartheta }\}\mathit{\pi }{\mu }^s}{\sum _{\mathit{i}=1}^{\mathit{r}}\{\prod _{\mathit{j}=\mathrm{\Im }+1}^{{\mathit{Q}}_{\mathit{i}}}\mathit{G}(q^{1-{\mathit{b}}_{\mathit{ji}}+{\xi }_{\mathit{ji}}\mathit{s}})\}\{\prod _{\mathit{j}=\mathrm{\wp }+1}^{{\mathit{P}}_{\mathit{i}}}\mathit{G}(q^{{\mathit{a}}_{\mathit{ji}}-{\vartheta }_{\mathit{ji}}\mathit{s}})\}}\end{array}$

(3.2) $\times \frac{1}{\mathit{G}({\mathit{q}}^{1-\mathit{s}})\mathit{sin\pi s}}\{\frac{\mathit{G}(\mathit{q})}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{\sigma }+\mathit{\delta }-\mathit{\varrho s}-1}{E}_q(\mathit{qy})d_q\mathit{y}\}{d}_q\mathit{s},$(3.2)

Due to Hann (1949), the integral representation as

$\frac{\mathit{G}(\mathit{q})}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{\sigma }+\mathit{\delta }-\mathit{\varrho s}-1}{E}_q(\mathit{qy})d_q\mathit{y}=\mathit{G}(q^{\mathit{\sigma }+\mathit{\delta }-\mathit{\varrho s}}).$

Utilizing the above integral in (3.2(3.2) $\times \frac{1}{\mathit{G}({\mathit{q}}^{1-\mathit{s}})\mathit{sin\pi s}}\{\frac{\mathit{G}(\mathit{q})}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{\sigma }+\mathit{\delta }-\mathit{\varrho s}-1}{E}_q(\mathit{qy})d_q\mathit{y}\}{d}_q\mathit{s},$(3.2) ), we get

$\mathit{\Lambda }=\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}(\mathit{\tau }{)}^{\delta }$

$\times \frac{1}{2\mathit{\pi w}}\frac{\{\prod _{\mathit{j}=1}^{\mathrm{\Im }}\mathit{G}(q^{b_j-{\xi }_j\mathit{s}})\}\{\prod _{\mathit{j}=1}^{\mathrm{\wp }}\mathit{G}(q^{1-a_j+{\vartheta }_j\mathit{s}})\mathit{\vartheta }\}\mathit{G}(q^{\mathit{\sigma }+\mathit{\delta }-\mathit{\varrho s}})\mathit{\pi }{\mu }^s}{\sum _{\mathit{i}=1}^{\mathit{r}}\{\prod _{\mathit{j}=\mathrm{\Im }+1}^{{\mathit{Q}}_{\mathit{i}}}\mathit{G}(q^{1-{\mathit{b}}_{\mathit{ji}}+{\xi }_{\mathit{ji}}\mathit{s}})\}\{\prod _{\mathit{j}=\mathrm{\wp }+1}^{{\mathit{P}}_{\mathit{i}}}\mathit{G}(q^{{\mathit{a}}_{\mathit{ji}}-{\vartheta }_{\mathit{ji}}\mathit{s}})\}\mathit{G}(q^{1-\mathit{s}})\mathit{sin\pi s}}{d}_q\mathit{s}$

In accordance with its description (1.11(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11) ), the intended outcome is obtained. Proof of the Theorem 3 is now complete.

Theorem 4.

If $\mathrm{\Re }(\mathit{\sigma })>0$, $\mathrm{\Re }(\mathit{\rho })>0$, $\mathrm{\Re }(\mathit{\varrho })>0$,$\mathrm{\Re }(\mathit{\rho }-\mathit{\sigma })>0$, $|\mathit{arg}\mathit{\mu }|<\mathit{\pi }$, τ an arbitrary number and assume $\mathrm{\Im },\mathrm{\wp },P_i,Q_i$ to be positive integers; $0\le \mathrm{\wp }\le P_i$, $0\le \mathrm{\Im }\le Q_i;$ and let γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, we have the following integral

$\begin{array}{ll}& \frac{\mathit{G}(\mathit{q})}{1-\mathit{q}}{\int }_0^1{y}^{\mathit{\sigma }-1}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\tau y};\mathit{q})(1-\mathit{qy}{)}_{\mathit{\rho }-\mathit{\sigma }-1}{I}_{P_i,Q_i}^{\mathrm{\Im },\mathrm{\wp }}\\ & [\mathit{\mu }{y}^{-\mathit{\varrho }};\mathit{q}\left|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] d_q\mathit{y}\end{array}$

(3.3) $\begin{array}{ll}=& \mathit{G}(\mathit{\rho }-\mathit{\sigma })\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{I}_{P_i+1,Q_i+1}^{\mathrm{\Im }+1,\mathrm{\wp }}\\ & [\mathit{\mu };\mathit{q}\left|\begin{array}{l}(\mathit{\delta }+\mathit{\rho },\mathit{\varrho }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (\mathit{\sigma }+\mathit{\delta },\mathit{\varrho }),(b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] .\end{array}$(3.3)

Proof.

In view of (1.11(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11) ) and on interchanging the order of integration which is justified in view of the aforesaid conditions, considering the left-hand side (3.3(3.3) $\begin{array}{ll}=& \mathit{G}(\mathit{\rho }-\mathit{\sigma })\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{I}_{P_i+1,Q_i+1}^{\mathrm{\Im }+1,\mathrm{\wp }}\\ & [\mathit{\mu };\mathit{q}\left|\begin{array}{l}(\mathit{\delta }+\mathit{\rho },\mathit{\varrho }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (\mathit{\sigma }+\mathit{\delta },\mathit{\varrho }),(b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] .\end{array}$(3.3) ) by ${\Lambda }^{\mathit{\prime }}$, transforms in to the integral

$\begin{array}{ll}& {\Lambda }^{\mathit{\prime }}=\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})\\ & \{\frac{\mathit{G}(\mathit{q})}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{\sigma }-1}\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}(\mathit{\tau }\mathit{y}{)}^{\delta } \\ & \phantom{\frac{\mathit{G}(\mathit{q})}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{\sigma }-1}\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}(\mathit{\tau }\mathit{y}{)}^{\delta }}(1-\mathit{qy}{)}_{\mathit{\rho }-\mathit{\sigma }-1}{d}_q\mathit{y}\}{\mu }^s{y}^{-\mathit{\varrho s}}{d}_q\mathit{s},\end{array}$

$\begin{array}{ll}& =\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}(\mathit{\tau }{)}^{\delta }\frac{1}{2\mathit{\pi w}}\\ & {\int }_{\mathit{C}}\frac{\{\prod _{\mathit{j}=1}^{\mathrm{\Im }}\mathit{G}(q^{b_j-{\xi }_j\mathit{s}})\}\{\prod _{\mathit{j}=1}^{\mathrm{\wp }}\mathit{G}(q^{1-a_j+{\vartheta }_j\mathit{s}})\mathit{\vartheta }\}\mathit{\pi }{\mu }^s}{\sum _{\mathit{i}=1}^{\mathit{r}}\{\prod _{\mathit{j}=\mathrm{\Im }+1}^{{\mathit{Q}}_{\mathit{i}}}\mathit{G}(q^{1-{\mathit{b}}_{\mathit{ji}}+{\xi }_{\mathit{ji}}\mathit{s}})\}\{\prod _{\mathit{j}=\mathrm{\wp }+1}^{{\mathit{P}}_{\mathit{i}}}\mathit{G}(q^{{\mathit{a}}_{\mathit{ji}}-{\vartheta }_{\mathit{ji}}\mathit{s}})\}}\end{array}$

(3.4) $\begin{array}{l}\times \frac{1}{\mathit{G}({\mathit{q}}^{1-\mathit{s}})\mathit{sin\pi s}}\{\frac{\mathit{G}(\mathit{q})}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{\sigma }+\mathit{\delta }-\mathit{\varrho s}-1}(1-\mathit{qy}{)}_{\mathit{\rho }-\mathit{\sigma }-1} \\ & \phantom{\frac{\mathit{G}(\mathit{q})}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{\sigma }+\mathit{\delta }-\mathit{\varrho s}-1}(1-\mathit{qy}{)}_{\mathit{\rho }-\mathit{\sigma }-1}}{d}_q\mathit{y}\}{d}_q\mathit{s},\end{array}$(3.4)

Due to Hann (1949), the integral representation as

$\frac{1}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{ϵ}-1}(1-\mathit{qy}{)}_{\mathit{\eta }-1}{d}_q\mathit{y}=\prod _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(1-{\mathit{q}}^{\mathit{ϵ}+\mathit{\eta }+\mathit{m}})(1-q^{1+\mathit{m}})}{(1-q^{\mathit{ϵ}+\mathit{m}})(1-q^{\mathit{\eta }+\mathit{m}})}.$

Utilizing the above integral formula in (3.4(3.4) $\begin{array}{l}\times \frac{1}{\mathit{G}({\mathit{q}}^{1-\mathit{s}})\mathit{sin\pi s}}\{\frac{\mathit{G}(\mathit{q})}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{\sigma }+\mathit{\delta }-\mathit{\varrho s}-1}(1-\mathit{qy}{)}_{\mathit{\rho }-\mathit{\sigma }-1} \\ & \phantom{\frac{\mathit{G}(\mathit{q})}{(1-\mathit{q})}{\int }_0^1{y}^{\mathit{\sigma }+\mathit{\delta }-\mathit{\varrho s}-1}(1-\mathit{qy}{)}_{\mathit{\rho }-\mathit{\sigma }-1}}{d}_q\mathit{y}\}{d}_q\mathit{s},\end{array}$(3.4) ), we obtain

$\mathit{\Lambda }=\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}(\mathit{\tau }{)}^{\delta }\prod _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(1-{\mathit{q}}^{\mathit{\delta }+\mathit{\rho }-\mathit{\varrho s}+\mathit{m}})(1-q^{1+\mathit{m}})}{(1-q^{\mathit{\sigma }+\mathit{\delta }-\mathit{\varrho s}+\mathit{m}})(1-q^{\mathit{\rho }-\mathit{\sigma }+\mathit{m}})}.$

$\times \frac{1}{2\mathit{\pi w}}{\int }_C\frac{\{\prod _{\mathit{j}=1}^{\mathrm{\Im }}\mathit{G}(q^{b_j-{\xi }_j\mathit{s}})\}\{\prod _{\mathit{j}=1}^{\mathrm{\wp }}\mathit{G}(q^{1-a_j+{\vartheta }_j\mathit{s}})\mathit{\vartheta }\}\mathit{G}(\mathit{q})\mathit{\pi }{\mu }^s}{\sum _{\mathit{i}=1}^{\mathit{r}}\{\prod _{\mathit{j}=\mathrm{\Im }+1}^{{\mathit{Q}}_{\mathit{i}}}\mathit{G}(q^{1-{\mathit{b}}_{\mathit{ji}}+{\xi }_{\mathit{ji}}\mathit{s}})\}\{\prod _{\mathit{j}=\mathrm{\wp }+1}^{{\mathit{P}}_{\mathit{i}}}\mathit{G}(q^{{\mathit{a}}_{\mathit{ji}}-{\vartheta }_{\mathit{ji}}\mathit{s}})\}\mathit{G}(q^{1-\mathit{s}})\mathit{sin\pi s}}{d}_q\mathit{s}$

In accordance with its description (1.11(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11) ), the intended outcome is obtained. Proof of the Theorem 4 is now complete.

Theorem 5.

If $\mathrm{\Re }(\mathit{\sigma })>0$, $|\mathit{arg}\mathit{\mu }|<\mathit{\pi }$, τ an arbitrary number. Suppose $\mathrm{\Im },\mathrm{\wp },P_i,Q_i$ to be positive integers; $0\le \mathrm{\wp }\le P_i$, $0\le \mathrm{\Im }\le Q_i;$ and γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, we have the following integral

$\begin{array}{ll}& \frac{1}{2\mathit{\pi w}}{\int }_C^{}{y}^{-\mathit{\sigma }}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\tau y};\mathit{q})e_q(\mathit{y})I_{P_i,Q_i}^{\mathrm{\Im },\mathrm{\wp }}\\ & [\mathit{\mu }{y}^{\mathit{\varrho }};\mathit{q}\left|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] d_q\mathit{y}\end{array}$

(3.5) $\begin{array}{ll}=& \mathit{G}(\mathit{q})\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{I}_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }}\\ & [\mathit{\mu };\mathit{q}\left|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}},(\mathit{\sigma }+\mathit{\delta },\mathit{\varrho })\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] .\end{array}$(3.5)

Proof.

In view of (1.11(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11) ) and on interchanging the order of integration that is allowed in view of the aforesaid conditions, considering the left-hand side (3.5(3.5) $\begin{array}{ll}=& \mathit{G}(\mathit{q})\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{I}_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }}\\ & [\mathit{\mu };\mathit{q}\left|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}},(\mathit{\sigma }+\mathit{\delta },\mathit{\varrho })\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] .\end{array}$(3.5) ) by ${\Lambda }^{\mathit{\prime \prime }}$, transforms in to the integral

$\begin{array}{ll}& {\Lambda }^{\mathit{\prime \prime }}=\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})\{\frac{1}{2\mathit{\pi w}}{\int }_0^1{y}^{-\mathit{\sigma }}\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q \\ & \phantom{\frac{1}{2\mathit{\pi w}}{\int }_0^1{y}^{-\mathit{\sigma }}\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q}{S}_{\mathit{\delta },\mathit{q}}(\mathit{\tau }\mathit{y}{)}^{\delta }{e}_q(\mathit{y})d_q\mathit{y}\}{\mu }^s{y}^{\mathit{\varrho s}}{d}_q\mathit{s},\end{array}$

$\begin{array}{ll}& =\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma \delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}(\mathit{\tau }{)}^{\delta }\frac{1}{2\mathit{\pi w}}\\ & {\int }_{\mathit{C}}\frac{\{\prod _{\mathit{j}=1}^{\mathrm{\Im }}\mathit{G}(q^{b_j-{\xi }_j\mathit{s}})\}\{\prod _{\mathit{j}=1}^{\mathrm{\wp }}\mathit{G}(q^{1-a_j+{\vartheta }_j\mathit{s}})\mathit{\vartheta }\}\mathit{\pi }{\mu }^s}{\sum _{\mathit{i}=1}^{\mathit{r}}\{\prod _{\mathit{j}=\mathrm{\Im }+1}^{{\mathit{Q}}_{\mathit{i}}}\mathit{G}(q^{1-{\mathit{b}}_{\mathit{ji}}+{\xi }_{\mathit{ji}}\mathit{s}})\}\{\prod _{\mathit{j}=\mathrm{\wp }+1}^{{\mathit{P}}_{\mathit{i}}}\mathit{G}(q^{{\mathit{a}}_{\mathit{ji}}-{\vartheta }_{\mathit{ji}}\mathit{s}})\}}\end{array}$

(3.6) $\times \frac{1}{\mathit{G}({\mathit{q}}^{1-\mathit{s}})\mathit{sin\pi s}}\{\frac{1}{2\mathit{\pi w}}{\int }_0^1{y}^{-\mathit{\sigma }+\mathit{\delta }+\mathit{\varrho s}}{e}_q(\mathit{y})d_q\mathit{y}\}{d}_q\mathit{s},$(3.6)

Due to Hann (1949), the integral representation as

$\frac{1}{2\mathit{\pi w}}{\int }_C^{}(\mathit{sy}{)}^{-\mathit{\sigma }-1}{e}_q(\mathit{sy})\mathit{ds}=\frac{y^{\sigma }}{(1-\mathit{q}{)}_{\sigma }}$

Utilizing the above integral in (3.6(3.6) $\times \frac{1}{\mathit{G}({\mathit{q}}^{1-\mathit{s}})\mathit{sin\pi s}}\{\frac{1}{2\mathit{\pi w}}{\int }_0^1{y}^{-\mathit{\sigma }+\mathit{\delta }+\mathit{\varrho s}}{e}_q(\mathit{y})d_q\mathit{y}\}{d}_q\mathit{s},$(3.6) ), making accordance with its description (1.11(1.11) $\begin{array}{lll}& & I_{{\mathit{P}}_{\mathit{i}},Q_{\mathit{i}}}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ]\\ & & =\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}\mathit{\psi }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.11) ), the intended outcome is obtained. Theorem 5 proof is now complete.

Remark: If we set in the Theorems 3-5, the bounded sequence $S_{\mathit{\delta },\mathit{q}}=1$ along with $\mathit{\mu }=\mathit{\delta }=0$, then for the q-polynomials one has $f_{\mathit{n},\mathit{\gamma }}(\mathit{\mu }\mathit{x};\mathit{q})=1$, and we have the findings stated by Saxena & Kumar (1995), (Equation (3.1)-(3.3), pp. 267).

4. Concluding observations

In this part, we look at a few implications of findings from the preceding section.

If we set r = 1, $P_1=\mathit{P};Q_1=\mathit{Q}$; and take (1.14(1.14) $\begin{array}{ll}& H_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}|\begin{array}{l}(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array} ]=\frac{1}{2\mathit{\pi w}}{\int }_{\mathit{C}}{\psi }^{\star }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.14) ) into consideration, then Theorem 1 to 5 generate Corollaries 1 to 5, accordingly.

Corollary 1.

Let $\mathrm{\Im },\mathrm{\wp },\mathit{P},\mathit{Q}$ to be positive integers; $0\le \mathrm{\wp }\le \mathit{P}$, $0\le \mathrm{\Im }\le \mathit{Q};$ λ > 0, and assume γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, then the subsequent bilinear q-generating relation follows:

$\begin{array}{ll}& \sum _{\mathit{n}=0}^{\mathrm{\infty }}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\mu }\mathit{x};\mathit{q})H_{P+1,Q}^{\mathrm{\Im },\mathrm{\wp }+1}\\ & [\mathit{y};\mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{n},\mathit{\lambda }),(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array}\right]\frac{t^n}{(\mathit{q};\mathit{q}{)}_n} \end{array}$

$=\frac{1}{(1-\mathit{t}{)}^{(\mathit{\kappa })}}\sum _{\mathit{\delta }=0}^{\mathrm{\infty }}\frac{S_{\mathit{\delta },\mathit{q}}}{(\mathit{q};\mathit{q}{)}_{\mathit{\gamma \delta }}}\frac{(\mu x{t}^{\gamma }{)}^{\delta }}{(1-\mathit{t}{q}^{\kappa }{)}^{(\mathit{\gamma \delta })}}$

$\begin{array}{ll}& \times H_{P+1,Q}^{\mathrm{\Im },\mathrm{\wp }+1}\\ & [\frac{y}{(1-\mathit{t}{\mathit{q}}^{\mathit{\kappa }+\mathit{\gamma \delta }}{)}^{\mathit{\lambda }}};\mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{\gamma \delta },\mathit{\lambda }),(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array}\right] ,\end{array}$

where µ and κ are arbitrary numbers, $0<\left|\mathit{q}\right|<1,\left|\mathit{t}\right|<1,$.

Corollary 2.

Let λ > 0, κ an arbitrary number; $0\le \mathrm{\Im }\le \mathit{Q},0\le \mathrm{\wp }\le \mathit{P}$ be satisfied by positive integers $\mathrm{\Im },\mathrm{\wp },\mathit{P},\mathit{Q}$. The q-generating connection for the I_q-function is thus given by

$\begin{array}{ll}& \sum _{\mathit{n}=0}^{\mathrm{\infty }}{H}_{P+1,Q}^{\mathrm{\Im },\mathrm{\wp }+1}[\mathit{y};\mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{n},\mathit{\lambda }),(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array}\right] \\ & \frac{t^n}{(\mathit{q};\mathit{q}{)}_n}\end{array}$

(4.1) $\begin{array}{ll}& =\frac{1}{(1-\mathit{t}{)}^{\mathit{\kappa }}}{H}_{P+1,Q}^{\mathrm{\Im },\mathrm{\wp }+1}\\ & [\frac{y}{(1-t{q}^{\kappa }{)}^{\mathit{\lambda }}};\mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa },\mathit{\lambda }),(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array}\right] ,\end{array}$(4.1)

where $0<|\mathit{q}|<1$ and $|\mathit{t}|<1$.

Corollary 3.

If $\mathrm{\Re }(\mathit{\sigma })>0$, $\mathrm{\Re }(\mathit{\varrho })>0$, $|\mathit{arg}\mathit{\mu }|<\mathit{\pi }$ and assume $\mathrm{\Im },\mathrm{\wp },\mathit{P},\mathit{Q}$ to be positive integers; $0\le \mathrm{\wp }\le \mathit{P}$, $0\le \mathrm{\Im }\le \mathit{Q};$ and suppose γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, we have the following integral

$\begin{array}{ll}& \frac{\mathit{G}(\mathit{q})}{1-\mathit{q}}{\int }_0^1{y}^{\mathit{\sigma }-1}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\tau y};\mathit{q})E_q(\mathit{qy})H_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}\\ & [\mathit{\mu }{y}^{-\mathit{\varrho }};\mathit{q}\left|\begin{array}{l}(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array}\right] d_q\mathit{y}\end{array}$

$\begin{array}{ll}& =\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{H}_{P,Q+1}^{\mathrm{\Im }+1,\mathrm{\wp }}\\ & [\mathit{\mu };\mathit{q}\left|\begin{array}{l}(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (\mathit{\sigma }+\mathit{\delta },\mathit{\varrho }),(b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array}\right] .\end{array}$

Corollary 4.

If $\mathrm{\Re }(\mathit{\sigma })>0$, $\mathrm{\Re }(\mathit{\rho })>0$, $\mathrm{\Re }(\mathit{\varrho })>0$, $\mathrm{\Re }(\mathit{\rho }-\mathit{\sigma })>0$, $|\mathit{arg}\mathit{\mu }|<\mathit{\pi }$; assume $\mathrm{\Im },\mathrm{\wp },\mathit{P},\mathit{Q}$ to be positive integers; $0\le \mathrm{\wp }\le \mathit{P}$, $0\le \mathrm{\Im }\le \mathit{Q};$ and suppose γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, we have the proceeding integral

$\begin{array}{ll}& \frac{\mathit{G}(\mathit{q})}{1-\mathit{q}}{\int }_0^1{y}^{\mathit{\sigma }-1}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\tau y};\mathit{q})(1-\mathit{qy}{)}_{\mathit{\rho }-\mathit{\sigma }-1}{H}_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}\\ & [\mathit{\mu }{y}^{-\mathit{\varrho }};\mathit{q}\left|\begin{array}{l}(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array}\right] d_q\mathit{y}\end{array}$

$\begin{array}{ll}& =\mathit{G}(\mathit{\rho }-\mathit{\sigma })\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{H}_{P+1,Q+1}^{\mathrm{\Im }+1,\mathrm{\wp }}\\ & [\mathit{\mu };\mathit{q}\left|\begin{array}{l}(\mathit{\delta }+\mathit{\rho },\mathit{\varrho }),(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (\mathit{\sigma }+\mathit{\delta },\mathit{\varrho }),(b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array}\right] .\end{array}$

Corollary 5.

If $\mathrm{\Re }(\mathit{\sigma })>0$, $|\mathit{arg}\mathit{\mu }|<\mathit{\pi }$ and assume $\mathrm{\Im },\mathrm{\wp },\mathit{P},\mathit{Q}$ to be positive integers; $0\le \mathrm{\wp }\le \mathit{P}$, $0\le \mathrm{\Im }\le \mathit{Q};$ let γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, we have the subsequent integral

$\begin{array}{ll}& \frac{1}{2\mathit{\pi w}}{\int }_C^{}{y}^{-\mathit{\sigma }}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\tau y};\mathit{q})e_q(\mathit{y})H_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}\\ & [\mathit{\mu }{y}^{\mathit{\varrho }};\mathit{q}\left|\begin{array}{l}(a_1,{\vartheta }_1),\dots ,(a_P,{\vartheta }_P)\\ (b_1,{\xi }_1),\dots ,(b_Q,{\xi }_Q)\end{array}\right] d_q\mathit{y}\end{array}$

$\begin{array}{ll}& =\mathit{G}(\mathit{q})\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{H}_{P+1,Q}^{\mathrm{\Im },\mathrm{\wp }}\\ & [\mathit{\mu };\mathit{q}\left|\begin{array}{l}(a_1,{\vartheta }_1),(a_P,{\vartheta }_P),(\mathit{\sigma }+\mathit{\delta },\mathit{\varrho })\\ (b_1,{\xi }_1),(b_Q,{\xi }_Q)\end{array}\right] .\end{array}$

Remark: If we set in the Corollary 3-5, the bounded sequence $S_{\mathit{\delta },\mathit{q}}=1$ along with $\mathit{\mu }=\mathit{\delta }=0$, then for the q-polynomials one has $f_{\mathit{n},\mathit{\gamma }}(\mathit{\mu }\mathit{x};\mathit{q})=1$, and we have the findings stated by Saxena et al. (1983) (pp. 140, Equation (3.1)-(3.3)).

If we set r = 1, $P_1=\mathit{P};Q_1=\mathit{Q}$; ${\vartheta }_{\mathit{j}}={\xi }_j=1$, $\mathit{j}=1,\dots ,\mathit{P};\mathit{i}=1,\dots ,\mathit{Q}$, $\mathit{\gamma }=\mathit{\lambda }=1$, and take (1.16(1.16) $\begin{array}{l}{G}_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}\left|\begin{array}{l}{a}_1,a_2,\dots ,a_P\\ b_1,b_2,\dots ,b_Q\end{array}\right] \\ & =\frac{1}{2\mathit{\pi i}}{\int }_{\mathit{C}}{\psi }^{\star \star }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.16) ) into considered, and now, Theorem 1, 2 produce Corollaries 6, 7, correspondingly.

Corollary 6.

Assume $\mathrm{\Im },\mathrm{\wp },\mathit{P},\mathit{Q}$ to be positive integers; $0\le \mathrm{\wp }\le \mathit{P}$, $0\le \mathrm{\Im }\le \mathit{Q}$. If ${{\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, then the subsequent bilinear q-generating relation follows:

$\begin{array}{l}\sum _{\mathit{n}=0}^{\mathrm{\infty }}{f}_{\mathit{n},1}(\mathit{\mu x};\mathit{q})G_{P+1,Q}^{\mathrm{\Im },\mathrm{\wp }+1}[\mathit{y};\mathit{q}\left|\begin{array}{l}1-\mathit{\kappa }-\mathit{\delta },a_1,a_P\\ b_1,b_Q\end{array}\right] \frac{t^n}{{(\mathit{q};\mathit{q})}_n}\end{array}$

(4.2) $\begin{array}{l}=\frac{1}{(1-\mathit{t}{)}^{(\mathit{\kappa })}}\sum _{\mathit{\delta }=0}^{\mathrm{\infty }}\frac{S_{\mathit{\delta },\mathit{q}}}{(\mathit{q};\mathit{q}{)}_{\delta }}\frac{(\mu xt{)}^{\delta }}{(1-\mathit{t}{q}^{\kappa }{)}^{\mathit{\delta }}}\\ & & G_{P+1,Q}^{\mathrm{\Im },\mathrm{\wp }+1}[\frac{y}{(1-\mathit{t}{\mathit{q}}^{\mathit{\kappa }+\mathit{\delta }})};\mathit{q}\left|\begin{array}{l}1-\mathit{\kappa }-\mathit{\delta },a_1,a_P\\ b_1,b_Q\end{array}\right] ,\end{array}$(4.2)

where κ is an arbitrary number, $|\mathit{t}|<1$ and $0<|\mathit{q}|<1$.

Corollary 7.

Suppose κ an arbitrary number and $\mathrm{\Im },\mathrm{\wp },\mathit{P},\mathit{Q}$ to be positive integers such that $0\le \mathrm{\wp }\le \mathit{P}$, $0\le \mathrm{\Im }\le \mathit{Q};$ then the subsequent bilinear q-generating relation follows:

$\sum _{\mathit{n}=0}^{\mathrm{\infty }}{G}_{P+1,Q}^{\mathrm{\Im },\mathrm{\wp }+1}[\mathit{y};\mathit{q}\left|\begin{array}{l}1-\mathit{\kappa }-\mathit{n},a_1,\dots ,a_P\\ b_1,\dots ,b_Q\end{array}\right] \frac{t^n}{(\mathit{q};\mathit{q}{)}_n}$

(4.3) $=\frac{1}{(1-\mathit{t}{)}^{(\mathit{\kappa })}}{G}_{P+1,Q}^{\mathrm{\Im },\mathrm{\wp }+1}[\frac{y}{(1-\mathit{t}{\mathit{q}}^{\mathit{\kappa }})};\mathit{q}\left|\begin{array}{l}1-\mathit{\kappa },a_1,a_P\\ b_1,b_Q\end{array}\right] ,$(4.3)

where $|\mathit{t}|<1,0<|\mathit{q}|<1$.

Again, if we set r = 1, $P_1=\mathit{P};Q_1=\mathit{Q}$; ϱ = 1 and take (1.16(1.16) $\begin{array}{l}{G}_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{x};\mathit{q}\left|\begin{array}{l}{a}_1,a_2,\dots ,a_P\\ b_1,b_2,\dots ,b_Q\end{array}\right] \\ & =\frac{1}{2\mathit{\pi i}}{\int }_{\mathit{C}}{\psi }^{\star \star }(\mathit{s};\mathit{q})x^s{d}_q\mathit{s},\end{array}$(1.16) ) into considered, then Theorem 3 to 5 gives Corollaries 8 to 10, correspondingly.

Corollary 8.

If $\mathrm{\Re }(\mathit{\sigma })>0$, $|\mathit{arg}\mathit{\mu }|<\mathit{\pi }$ and assume $\mathrm{\Im },\mathrm{\wp },\mathit{P},\mathit{Q}$ to be positive integers; $0\le \mathrm{\wp }\le \mathit{P}$, $0\le \mathrm{\Im }\le \mathit{Q};$ in addition, γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, we have the following integral

$\begin{array}{l}\frac{\mathit{G}(\mathit{q})}{1-\mathit{q}}{\int }_0^1{y}^{\mathit{\sigma }-1}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\tau y};\mathit{q})E_q(\mathit{qy})\\ & & G_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{\mu }{y}^{-1};\mathit{q}\left|\begin{array}{l}{a}_1,\dots ,a_P\\ b_1,\dots ,b_Q\end{array}\right] d_q\mathit{y}\end{array}$

$\begin{array}{l}=\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{G}_{P,Q+1}^{\mathrm{\Im }+1,\mathrm{\wp }}[\mathit{\mu };\mathit{q}\left|\begin{array}{l}{a}_1,a_P\\ \mathit{\sigma }+\mathit{\delta },b_1,b_Q\end{array}\right] .\end{array}$

Corollary 9.

If $\mathrm{\Re }(\mathit{\sigma })>0$, $\mathrm{\Re }(\mathit{\rho })>0$, $\mathrm{\Re }(\mathit{\rho }-\mathit{\sigma })>0$, $|\mathit{arg}\mathit{\mu }|<\mathit{\pi }$, assume $\mathrm{\Im },\mathrm{\wp },\mathit{P},\mathit{Q}$ to be positive integers; $0\le \mathrm{\wp }\le \mathit{P}$, $0\le \mathrm{\Im }\le \mathit{Q};$ and γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, we have the proceeding integral

$\begin{array}{l}\frac{\mathit{G}(\mathit{q})}{1-\mathit{q}}{\int }_0^1{y}^{\mathit{\sigma }-1}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\tau y};\mathit{q})(1-\mathit{qy}{)}_{\mathit{\rho }-\mathit{\sigma }-1}\\ & G_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{\mu }{y}^{-1};\mathit{q}\left|\begin{array}{l}{a}_1,\dots ,a_P\\ b_1,\dots ,b_Q\end{array}\right] d_q\mathit{y}\end{array}$

$\begin{array}{l}=\mathit{G}(\mathit{\rho }-\mathit{\sigma })\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }\\ & G_{P+1,Q+1}^{\mathrm{\Im }+1,\mathrm{\wp }}[\mathit{\mu };\mathit{q}\left|\begin{array}{l}\mathit{\delta }+\mathit{\rho },a_1,a_P\\ \mathit{\sigma }+\mathit{\delta },b_1,b_Q\end{array}\right] .\end{array}$

Corollary 10.

If $\mathrm{\Re }(\mathit{\sigma })>0$, $|\mathit{arg}\mathit{\mu }|<\mathit{\pi }$, assume $\mathrm{\Im },\mathrm{\wp },\mathit{P},\mathit{Q}$ to be positive integers; $0\le \mathrm{\wp }\le \mathit{P}$, $0\le \mathrm{\Im }\le \mathit{Q};$ γ to be any arbitrary positive integer. If ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$ is an arbitrary bounded sequence, we have the subsequent integral

$\frac{1}{2\mathit{\pi w}}{\int }_C^{}{y}^{-\mathit{\sigma }}{f}_{\mathit{n},\mathit{\gamma }}(\mathit{\tau y};\mathit{q})e_q(\mathit{y})G_{P,Q}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{\mu y};\mathit{q}\left|\begin{array}{l}{a}_1,\dots ,a_P\\ b_1,\dots ,b_Q\end{array}\right] d_q\mathit{y}$

$=\mathit{G}(\mathit{q})\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{\tau }^{\delta }{G}_{P+1,Q}^{\mathrm{\Im },\mathrm{\wp }}[\mathit{\mu };\mathit{q}\left|\begin{array}{l}{a}_1,a_P,\mathit{\sigma }+\mathit{\delta }\\ b_1,b_Q\end{array}\right] .$

Furthermore, it is worth noting that in light of subsequent limiting cases:

(4.4) $\underset{q\to 1^{-}}{lim}{\mathrm{\Gamma }}_q(\mathit{\varpi })=\mathrm{\Gamma }(\mathit{\varpi })\mathit{and}\underset{q\to 1^{-}}{lim}\frac{(q^{\varpi };\mathit{q}{)}_n}{(1-\mathit{q}{)}^n}=(\mathit{\varpi }{)}_n,$(4.4)

where

(4.5) $(\mathit{\varpi }{)}_n=\mathit{\varpi }(\mathit{\varpi }+1)(\mathit{\varpi }+\mathit{n}-1),$(4.5)

the q-generating relation (2.1(2.1) $\begin{array}{ll}& \times I_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}[\frac{y}{(1-\mathit{t}{\mathit{q}}^{\mathit{\kappa }+\mathit{\gamma \delta }}{)}^{(\mathit{\lambda })}}; \\ & \mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{\gamma \delta },\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] ,\end{array}$(2.1) ) of Theorem 1 provides the q-extension of the present finding acknowledged to Raina (1976) (Equation (2.1), p. 301).

5. Conclusion and future works

Our main conclusion (Theorem 1) can be used to deduce expected bilinear q-generating relations for the product of orthogonal q-polynomials and the I_q-function by giving appropriate special values to the sequence ${\left\{{\mathit{S}}_{\mathit{\delta },\mathit{q}}\right\}}_{\mathit{\delta }=0}^{\mathrm{\infty }}$. We use the illustration below to demonstrate this.

(i) Setting γ = 1 and

(5.1) $S_{\mathit{\delta },\mathit{q}}=\frac{(-1{)}^{\delta }{q}^{\mathit{\delta }(\mathit{\delta }-1)}(\mathit{\vartheta q};\mathit{q}{)}_n}{(\mathit{\vartheta q};\mathit{q}{)}_{\delta }(\mathit{q};\mathit{q}{)}_n},$(5.1)

we discover from (1.10(1.10) $f_{\mathit{n},\mathit{\gamma }}(\mathit{x};\mathit{q})=\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{x}^{\delta },(\mathit{n}\in \mathbb{N}),$(1.10) ) that

$f_{\mathit{n},1}(\mathit{x};\mathit{q})=L_n^{(\vartheta )}(\mathit{x};\mathit{q}),$

where the q-Laguerre polynomial Srivastava & Agarwal (1989) is denoted by

(5.2) $L_n^{(\vartheta )}(\mathit{x};\mathit{q})=\frac{(\vartheta q;q{)}_n}{(\mathit{q};\mathit{q}{)}_n}{}_1{\mathrm{\Phi }}_1[\begin{array}{l}{q}^{-\mathit{n}};\\ \mathit{\vartheta q};\end{array}-\mathit{x}{q}^n].$(5.2)

Theorem 1 thus produces the q-generating connection between the I_q-function and the q-Laguerre polynomial as follows in light of the aforementioned relations:

$\begin{array}{ll}& \sum _{\mathit{n}=0}^{\mathrm{\infty }}{L}_n^{(\vartheta )}(\mathit{\mu }\mathit{x};\mathit{q})I_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}\\ & [\mathit{y};\mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{n},\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] \frac{t^n}{(\mathit{q};\mathit{q}{)}_n}\end{array}$

$=\frac{(\vartheta q;q{)}_n}{(1-\mathit{t}{)}^{(\mathit{\kappa })}(\mathit{q};\mathit{q}{)}_n}\sum _{\mathit{\delta }=0}^{\mathrm{\infty }}\frac{(-1{)}^{\delta }{q}^{\mathit{\delta }(\mathit{\delta }-1)}}{(\mathit{q};\mathit{q}{)}_{\delta }(\mathit{\vartheta q};\mathit{q}{)}_{\delta }}\frac{(\mu xt{)}^{\delta }}{(1-\mathit{t}{q}^{\kappa }{)}^{\mathit{\delta }}}$

(5.3) $\begin{array}{l}\times I_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}[\frac{y}{(1-\mathit{t}{\mathit{q}}^{\mathit{\kappa }+\mathit{\delta }}{)}^{\mathit{\lambda }}}; \\ & \mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{\delta },\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] .\end{array}$(5.3)

(ii) Putting γ = 1 and

(5.4) $S_{\mathit{\delta },\mathit{q}}=\frac{(\vartheta q;q{)}_n(\mathit{\vartheta \xi }{q}^{\mathit{n}+1};\mathit{q}{)}_{\delta }(-1{)}^{\delta }{q}^{\mathit{\delta }(\mathit{\delta }+1)/2-\mathit{n\delta }}}{(\mathit{\vartheta q};\mathit{q}{)}_{\delta }(\mathit{q};\mathit{q}{)}_n},$(5.4)

we get from (1.10(1.10) $f_{\mathit{n},\mathit{\gamma }}(\mathit{x};\mathit{q})=\sum _{\mathit{\delta }=0}^{[\mathit{n}/\gamma ]}{\left[\begin{array}{l}\mathit{n}\\ \mathit{\gamma }\mathit{\delta }\end{array}\right]}_q{S}_{\mathit{\delta },\mathit{q}}{x}^{\delta },(\mathit{n}\in \mathbb{N}),$(1.10) ) that

$f_{\mathit{n},1}(\mathit{x};\mathit{q})=P_n^{(\vartheta ,\xi )}(\mathit{x};\mathit{q}),$

where the q-Jacobi polynomial (Mathai & Saxena; 1978) is represented by the symbol as

(5.5) $P_n^{(\vartheta ,\xi )}(\mathit{x};\mathit{q})=\frac{(\vartheta q;q{)}_n}{(\mathit{q};\mathit{q}{)}_n}{}_2{\mathrm{\Phi }}_1\left[\begin{array}{l}{q}^{-\mathit{n}},\mathit{\vartheta \xi }{q}^{\mathit{n}+1};\\ \mathit{\vartheta q};\end{array}\mathit{xq}\right].$(5.5)

Theorem 1 then gives the q-generating connection between the I_q-function and q-Jacobi polynomial, precisely

$\begin{array}{ll}& \sum _{\mathit{n}=0}^{\mathrm{\infty }}{P}_n^{(\vartheta ,\xi )}(\mathit{\mu }\mathit{x};\mathit{q})I_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}\\ & [\mathit{y};\mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{n},\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] \frac{t^n}{(\mathit{q};\mathit{q}{)}_n}\end{array}$

$=\frac{(\vartheta q;q{)}_n}{(1-\mathit{t}{)}^{(\mathit{\kappa })}(\mathit{q};\mathit{q}{)}_n}\sum _{\mathit{\delta }=0}^{\mathrm{\infty }}\frac{(\mathit{\vartheta \xi }{\mathit{q}}^{\mathit{n}+1})(-1{)}^{\delta }{q}^{\mathit{\delta }(\mathit{\delta }-1)/2-\mathit{n\delta }}}{(\mathit{q};\mathit{q}{)}_{\delta }(\mathit{\vartheta q};\mathit{q}{)}_{\delta }}\frac{(\mu xt{)}^{\delta }}{(1-\mathit{t}{q}^{\kappa }{)}^{\mathit{\delta }}}$

(5.6) $\begin{array}{ll}& \times I_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}[\frac{y}{(1-\mathit{t}{\mathit{q}}^{\mathit{\kappa }+\mathit{\delta }}{)}^{\mathit{\lambda }}}; \mathit{q}\\ & |\begin{array}{l}(1-\mathit{\kappa }-\mathit{\delta },\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array} ].\end{array}$(5.6)

The research monograph by Koekoek et al. (2010) and Srivastava & Agarwal (1989) provides a thorough description of several hypergeometric orthogonal q-polynomials. The mathematical descriptions of the q-Jacobi polynomials and q-Laguerre provided by the Equations (5.5)(5.5) $P_n^{(\vartheta ,\xi )}(\mathit{x};\mathit{q})=\frac{(\vartheta q;q{)}_n}{(\mathit{q};\mathit{q}{)}_n}{}_2{\mathrm{\Phi }}_1\left[\begin{array}{l}{q}^{-\mathit{n}},\mathit{\vartheta \xi }{q}^{\mathit{n}+1};\\ \mathit{\vartheta q};\end{array}\mathit{xq}\right].$(5.5) and (5.2(5.2) $L_n^{(\vartheta )}(\mathit{x};\mathit{q})=\frac{(\vartheta q;q{)}_n}{(\mathit{q};\mathit{q}{)}_n}{}_1{\mathrm{\Phi }}_1[\begin{array}{l}{q}^{-\mathit{n}};\\ \mathit{\vartheta q};\end{array}-\mathit{x}{q}^n].$(5.2) ), respectively, differ slightly from those provided in the seminal work (Koekoek et al., 2010). Consequently, by taking into account the definitions of the q-polynomials provided in Koekoek et al. (2010), one might arrive to results of a similar nature.

We conclude by noting that the general nature of the q-generating relation (2.1(2.1) $\begin{array}{ll}& \times I_{P_i+1,Q_i}^{\mathrm{\Im },\mathrm{\wp }+1}[\frac{y}{(1-\mathit{t}{\mathit{q}}^{\mathit{\kappa }+\mathit{\gamma \delta }}{)}^{(\mathit{\lambda })}}; \\ & \mathit{q}\left|\begin{array}{l}(1-\mathit{\kappa }-\mathit{\gamma \delta },\mathit{\lambda }),(a_{\mathit{j}},{\vartheta }_{\mathit{j}}{)}_{1,\mathrm{\wp }},(a_{\mathit{ji}},{\vartheta }_{\mathit{ji}}{)}_{\mathrm{\wp }+1,{\mathit{P}}_{\mathit{i}}}\\ (b_{\mathit{j}},{\xi }_{\mathit{j}}{)}_{1,\mathrm{\Im }},(b_{\mathit{ji}},{\xi }_{\mathit{ji}}{)}_{\mathrm{\Im }+1,{\mathit{Q}}_{\mathit{i}}}\end{array}\right] ,\end{array}$(2.1) ) will lead to a number of generating relationships for the product of the q-analogue of the Fox’s H-functions and the orthogonal q-polynomials by appropriately assigning values to the sequence ${\left\{{\mathit{S}}_{\mathit{k},\mathit{q}}\right\}}_{\mathit{k}=0}^{\mathrm{\infty }}$.

Disclosure statement

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