On q-Horn Hypergeometric Functions H₆ and H₇

Abstract

This work aims to construct various properties for basic Horn functions ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$ under conditions on the numerator and denominator parameters, such as several q-contiguous function relations, q-differential relations, and q-differential equations. Special cases of our main results are also demonstrated.

1. Introduction

The theory of quantum calculus or q-calculus has a wide range of applications in several fields of mathematics, engineering, physics, partition theory, number theory, Lie theory, combinatorial analysis, integral transforms, fractional calculus, and quantum theory, etc. Several authors have contributed works on this subject: (see for example, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]). Sahai and Verma [16,17], Guo and Schlosser [18], Verma and Sahai [19], Verma and Yadav [20], Wei and Gong [21] studied and investigated some properties for various families of the q-hypergeometric, q-Appell and q-Lauricella series by applying operators of quantum calculus. In [22], Ernst obtained the q-analogues of Srivastava’s triple hypergeometric functions. Araci et al. [23,24,25] studied some properties of q-Bernoulli, q-Euler, and q-Frobenius–Euler polynomials based on q-exponential functions. Duran et al. [26,27] investigated q-Bernoulli, q-Genocchi, and q-Euler polynomials and introduced the q-analogues of familiar earlier formulas. In [28], Pathan et al. derived the certain new formulas for the classical Horn’s hypergeometric functions $H_1$, $H_2,\dots H_{11}$. In [29,30], the author introduced the $(p,q)$-Humbert, $(p,q)$-Bessel functions. In [31], Shehata has earlier investigated the results for basic Horn hypergeometric functions $H_3$ and $H_4$. The reason of interest for this family of basic Horn’s hypergeometric functions is due to their intrinsic mathematical physics importance.

Throughout this work, we assume that the expression $0<\left|q\right|<1$, $q\in \mathbb{C}$, we use the following abbreviated notations: let $\mathbb{C}$, $\mathbb{N}$, and ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}=\{0,1,2,\dots \}$ be the sets of complex, natural and non-negative numbers.

The q-shifted factorial (q-Pochhammer symbol) ${(\eta ;q)}_m$ is defined by (see [32]):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {(\eta ;q)}_m& =\left\{\begin{array}{cc}(1-\eta )(1-\eta q)\dots (1-\eta q^{m-1}),\hfill & m\in \mathbb{N},\eta \in \mathbb{C}\backslash \{1,q^{-1},q^{-2},\dots ,q^{1-m}\};\hfill \\ 1,\hfill & m=0,\eta \in \mathbb{C},\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(1)

for negative subscripts,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {(\eta ;q)}_{-m}& =\frac{1}{{(\eta q^{-m};q)}_m}=\frac{1}{(1-\eta q^{-1})(1-\eta q^{-2})\dots (1-\eta q^{-m})}=\frac{{(-{\eta }^{-1}q)}^m{q}^{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}}{{({\eta }^{-1}q;q)}_m},\hfill \\ & \eta \ne q^{\pm 1},q^{\pm 2},q^{\pm 3},q^{\pm 4},\dots ,q^{\pm m},m=1,2,3,\dots .\hfill \end{array}\end{array}$$

(2)

For $\eta \in \mathbb{C}$, the q-number or q-bracket ${\left[\eta \right]}_q$ is defined as (see [32])

$$\begin{array}{c}\hfill {\left[\eta \right]}_q=\left\{\begin{array}{cc}\frac{1-q^{\eta }}{1-q},\hfill & \eta \in \mathbb{C};\hfill \\ 1,\hfill & \eta =0.\hfill \end{array}\right..\end{array}$$

(3)

Let m be a non-negative integer number, the q-number ${\left[m\right]}_q$ and q-factorial ${\left[m\right]}_q!$ are defined by (see [13,27])

$$\begin{array}{c}\hfill {\left[m\right]}_q=\left\{\begin{array}{cc}\frac{1-q^m}{1-q},\hfill & m\in \mathbb{N};\hfill \\ 1,\hfill & m=0\hfill \end{array}\right.\end{array}$$

(4)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left[m\right]}_q!=\left\{\begin{array}{cc}\prod _{r=1}^m{\left[r\right]}_q={\left[m\right]}_q{[m-1]}_q\dots {\left[2\right]}_q{\left[1\right]}_q=\frac{{(q;q)}_m}{{(1-q)}^m},\hfill & m\in \mathbb{N};\hfill \\ 1,\hfill & m=0.\hfill \end{array}\right.\end{array}\end{array}$$

(5)

We recall the notations for $m,k\in \mathbb{N}$, $\eta \in \mathbb{C}$, which are used in the sequel (see [32])

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {(\eta ;q)}_m=& (1-\eta ){(\eta q;q)}_{m-1}\hfill \\ \hfill =& (1-\eta q^{m-1}){(\eta ;q)}_{m-1},\hfill \end{array}\end{array}$$

(6)

$$\begin{array}{cc}\hfill {(\eta q;q)}_m=& \frac{1-\eta q^m}{1-\eta }{(\eta ;q)}_m\hfill \\ \hfill =& (1-\eta q^m){(\eta q;q)}_{m-1},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {(\eta q^{-1};q)}_m=& \frac{1-\eta q^{-1}}{1-\eta q^{m-1}}{(\eta ;q)}_m\hfill \\ \hfill =& (1-\eta q^{-1}){(\eta ;q)}_{m-1}\hfill \end{array}$$

(8)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {(\eta ;q)}_{m+k}& ={(\eta ;q)}_m{(\eta q^m;q)}_k\hfill \\ & ={(\eta ;q)}_k{(\eta q^k;q)}_m.\hfill \end{array}\end{array}$$

(9)

The q-difference operator $D_{z,q}$ of a function f at $z\ne 0\in \mathbb{C}$ is defined as (see [33]).

$$\begin{array}{c}\hfill D_{z,q}f\left(z\right)=\frac{f\left(z\right)-f\left(qz\right)}{(1-q)z},z\ne 0,\end{array}$$

(10)

and $D_{z,q}f\left(0\right)=\frac{df\left(z\right)}{dz}{|}_{z=0}=f^{\prime }\left(0\right)$, provided that f is differentiable at $z=0$, and defined differential operator ${\theta }_{z,q}=z{D}_{z,q}$.

For $0<\left|q\right|<1$, $q\in \mathbb{C}$ and (1), we give the definition of the basic Horn functions ${\mathbf{H}}_6$, ${\mathbf{H}}_7$, ${\mathbb{H}}_6$ and ${\mathbb{H}}_7$ as follows

$$\begin{array}{c}\hfill {\mathbf{H}}_6(\alpha ;\beta ;q,x,y)=\sum _{r,s=0}^{\infty }\frac{{(\alpha ;q)}_{2r+s}}{{(\beta ;q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s,\beta \ne 1,q^{-1},q^{-2},\dots ,\end{array}$$

(11)

$$\begin{array}{c}\hfill {\mathbf{H}}_7(\alpha ;\beta ,\gamma ;q,x,y)=\sum _{r,s=0}^{\infty }\frac{{(\alpha ;q)}_{2r+s}}{{(\beta ;q)}_r{(\gamma ;q)}_s{(q;q)}_r{(q;q)}_s}{x}^r{y}^s,\beta ,\gamma \ne 1,q^{-1},q^{-2},\dots ,\end{array}$$

(12)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,y)=\sum _{r,s=0}^{\infty }\frac{{(q^{\alpha };q)}_{2r+s}}{{(q^{\beta };q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s,q^{\beta }\ne 1,q^{-1},q^{-2},\dots \end{array}\end{array}$$

(13)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,y)=\sum _{r,s=0}^{\infty }\frac{{(q^{\alpha };q)}_{2r+s}}{{(q^{\beta };q)}_r{(q^{\gamma };q)}_s{(q;q)}_r{(q;q)}_s}{x}^r{y}^s,q^{\beta },q^{\gamma }\ne 1,q^{-1},q^{-2},\dots .\end{array}\end{array}$$

(14)

Note that for $q⟶1^{-}$, the basic Horn functions ${\mathbf{H}}_6(\alpha ;\beta ;q,x,y)$ and ${\mathbf{H}}_7(\alpha ;\beta ,\gamma ;q,x,y)$ reduces to Horn functions ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$ [34].

So as to simplify the following notations, we are writing ${\mathbf{H}}_6$ for the function ${\mathbf{H}}_6(\alpha ;\beta ;q,x,y)$, ${\mathbf{H}}_6\left(\alpha q^{\pm 1}\right)$ for the function ${\mathbf{H}}_6(\alpha q^{\pm 1};\beta ;q,x,y)$, ${\mathbf{H}}_6\left(\beta q^{\pm 1}\right)$ for the function ${\mathbf{H}}_6(\alpha ;\beta q^{\pm 1};q,x,y)$, ${\mathbf{H}}_7$ for the function ${\mathbf{H}}_7(\alpha ;\beta ,\gamma ;q,x,y)$, ${\mathbf{H}}_7\left(\gamma q^{\pm 1}\right)$ stands for the function ${\mathbf{H}}_7(\alpha ;\beta ,\gamma q^{\pm 1};q,x,y)$, ${\mathbb{H}}_6$ for the function ${\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,y)$, ${\mathbb{H}}_6\left(q^{\alpha \pm 1}\right)$ for the function ${\mathbb{H}}_6(q^{\alpha \pm 1};q^{\beta };q,x,y)$, ${\mathbb{H}}_6\left(q^{\beta \pm 1}\right)$ for the function ${\mathbb{H}}_6(q^{\alpha };q^{\beta \pm 1};q,x,y)$, and ${\mathbb{H}}_7$ for the function ${\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,y)$, … etc.

Our present study is primarily motivated by the former works in quantum calculus. We express a family of extended forms for the functions ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$. In Section 2, the q-contiguous relations, q-differential relations, and q-differential equations for the functions ${\mathbf{H}}_6$, ${\mathbb{H}}_6$, ${\mathbf{H}}_7$ and ${\mathbb{H}}_7$ under conditions on the numerator and denominator parameters are derived. Finally, some concluding remarks for the functions ${\mathbf{H}}_6$, ${\mathbb{H}}_6$, ${\mathbf{H}}_7$ and ${\mathbb{H}}_7$ are determined in Section 3.

2. Main Result

Here we establish various properties as well as the q-contiguous function relations and q-differential equations for the functions ${\mathbf{H}}_6$ with $\beta \ne 1,q^{-1},q^{-2},\dots$, ${\mathbf{H}}_7$ with $\alpha ,\beta \ne 1,q^{-1},q^{-2},\dots$, ${\mathbb{H}}_6$ with $q^{\beta }\ne 1,q^{-1},q^{-2},\dots$, and ${\mathbb{H}}_7$ with $q^{\beta },q^{\gamma }\ne 1,q^{-1},q^{-2},\dots$ which will be useful in the sequel.

Theorem 1.

The relations of the functions ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$ with the numerator parameter α

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6\left(\alpha q\right)={\mathbf{H}}_6+\frac{\alpha x(1-\alpha q)}{1-\beta }{\mathbf{H}}_6(\alpha q^2;\beta q;q,x,y)\hfill \\ \hfill +& \frac{\alpha xq(1-\alpha q)}{1-\beta }{\mathbf{H}}_6(\alpha q^2;\beta q;q,xq,y)+\frac{\alpha y}{1-\beta }{\mathbf{H}}_6(\alpha q;\beta q;x{q}^2,y),\beta \ne 1,\hfill \end{array}\end{array}$$

(15)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6\left(\alpha q\right)={\mathbf{H}}_6+\frac{\alpha y}{1-\beta }{\mathbf{H}}_6(\alpha q;\beta q;q,x,y)+\frac{\alpha xq(1-\alpha q)}{1-\beta }{\mathbf{H}}_6(\alpha q^2;\beta q;q,xq,yq)\hfill \\ \hfill +& \frac{\alpha x(1-\alpha q)}{1-\beta }{\mathbf{H}}_6(\alpha q^2;\beta q;q,x,yq),\beta \ne 1,\hfill \end{array}\end{array}$$

(16)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7\left(\alpha q\right)={\mathbf{H}}_7+\frac{\alpha x(1-\alpha q)}{1-\beta }{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma ;q,x,y)+\frac{\alpha y}{1-\gamma }{\mathbf{H}}_7(\alpha q;\beta ,\gamma q;q,x{q}^2,y)\hfill \\ \hfill +& \frac{\alpha xq(1-\alpha q)}{1-\beta }{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma ;q,xq,y),\beta ,\gamma \ne 1\hfill \end{array}\end{array}$$

(17)

and

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7\left(\alpha q\right)={\mathbf{H}}_7+\frac{\alpha y}{1-\gamma }{\mathbf{H}}_7(\alpha q;b,\gamma q;q,x,y)+\frac{\alpha xq(1-\alpha q)}{1-\beta }{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma ;q,xq,yq)\hfill \\ \hfill +& \frac{\alpha x(1-\alpha q)}{1-\beta }{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma ;q,x,yq),\beta ,\gamma \ne 1.\hfill \end{array}\end{array}$$

(18)

Proof. 

To prove (15). Replacing $\alpha$ by $\alpha q$ in (11) and using (6)–(7), we get

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6\left(\alpha q\right)-{\mathbf{H}}_6=\sum _{r,s=0}^{\infty }\frac{1}{{(\beta ;q)}_{r+s}{(q;q)}_r{(q;q)}_s}[{(\alpha q;q)}_{2r+s}-{(\alpha ;q)}_{2r+s}]x^r{y}^s\hfill \\ & =\sum _{r,s=0}^{\infty }\frac{\alpha (1-q^{2r+s}){(\alpha q;q)}_{2r+s-1}}{{(\beta ;q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \sum _{r=1,s=0}^{\infty }\frac{\alpha {(\alpha q;q)}_{2r+s-1}}{{(\beta ;q)}_{r+s}{(q;q)}_{r-1}{(q;q)}_s}{x}^r{y}^s+\sum _{r=1,s=0}^{\infty }\frac{\alpha q^r{(\alpha q;q)}_{2r+s-1}}{{(\beta ;q)}_{r+s}{(q;q)}_{r-1}{(q;q)}_s}{x}^r{y}^s\hfill \\ & +\sum _{r=0,s=1}^{\infty }\frac{\alpha q^{2r}{(\alpha q;q)}_{2r+s-1}}{{(\beta ;q)}_{r+s}{(q;q)}_r{(q;q)}_{s-1}}{x}^r{y}^s\hfill \\ \hfill =& \frac{\alpha x(1-\alpha q)}{1-\beta }{\mathbf{H}}_6(\alpha q^2;\beta q;q,x,y)+\frac{\alpha xq(1-\alpha q)}{1-\beta }{\mathbf{H}}_6(\alpha q^2;\beta q;q,xq,y)\hfill \\ & +\frac{\alpha y}{1-\beta }{\mathbf{H}}_6(\alpha q;\beta q;q,x{q}^2,y),\beta \ne 1.\hfill \end{array}\end{array}$$

Similarly, by using the relation

$$1-q^{2r+s}=1-q^s+q^s(1-q^r)+q^{r+s}(1-q^r),$$

we get (16). Equations (17) and (18) can be proven on as same lines as the Equation (15). □

Theorem 2.

The functions ${\mathbf{H}}_7$ and ${\mathbf{H}}_6$ satisfy the q-derivative equations

$$\begin{array}{c}\hfill D_{x,q}^r{\mathbf{H}}_7=\frac{{(\alpha ;q)}_{2r}}{{(\beta ;q)}_r{(1-q)}^r}{\mathbf{H}}_7(\alpha q^{2r};\beta q^r,\gamma ;q,x,y),D_{x,q}=\frac{\partial }{\partial x},\end{array}$$

(19)

$$\begin{array}{c}\hfill D_{y,q}^r{\mathbf{H}}_7(\alpha ;\beta ,\gamma ;q,x,y)=\frac{{(\alpha ;q)}_r}{{(\gamma ;q)}_r{(1-q)}^r}{\mathbf{H}}_7(\alpha q^r;\beta ,\gamma q^r;q,x,y),D_{y,q}=\frac{\partial }{\partial y},\end{array}$$

(20)

$$\begin{array}{c}\hfill D_{x,q}^r{\mathbf{H}}_6=\frac{{(\alpha ;q)}_{2r}}{{(\beta ;q)}_r{(1-q)}^r}{\mathbf{H}}_6(\alpha q^{2r};\beta q^r;q,x,y)\end{array}$$

(21)

and

$$\begin{array}{c}\hfill D_{y,q}^r{\mathbf{H}}_6=\frac{{(\alpha ;q)}_r}{{(\beta ;q)}_r{(1-q)}^r}{\mathbf{H}}_6(\alpha q^r;\beta q^r;q,x,y).\end{array}$$

(22)

Proof. 

From (12) and (10), we get q-derivatives of ${\mathbf{H}}_7$ with respect to x and y as follows

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{x,q}{\mathbf{H}}_7=& \sum _{r,s=0}^{\infty }\left[\frac{1-q^r}{1-q}\right]\frac{{(\alpha ;q)}_{2r+s}}{{(\beta ;q)}_r{(\gamma ;q)}_s{(q;q)}_{r-1}{(q;q)}_s}{x}^{r-1}{y}^s\hfill \\ \hfill =& \sum _{r=1,s=0}^{\infty }\left[\frac{1}{1-q}\right]\frac{{(\alpha ;q)}_{2r+s}}{{(\beta ;q)}_r{(\gamma ;q)}_s{(q;q)}_{r-1}{(q;q)}_s}{x}^{r-1}{y}^s\hfill \\ \hfill =& \sum _{r,s=0}^{\infty }\left[\frac{1}{1-q}\right]\frac{{(\alpha ;q)}_{2m+n+2}}{{(\beta ;q)}_{m+1}{(\gamma ;q)}_n{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \frac{(1-\alpha )(1-\alpha q)}{(1-\beta )(1-q)}{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma ;q,x,y)\hfill \end{array}\end{array}$$

(23)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{y,q}{\mathbf{H}}_7=& \sum _{r,s=0}^{\infty }\left[\frac{1-q^s}{1-q}\right]\frac{{(\alpha ;q)}_{2r+s}}{{(\beta ;q)}_r{(\gamma ;q)}_s{(q;q)}_r{(q;q)}_s}{x}^r{y}^{s-1}\hfill \\ \hfill =& \sum _{r=0,s=1}^{\infty }\left[\frac{1}{1-q}\right]\frac{{(\alpha ;q)}_{2r+s}}{{(\beta ;q)}_r{(\gamma ;q)}_s{(q;q)}_r{(q;q)}_{s-1}}{x}^r{y}^{s-1}\hfill \\ \hfill =& \sum _{r,s=0}^{\infty }\left[\frac{1}{1-q}\right]\frac{{(\alpha ;q)}_{2r+s+1}}{{(\beta ;q)}_m{(\gamma ;q)}_{n+1}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \frac{(1-\alpha )}{(1-\gamma )(1-q)}{\mathbf{H}}_7(\alpha q;\beta ,\gamma q;q,x,y).\hfill \end{array}\end{array}$$

(24)

Iterating this technique r times on ${\mathbf{H}}_7$, we obtain (19) and (20). In the same way, the Equations (21) and (22) can be proven. □

Theorem 3.

The functions ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$ satisfy the q-derivative formulas:

$$\begin{array}{c}\hfill \begin{array}{cc}& [\alpha {\theta }_{x,q}+\frac{1-\alpha }{1-q}]{\mathbf{H}}_6+\alpha {\theta }_{x,q}{\mathbf{H}}_6\left(xq\right)+\alpha {\theta }_{y,q}{\mathbf{H}}_6\left(x{q}^2\right)=\frac{1-\alpha }{1-q}{\mathbf{H}}_6\left(\alpha q\right),\hfill \end{array}\end{array}$$

(25)

where ${\theta }_{x,q}=x{D}_{x,q}$ and ${\theta }_{y,q}=y{D}_{y,q}$ are differential operators,

$$\begin{array}{c}\hfill \begin{array}{cc}& [\alpha {\theta }_{y,q}+\frac{1-\alpha }{1-q}]{\mathbf{H}}_6+\alpha {\theta }_{x,q}{\mathbf{H}}_6\left(yq\right)+\alpha {\theta }_{x,q}{\mathbf{H}}_6(xq,yq)=\frac{1-\alpha }{1-q}{\mathbf{H}}_6\left(\alpha q\right),\hfill \end{array}\end{array}$$

(26)

$$\begin{array}{c}\hfill \begin{array}{cc}& [\alpha q^{-1}{\theta }_{x,q}+\frac{1-\alpha q^{-1}}{1-q}]{\mathbf{H}}_6\left(\alpha q^{-1}\right)+\alpha q^{-1}{\theta }_{y,q}{\mathbf{H}}_6(\alpha q^{-1},xq,yq)\hfill \\ & +\alpha q^{-1}{\theta }_{y,q}{\mathbf{H}}_6(\alpha q^{-1},xq)=\frac{1-\alpha q^{-1}}{1-q}{\mathbf{H}}_6,\hfill \end{array}\end{array}$$

(27)

$$\begin{array}{c}\hfill \begin{array}{cc}& [\alpha q^{-1}{\theta }_{y,q}+\frac{1-\alpha q^{-1}}{1-q}]{\mathbf{H}}_6\left(\alpha q^{-1}\right)+\alpha q^{-1}{\theta }_{x,q}{\mathbf{H}}_6(\alpha q^{-1},xq,yq)\hfill \\ & +\alpha q^{-1}{\theta }_{x,q}{\mathbf{H}}_6(\alpha q^{-1},yq)=\frac{1-\alpha q^{-1}}{1-q}{\mathbf{H}}_6,\hfill \end{array}\end{array}$$

(28)

$$\begin{array}{c}\hfill \begin{array}{cc}& [\alpha {\theta }_{x,q}+\frac{1-\alpha }{1-q}]{\mathbf{H}}_7+\alpha {\theta }_{x,q}{\mathbf{H}}_7\left(xq\right)+\alpha {\theta }_{y,q}{\mathbf{H}}_7\left(x{q}^2\right)=\frac{1-\alpha }{1-q}{\mathbf{H}}_7\left(\alpha q\right),\hfill \end{array}\end{array}$$

(29)

$$\begin{array}{c}\hfill \begin{array}{cc}& [\alpha {\theta }_{y,q}+\frac{1-\alpha }{1-q}]{\mathbf{H}}_7+\alpha {\theta }_{x,q}{\mathbf{H}}_7\left(yq\right)+\alpha {\theta }_{x,q}{\mathbf{H}}_7(xq,yq)=\frac{1-\alpha }{1-q}{\mathbf{H}}_7\left(\alpha q\right),\hfill \end{array}\end{array}$$

(30)

$$\begin{array}{c}\hfill \begin{array}{cc}& [\alpha q^{-1}{\theta }_{x,q}+\frac{1-\alpha q^{-1}}{1-q}]{\mathbf{H}}_7\left(\alpha q^{-1}\right)+\alpha q^{-1}{\theta }_{y,q}{\mathbf{H}}_7(\alpha q^{-1},xq)\hfill \\ & +\alpha q^{-1}{\theta }_{y,q}{\mathbf{H}}_7(\alpha q^{-1},xq,yq)=\frac{1-\alpha q^{-1}}{1-q}{\mathbf{H}}_7\hfill \end{array}\end{array}$$

(31)

and

$$\begin{array}{c}\hfill \begin{array}{cc}& [\alpha q^{-1}{\theta }_{y,q}+\frac{1-\alpha q^{-1}}{1-q}]{\mathbf{H}}_7\left(\alpha q^{-1}\right)+\alpha q^{-1}{\theta }_{x,q}{\mathbf{H}}_7(\alpha q^{-1},yq)\hfill \\ & +\alpha q^{-1}{\theta }_{x,q}{\mathbf{H}}_7(\alpha q^{-1},xq,yq)=\frac{1-\alpha q^{-1}}{1-q}{\mathbf{H}}_7.\hfill \end{array}\end{array}$$

(32)

Proof. 

Using (23) and (24) for ${\mathbf{H}}_6$, we get the results (25)–(28). Equations (29)–(32) would run a parallel to Equations (25)–(28). □

Theorem 4.

The relations of ${\mathbf{H}}_7$ and ${\mathbf{H}}_6$ with denominator parameters β and γ hold true

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7\left(\beta q^{-1}\right)={\mathbf{H}}_7+\frac{\beta x(1-\alpha )(1-\alpha q)}{(q-\beta )(1-\beta )}{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma ;q,x,y);\beta ,\beta q^{-1}\ne 1,\hfill \end{array}\end{array}$$

(33)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7\left(\gamma q^{-1}\right)={\mathbf{H}}_7+\frac{\gamma y(1-\alpha )}{(q-\gamma )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta ,\gamma q;q,x,y);\gamma ,\gamma q^{-1}\ne 1,\hfill \end{array}\end{array}$$

(34)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6\left(\beta q^{-1}\right)={\mathbf{H}}_6+\frac{\beta x(1-\alpha )(1-\alpha q)}{(q-\beta )(1-\beta )}{\mathbf{H}}_6(\alpha q^2;\beta q;q,x,y)\hfill \\ & +\frac{\beta (1-\alpha )y}{(q-\beta )(1-\beta )}{\mathbf{H}}_6(\alpha q;\beta q;q,xq,y),\beta ,\beta q^{-1}\ne 1\hfill \end{array}\end{array}$$

(35)

and

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6\left(\beta q^{-1}\right)={\mathbf{H}}_6+\frac{\beta y(1-\alpha )}{(q-\beta )(1-\beta )}{\mathbf{H}}_6(\alpha q;\beta q;q,x,y)\hfill \\ & +\frac{\beta x(1-\alpha )(1-\alpha q)}{(q-\beta )(1-\beta )}{\mathbf{H}}_6(\alpha q^2;\beta q;q,x,yq),\beta ,\beta q^{-1}\ne 1,\hfill \end{array}\end{array}$$

(36)

Proof. 

Using the identities (6), (8) and replacing $\beta$ by $\beta q^{-1}$ in (12), we get

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7\left(\beta q^{-1}\right)-{\mathbf{H}}_7=\sum _{r,s=0}^{\infty }\frac{{(\alpha ;q)}_{2r+s}}{{(\gamma ;q)}_n{(q;q)}_r{(q;q)}_s}\frac{{(\beta ;q)}_r-{(\beta q^{-1};q)}_r}{{(\beta q^{-1};q)}_r{(\beta ;q)}_r}{x}^r{y}^s\hfill \\ \hfill =& \sum _{r,s=0}^{\infty }\frac{\beta (1-q^r)}{q-\beta }\frac{{(\alpha ;q)}_{2r+s}}{{(\beta ;q)}_r{(\gamma ;q)}_s{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \frac{\beta x(1-\alpha )(1-\alpha q)}{(q-\beta )(1-\beta )}{\mathbf{H}}_7(\alpha q^2;\beta q;q,x,y),\beta ,\beta q^{-1}\ne 1.\hfill \end{array}\end{array}$$

The proof (34) is a very similar to those of Equation (33). Similarly, by using the relation $\frac{1-q^{r+s}}{q-\gamma }=\frac{1-q^r}{q-\gamma }+q^r\frac{1-q^s}{q-\gamma }$ and $\frac{1-q^{r+s}}{q-\gamma }=\frac{1-q^s}{q-\gamma }+q^s\frac{1-q^r}{q-\gamma }$, we get (35) and (36). □

Theorem 5.

The q-contiguous relations hold true for the denominator parameters β and γ of the functions ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$

$$\begin{array}{cc}& {\mathbf{H}}_6\left(\beta q^{-1}\right)=\frac{\beta }{\beta -q}{\mathbf{H}}_6(\alpha ;\beta ;q,xq,yq)-\frac{q}{\beta -q}{\mathbf{H}}_6(\alpha ;\beta ;q,x,y),\beta \ne p,\end{array}$$

(37)

$$\begin{array}{c}\hfill {\mathbf{H}}_7\left(\beta q^{-1}\right)=\frac{\beta }{\beta -q}{\mathbf{H}}_7(\alpha ;\beta ,\gamma ;q,xq,y)-\frac{q}{\beta -q}{\mathbf{H}}_7(\alpha ;\beta ,\gamma ;q,x,y),\beta \ne p\end{array}$$

(38)

and

$$\begin{array}{c}\hfill {\mathbf{H}}_7\left(\gamma q^{-1}\right)=\frac{\gamma }{\gamma -q}{\mathbf{H}}_7(\alpha ;\beta ,\gamma ;q,x,yq)-\frac{q}{\gamma -q}{\mathbf{H}}_7(\alpha ;\beta ,\gamma ;q,x,y),\gamma q^{-1}\ne 1,\end{array}$$

(39)

Proof. 

Using the definition of ${\mathbf{H}}_6$ in (11) with the relation $\frac{1}{{(\beta q^{-1};q)}_{r+s}}=\frac{1}{{(\beta ;q)}_{r+s}}[\frac{\beta }{\beta -q}{q}^{r+s}-\frac{q}{\beta -q}]$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6\left(\beta q^{-1}\right)=\sum _{r,s=0}^{\infty }[\frac{\beta }{\beta -q}{q}^{r+s}-\frac{q}{\beta -q}]\frac{{(\alpha ;q)}_{2r+s}}{{(\beta ;q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \frac{\beta }{\beta -q}{\mathbf{H}}_6(\alpha ;\beta ;q,xq,yq)-\frac{q}{\beta -q}{\mathbf{H}}_6(\alpha ;\beta ;q,x,y);\beta \ne q.\hfill \end{array}\end{array}$$

The proof of the Equations (38) and (39) are on the same lines as of Equation (37). □

Theorem 6.

The q-derivative formulas for ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$ are satisfied:

$$\begin{array}{c}\hfill [\beta q^{-1}{\theta }_{x,q}+\frac{1-\beta q^{-1}}{1-q}]{\mathbf{H}}_6+\beta q^{-1}{\theta }_{y,q}{\mathbf{H}}_6\left(xq\right)=\frac{1-\beta q^{-1}}{1-q}{\mathbf{H}}_6\left(\beta q^{-1}\right),\end{array}$$

(40)

$$\begin{array}{cc}& [\beta q^{-1}{\theta }_{y,q}+\frac{1-\beta q^{-1}}{1-q}]{\mathbf{H}}_6+\beta q^{-1}{\theta }_{x,q}{\mathbf{H}}_6\left(yq\right)=\frac{1-\beta q^{-1}}{1-q}{\mathbf{H}}_6\left(\beta q^{-1}\right),\end{array}$$

(41)

$$\begin{array}{c}\hfill [\beta q^{-1}{\theta }_{x,q}+\frac{1-\beta q^{-1}}{1-q}]{\mathbf{H}}_7=\frac{1-\beta q^{-1}}{1-q}{\mathbf{H}}_7\left(\beta q^{-1}\right)\end{array}$$

(42)

and

$$\begin{array}{c}\hfill [\gamma q^{-1}{\theta }_{y,q}+\frac{1-\gamma q^{-1}}{1-q}]{\mathbf{H}}_7=\frac{1-\gamma q^{-1}}{1-q}{\mathbf{H}}_7\left(\gamma q^{-1}\right).\end{array}$$

(43)

Proof. 

By using Equations (23)–(24), we obtain Equations (40)–(43). □

Theorem 7.

For $\beta ,\beta q\ne 1$, the relations for ${\mathbf{H}}_6$ hold true

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6(\alpha q;\beta q;q,x,y)={\mathbf{H}}_6+\frac{(\alpha -\beta )x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,x,y)\hfill \\ & +\frac{(\alpha -\beta )y}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q;\beta q^2;q,xq,y)+\frac{\alpha xq(1-\alpha q)}{(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,xq,yq),\hfill \end{array}\end{array}$$

(44)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6(\alpha q;\beta q;q,x,y)={\mathbf{H}}_6+\frac{\alpha y}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q;\beta q^2;q,x,y)\hfill \\ & +\frac{\alpha xq(1-\alpha q)}{(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,xq,yq)+\frac{\alpha x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,x,yq)\hfill \\ & -\frac{\beta x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,x,y)-\frac{\beta y}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q;\beta q^2;q,xq,y),\hfill \end{array}\end{array}$$

(45)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6(\alpha q;\beta q;q,x,y)={\mathbf{H}}_6+\frac{\alpha x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,x,y)\hfill \\ & +\frac{\alpha y}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q;\beta q^2;q,xq,y)+\frac{\alpha xq(1-\alpha q)}{(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,xq,yq)\hfill \\ & -\frac{\beta y}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q;\beta q^2;q,x,y)-\frac{\beta x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,x,yq)\hfill \end{array}\end{array}$$

(46)

and

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6(\alpha q;\beta q;q,x,y)={\mathbf{H}}_6+\frac{(\alpha -\beta )y}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q;\beta q^2;q,x,y)\hfill \\ & +\frac{(\alpha -\beta )x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,x,yq)+\frac{\alpha xq(1-\alpha q)}{(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,xq,yq).\hfill \end{array}\end{array}$$

(47)

Proof. 

To prove (44). In (11), replacing $\alpha$ by $\alpha q$ and $\beta$ by $\beta q$, and by using relation (7), we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6(\alpha q;\beta q;q,x,y)-{\mathbf{H}}_6=\sum _{r,s=0}^{\infty }\frac{1}{{(q;q)}_r{(q;q)}_s}[\frac{{(\alpha q;q)}_{2r+s}}{{(\beta q;q)}_{r+s}}-\frac{{(\alpha ;q)}_{2r+s}}{{(\beta ;q)}_{r+s}}]x^r{y}^s\hfill \\ \hfill =& \sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s-1}}{{(q;q)}_r{(q;q)}_s}\left[\frac{\alpha (1-q^{2r+s})-\beta (1-q^{r+s})-\alpha \beta q^{r+s}(1-q^r)}{(1-\beta ){(\beta q;q)}_{r+s}}\right]x^r{y}^s\hfill \\ \hfill =& \sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s-1}}{{(q;q)}_r{(q;q)}_s}\left[\frac{(\alpha -\beta )(1-q^r)+(\alpha -\beta )q^r(1-q^s)+\alpha (1-\beta )q^{r+s}(1-q^r)}{(1-\beta ){(\beta q;q)}_{r+s}}\right]x^r{y}^s\hfill \\ \hfill =& \frac{(\alpha -\beta )}{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s+1}}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_{r+s+1}}{x}^{r+1}{y}^s+\frac{(\alpha -\beta )}{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{q^r{(\alpha q;q)}_{2r+s}}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_{r+s+1}}{x}^r{y}^{s+1}\hfill \\ & +\alpha \sum _{r,s=0}^{\infty }\frac{q^{r+s+1}{(\alpha q;q)}_{2r+s+1}}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_{r+s+1}}{x}^{r+1}{y}^s\hfill \\ \hfill =& \frac{(\alpha -\beta )x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,x,y)+\frac{(\alpha -\beta )y}{(1-\beta )(1-\beta q)}{\mathbf{H}}_6(\alpha q;\beta q^2;q,xq,y)\hfill \\ & +\frac{\alpha xq(1-\alpha q)}{(1-\beta q)}{\mathbf{H}}_6(\alpha q^2;\beta q^2;q,xq,yq).\hfill \end{array}\end{array}$$

Using the relations

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \alpha (1-q^s)+\alpha q^s(1-q^r)+\alpha (1-\beta )q^{r+s}(1-q^r)-\beta (1-q^r)-\beta q^r(1-q^s),\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \alpha (1-q^r)+\alpha q^r(1-q^s)+\alpha (1-\beta )q^{r+s}(1-q^r)-\beta (1-q^s)-\beta q^s(1-q^r)\end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill (\alpha -\beta )(1-q^s)+(\alpha -\beta )q^s(1-q^r)+\alpha (1-\beta )q^{r+s}(1-q^r),\end{array}\end{array}$$

we prove in a similar way that of Equations (45)–(47) would run a parallel to Equation (44). □

Theorem 8.

The following results of ${\mathbf{H}}_7$ are valid:

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7(\alpha q;\beta q,\gamma ;q,x,y)={\mathbf{H}}_7+\frac{(\alpha -\beta )x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,x,y)\hfill \\ & +\frac{\alpha y}{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,xq,y)+\frac{\alpha xq(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,xq,yq)\hfill \\ & -\frac{\alpha \beta xq(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,xq,y)-\frac{\alpha \beta }{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,x{q}^2,y),\beta ,\beta q,\gamma \ne 1,\hfill \end{array}\end{array}$$

(48)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7(\alpha q;\beta q,\gamma ;q,x,y)={\mathbf{H}}_7+\frac{(\alpha -\beta )x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,x,y)\hfill \\ & +\frac{\alpha y}{(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,xq,y)+\frac{\alpha xq(1-\alpha q)}{(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,xq,yq),\beta ,\beta q,\gamma \ne 1,\hfill \end{array}\end{array}$$

(49)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7(\alpha q;\beta q,\gamma ;q,x,y)={\mathbf{H}}_7+\frac{\alpha y}{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,x,y)\hfill \\ & +\frac{\alpha x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,x,yq)+\frac{\alpha xq(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,xq,yq)\hfill \\ & -\frac{\beta x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,x,y)-\frac{\alpha \beta xq(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,xq,y)\hfill \\ & -\frac{\alpha \beta y}{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,x{q}^2,y),\beta ,\beta q,\gamma \ne 1,\hfill \end{array}\end{array}$$

(50)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7(\alpha q;\beta q,\gamma ;q,x,y)={\mathbf{H}}_7+\frac{\alpha y}{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,x,y)\hfill \\ & +\frac{\alpha x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,x,yq)+\frac{\alpha xq(1-\alpha q)}{(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,xq,yq)\hfill \\ & -\frac{\beta x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,x,y)-\frac{\alpha \beta y}{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,xq,y),\beta ,\beta q,\gamma \ne 1,\hfill \end{array}\end{array}$$

(51)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7(\alpha q;\beta q,\gamma ;q,x,y)={\mathbf{H}}_7+\frac{\alpha y}{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,x,y)\hfill \\ & +\frac{\alpha x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,x,yq)+\frac{\alpha xq(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,xq,yq)\hfill \\ & -\frac{\beta x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,x,y)-\frac{\alpha \beta xq(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,xq,y)\hfill \\ & -\frac{\alpha \beta y}{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,x{q}^2,y),\beta ,\beta q,\gamma \ne 1,\hfill \end{array}\end{array}$$

(52)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7(\alpha q;\beta ,\gamma q;q,x,y)={\mathbf{H}}_7+\frac{\alpha x(1-\alpha q)}{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma q;q,x,y)\hfill \\ & +\frac{\alpha y}{(1-\gamma )(1-\gamma q)}{\mathbf{H}}_7(\alpha q;\beta ,\gamma q^2;q,xq,y)+\frac{\alpha xq(1-\alpha q)}{(1-\beta )}{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma q;q,xq,yq)\hfill \\ & -\frac{\beta y}{(1-\gamma )(1-\gamma q)}{\mathbf{H}}_7(\alpha q;\beta ,\gamma q^2;q,x,y)-\frac{\alpha \gamma x(1-\alpha q)}{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma q;q,x,yq),\beta ,\gamma ,\gamma q\ne 1\hfill \end{array}\end{array}$$

(53)

and

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7(\alpha q;\beta ,\gamma q;q,x,y)={\mathbf{H}}_7+\frac{(\alpha -\gamma )y}{(1-\gamma )(1-\gamma q)}{\mathbf{H}}_7(\alpha q;\beta ,\gamma q^2;q,x,y)\hfill \\ & +\frac{\alpha x(1-\alpha q)}{(1-\beta )}{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma q;q,x,yq)+\frac{\alpha xq(1-\alpha q)}{(1-\beta )}{\mathbf{H}}_7(\alpha q^2;\beta q,\gamma q;q,xq,yq),\beta ,\gamma ,\gamma q\ne 1.\hfill \end{array}\end{array}$$

(54)

Proof. 

To prove (48). In Equation (12), replacing $\alpha$ and $\beta$ by $\alpha q$ and $\beta q$, respectively, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7(\alpha q;\beta q,\gamma ;q,x,y)-{\mathbf{H}}_7\hfill \\ & =\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s-1}{(\beta q;q)}_{r-1}}{{(q;q)}_r{(q;q)}_s{(\gamma ;q)}_s}\left[\frac{(1-\alpha q^{2r+s})(1-\beta )-(1-\alpha )(1-\beta q^r)}{(1-\beta q^r){(\beta q;q)}_{r-1}{(\beta ;q)}_r}\right]x^r{y}^s\hfill \\ \hfill =& \sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s-1}}{{(q;q)}_r{(q;q)}_s{(\gamma ;q)}_s}\left[\frac{\alpha (1-q^{2r+s})-\beta (1-q^r)-\alpha \beta q^r(1-q^{r+s})}{(1-\beta q^r){(\beta ;q)}_r}\right]x^r{y}^s\hfill \\ \hfill =& \frac{(\alpha -\beta )}{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s-1}(1-q^r)}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_r{(\gamma ;q)}_s}{x}^r{y}^s+\frac{\alpha }{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s-1}{q}^r(1-q^s)}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_r{(\gamma ;q)}_s}{x}^r{y}^s\hfill \\ \hfill +& \frac{\alpha }{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s-1}{q}^{r+s}(1-q^r)}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_r{(\gamma ;q)}_s}{x}^r{y}^s-\frac{\alpha \beta }{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s-1}{q}^r(1-q^r)}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_r{(\gamma ;q)}_s}{x}^r{y}^s\hfill \\ & -\frac{\alpha \beta }{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s-1}{q}^{2s}(1-q^s)}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_r{(\gamma ;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \frac{(\alpha -\beta )}{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s+1}{q}^{r+s+1}}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_{r+1}{(\gamma ;q)}_s}{x}^{r+1}{y}^s+\frac{\alpha }{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s+1}}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_{r+1}{(\gamma ;q)}_s}{x}^{r+1}{y}^s\hfill \\ & +\frac{\alpha }{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s}{q}^r}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_r{(\gamma ;q)}_{s+1}}{x}^r{y}^{s+1}-\frac{\alpha \beta }{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s+1}{q}^{r+1}}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_{r+1}{(\gamma ;q)}_s}{x}^{r+1}{y}^s\hfill \\ & -\frac{\alpha \beta }{(1-\beta )}\sum _{r,s=0}^{\infty }\frac{{(\alpha q;q)}_{2r+s}{q}^{2s}}{{(q;q)}_r{(q;q)}_s{(\beta q;q)}_r{(\gamma ;q)}_{s+1}}{x}^r{y}^{s+1}\hfill \\ \hfill =& \frac{(\alpha -\beta )x(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,x,y)+\frac{\alpha y}{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,xq,y)\hfill \\ \hfill +& \frac{\alpha xq(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,xq,yq)-\frac{\alpha \beta xq(1-\alpha q)}{(1-\beta )(1-\beta q)}{\mathbf{H}}_7(\alpha q^2;\beta q^2,\gamma ;q,xq,y)\hfill \\ & -\frac{\alpha \beta }{(1-\beta )(1-\gamma )}{\mathbf{H}}_7(\alpha q;\beta q,\gamma q;q,x{q}^2,y).\hfill \end{array}\end{array}$$

Using the relations

$$\begin{array}{c}\hfill \begin{array}{c}\hfill (\alpha -\beta )(1-q^r)+\alpha (1-\beta )q^r(1-q^s)+\alpha (1-\beta )q^{r+s}(1-q^r),\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \alpha (1-q^s)+\alpha q^s(1-q^r)+\alpha q^{r+s}(1-q^r)-\beta (1-q^r)-\alpha \beta q^r(1-q^r)-\alpha \beta q^{2s}(1-q^s),\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \alpha (1-q^s)+\alpha q^s(1-q^r)+\alpha (1-\beta )q^{r+s}(1-q^r)-\beta (1-q^r)-\alpha \beta q^r(1-q^s),\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \alpha (1-q^r)+\alpha q^r(1-q^s)+\alpha (1-\gamma )q^{r+s}(1-q^r)-\gamma (1-q^s)-\alpha \gamma q^s(1-q^r)\end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill (\alpha -\gamma )(1-q^s)+\alpha (1-\gamma )q^s(1-q^r)+\alpha (1-\gamma )q^{r+s}(1-q^r)\end{array}\end{array}$$

and simplifying, we obtain (49)–(54). □

Theorem 9.

The q-contiguous relations for ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$ hold true

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6\left(\beta q\right)={\mathbf{H}}_6+\frac{\beta }{1-\beta }{\mathbf{H}}_6(\alpha ;\beta q;q,xq,yq)-\frac{\beta }{1-\beta }{\mathbf{H}}_6(\alpha ;\beta q;q,x,y),\beta \ne 1\hfill \end{array}\end{array}$$

(55)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7\left(\beta q\right)={\mathbf{H}}_7+\frac{\beta }{1-\beta }{\mathbf{H}}_7(\alpha ;\beta q,\gamma ;q,xq,y)-\frac{\beta }{1-\beta }{\mathbf{H}}_7(\alpha ;\beta q,\gamma ;q,x,y),\beta \ne 1\hfill \end{array}\end{array}$$

(56)

and

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_7\left(\gamma q\right)={\mathbf{H}}_7+\frac{\gamma }{1-\gamma }{\mathbf{H}}_7(\alpha ;\beta ,\gamma q;q,x,qy)-\frac{\gamma }{1-\gamma }{\mathbf{H}}_7(\alpha ;\beta ,\gamma q;q,x,y),\gamma \ne 1.\hfill \end{array}\end{array}$$

(57)

Proof. 

By using the definition of ${\mathbf{H}}_6$, we get

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{H}}_6\left(\beta q\right)-{\mathbf{H}}_6=\sum _{r,s=0}^{\infty }\frac{{(\alpha ;q)}_{2r+s}{(\beta q;q)}_{r+s-1}}{{(q;q)}_r{(q;q)}_s}\left[\frac{(1-\beta )-(1-\beta q^{r+s})}{(1-\beta q^{r+s}){(\beta q;q)}_{r+s-1}{(\beta ;q)}_{r+s}}\right]x^r{y}^s\hfill \\ \hfill =& \frac{\beta }{1-\beta }{\mathbf{H}}_6(\alpha ;\beta q;q,xq,yq)-\frac{\beta }{1-\beta }{\mathbf{H}}_6(\alpha ;\beta q;q,x,y).\hfill \end{array}\end{array}$$

We prove in a similar way of (56) and (57). □

Theorem 10.

For $\alpha \ne 1$, the basic Horn hypergeometric functions ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$ with respect to parameters satisfy the difference equations

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{\alpha ,q}{\mathbf{H}}_6=& -\frac{1}{1-\alpha }[{\theta }_{x,q}{\mathbf{H}}_6+q{\theta }_{x,q}{\mathbf{H}}_6\left(qx\right)+{\theta }_{y,q}{\mathbf{H}}_6\left(q^{2}x\right)],\hfill \end{array}\end{array}$$

(58)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{\alpha ,q}{\mathbf{H}}_6& =-\frac{1}{1-\alpha }[{\theta }_{y,q}{\mathbf{H}}_6+{\theta }_{x,q}{\mathbf{H}}_6\left(qy\right)+q{\theta }_{x,q}{\mathbf{H}}_6(qx,qy)],\hfill \end{array}\end{array}$$

(59)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{\alpha ,q}{\mathbf{H}}_7=& -\frac{1}{1-\alpha }[{\theta }_{x,q}{\mathbf{H}}_7+q{\theta }_{x,q}{\mathbf{H}}_7\left(qx\right)+{\theta }_{y,q}{\mathbf{H}}_7\left(q^{2}x\right)]\hfill \end{array}\end{array}$$

(60)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{\alpha ,q}{\mathbf{H}}_7& =-\frac{1}{1-\alpha }[{\theta }_{y,q}{\mathbf{H}}_7+{\theta }_{x,q}{\mathbf{H}}_7\left(qy\right)+q{\theta }_{x,q}{\mathbf{H}}_6(qx,qy)].\hfill \end{array}\end{array}$$

(61)

Proof. 

Applying the q-difference operator $D_{\alpha ,q}$ and using (7), (23) and (24), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{\alpha ,q}{\mathbf{H}}_6=& \sum _{r,s=0}^{\infty }\frac{{(\alpha ;q)}_{2r+s}-{(\alpha q;q)}_{2r+s}}{(1-q)\alpha {(\beta ;q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \sum _{r,s=0}^{\infty }[1-\frac{1-\alpha q^{2r+s}}{1-\alpha }]\frac{{(\alpha ;q)}_{2r+s}}{(1-q)\alpha {(\beta ;q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& -\frac{1}{1-\alpha }\sum _{r,s=0}^{\infty }[\frac{1-q^r}{1-q}+q^r\frac{1-q^r}{1-q}+q^{2r}\frac{1-q^s}{1-q}]\frac{{(\alpha ;q)}_{2r+s}}{{(\beta ;q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& -\frac{1}{1-\alpha }[{\theta }_{x,q}{\mathbf{H}}_6+q{\theta }_{x,q}{\mathbf{H}}_6\left(qx\right)+{\theta }_{y,q}{\mathbf{H}}_6\left(q^{2}x\right)].\hfill \end{array}\end{array}$$

By using the relation $1-q^{2r+s}=1-q^s+q^s(1-q^r)+q^{n+m}(1-q^r)$ and after simplification of the resulting equation, we arrive at the Equation (59). Similarly, by the same technique we obtain (60) and (61). □

Theorem 11.

The functions ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$ with respect to parameters satisfy the difference equations

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{\beta ,q}{\mathbf{H}}_6=& \frac{1}{1-\beta }[{\theta }_{x,q}{\mathbf{H}}_6\left(\beta q\right)+{\theta }_{y,q}{\mathbf{H}}_6(\beta q,qx)],\beta \ne 1,\hfill \end{array}\end{array}$$

(62)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{\beta ,q}{\mathbf{H}}_6=& \frac{1}{1-\beta }[{\theta }_{y,q}{\mathbf{H}}_6\left(\beta q\right)+{\theta }_{x,q}{\mathbf{H}}_6(\beta q,qy)],\beta \ne 1\hfill \end{array}\end{array}$$

(63)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{\beta ,q}{\mathbf{H}}_7=& \frac{1}{1-\beta }{\theta }_{x,q}{\mathbf{H}}_7\left(\beta q\right),\beta \ne 1\hfill \end{array}\end{array}$$

(64)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{\gamma ,q}{\mathbf{H}}_7=& \frac{1}{1-\gamma }{\theta }_{x,q}{\mathbf{H}}_7\left(\gamma q\right),\gamma \ne 1.\hfill \end{array}\end{array}$$

(65)

Proof. 

From the q-difference operator (10), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_{\beta ,q}{\mathbf{H}}_6=& \sum _{r,s=0}^{\infty }[1-\frac{1-\beta }{1-\beta q^{r+s}}]\frac{{(\alpha ;q)}_{2r+s}}{(1-q)\beta {(\beta ;q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \sum _{r,s=0}^{\infty }\left[\frac{1-q^{r+s}}{1-\beta q^{r+s}}\right]\frac{{(\alpha ;q)}_{2r+s}}{(1-q){(\beta ;q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \frac{1}{1-\beta }[{\theta }_{x,q}{\mathbf{H}}_6\left(\beta q\right)+{\theta }_{y,q}{\mathbf{H}}_6(\beta q,qx)].\hfill \end{array}\end{array}$$

By use of the relation $1-q^{r+s}=1-q^s+q^s(1-q^r)$ and after simplifying and rearranging the terms, we obtain (63). The proofs (64) and (65) are similar technique for parameters $\beta$ and $\gamma$ to the proof (62). □

Theorem 12.

For the functions ${\mathbb{H}}_6$ and ${\mathbb{H}}_7$, we have the relations

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6=& \frac{(1-q^{\alpha })(1-q^{\alpha +1})}{(1-q^{\beta })(1-q)}x{\mathbb{H}}_6(q^{\alpha +2};q^{\beta +1};q,x,y),\hfill \end{array}\end{array}$$

(66)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6=& \frac{(1-q^{\alpha })}{(1-q^{\beta })(1-q)}y{\mathbb{H}}_6(q^{\alpha +1};q^{\beta +1};q,x,y),\hfill \end{array}\end{array}$$

(67)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7=\frac{(1-q^{\alpha })(1-q^{\alpha +1})}{(1-q^{\beta })(1-q)}x{\mathbb{H}}_7(q^{\alpha +2};q^{\beta +1},q^{\gamma };q,x,y)\end{array}\end{array}$$

(68)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7=& \frac{(1-q^{\alpha })}{(1-q^{\gamma })(1-q)}y{\mathbb{H}}_7(q^{\alpha +1};q^{\beta },q^{\gamma +1};q,x,y).\hfill \end{array}\end{array}$$

(69)

Proof. 

Applying the operator ${\theta }_{x,q}$ to both sides of (13) with respect to x, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6=& \sum _{r,s=0}^{\infty }\left[\frac{1-q^r}{1-q}\right]\frac{{(q^{\alpha };q)}_{2r+s}}{{(q^{\beta };q)}_{r+s}{(q;q)}_{r-1}{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \sum _{r=1,s=0}^{\infty }\left[\frac{1}{1-q}\right]\frac{{(q^{\alpha };q)}_{2r+s}}{{(q^{\beta };q)}_{r+s}{(q;q)}_{r-1}{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& \frac{(1-q^{\alpha })(1-q^{\alpha +1})}{(1-q^{\beta })(1-q)}x{\mathbb{H}}_6(q^{\alpha +2};q^{\beta +1};q,x,y).\hfill \end{array}\end{array}$$

By the same way, the proof of Equations (67)–(69) are similar lines to the proof of Equation (66). □

Theorem 13.

The functions ${\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,y)$ and ${\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,y)$ satisfies the q-differential relations

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {[2{\theta }_{x,q}+{\theta }_{y,q}+\alpha ]}_q{\mathbb{H}}_6={\left[\alpha \right]}_q{\mathbb{H}}_6\left(q^{\alpha +1}\right),\end{array}\end{array}$$

(70)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {[{\theta }_{x,q}+{\theta }_{y,q}+\beta -1]}_q{\mathbb{H}}_6={[\beta -1]}_q{\mathbb{H}}_6\left(q^{\beta -1}\right),\end{array}\end{array}$$

(71)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {[2{\theta }_{x,q}+{\theta }_{y,q}+\alpha ]}_q{\mathbb{H}}_7={\left[\alpha \right]}_q{\mathbb{H}}_7\left(q^{\alpha +1}\right),\end{array}\end{array}$$

(72)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {[{\theta }_{x,q}+\beta -1]}_q{\mathbb{H}}_7={[\beta -1]}_q{\mathbb{H}}_7\left(q^{\beta -1}\right)\end{array}\end{array}$$

(73)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {[{\theta }_{y,q}+\gamma -1]}_q{\mathbb{H}}_7={[\gamma -1]}_q{\mathbb{H}}_7\left(q^{\gamma -1}\right).\end{array}\end{array}$$

(74)

Proof. 

For proving the theorem, we start from the definitions of (13) and (14), using the relation

$$\begin{array}{c}\hfill {(q^{\alpha +1};q)}_{2r+s}=\frac{1-q^{\alpha +2r+s}}{1-q^{\alpha }}{(q^{\alpha };q)}_{2r+s}=\frac{{[\alpha +2r+s]}_q}{{\left[\alpha \right]}_q}{(q^{\alpha };q)}_{2r+s}\end{array}$$

and applying to the q-derivatives operators (10) and (4) to get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {[2{\theta }_{x,q}+{\theta }_{y,q}+\alpha ]}_q{\mathbb{H}}_6=& \sum _{r,s=0}^{\infty }\frac{{[a+2r+s]}_q{(q^{\alpha };q)}_{2r+s}}{{(q^{\beta };q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ \hfill =& {\left[\alpha \right]}_q\sum _{r,s=0}^{\infty }\frac{{(q^{\alpha +1};q)}_{2r+s}}{{(q^{\beta };q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s={\left[\alpha \right]}_q{\mathbb{H}}_6\left(q^{\alpha +1}\right).\hfill \end{array}\end{array}$$

Using the relation

$$\begin{array}{c}\hfill {(q^{\beta };q)}_{r+s}=\frac{1-q^{\beta +r+s-1}}{1-q^{\beta -1}}{(q^{\beta -1};q)}_{r+s}=\frac{{[\beta +r+s-1]}_q}{{[\beta -1]}_q}{(q^{\beta -1};q)}_{r+s},\end{array}$$

we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& {[{\theta }_{x,q}+{\theta }_{y,q}+\beta -1]}_q{\mathbb{H}}_6=\sum _{r,s=0}^{\infty }\frac{{[\beta +r+s-1]}_q{(q^{\alpha };q)}_{2r+s}}{{(q^{\beta };q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ & ={[\beta -1]}_q\sum _{r,s=0}^{\infty }\frac{{[\beta +r+s-1]}_q{(q^{\alpha };q)}_{2r+s}}{{[\beta +r+s-1]}_q{(q^{\beta -1};q)}_{r+s}{(q;q)}_r{(q;q)}_s}{x}^r{y}^s={[\beta -1]}_q{\mathbb{H}}_6\left(q^{\beta -1}\right).\hfill \end{array}\end{array}$$

This is the proof of Equation (70), and using the same procedure leads to the results (71)–(74). We omit the details. □

Theorem 14.

For the functions ${\mathbb{H}}_6$ and ${\mathbb{H}}_7$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbb{H}}_7(q^{\alpha +1};q^{\beta },q^{\gamma };q,x,y)={\mathbb{H}}_7+\frac{1-q}{1-q^{\alpha }}[q^{\alpha }{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7\hfill \\ & +q^{\alpha }{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,qy)+q^{\alpha }{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,qy)],\hfill \end{array}\end{array}$$

(75)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbb{H}}_7(q^{\alpha +1};q^{\beta },q^{\gamma };q,x,y)={\mathbb{H}}_7+\frac{1-q}{1-q^{\alpha }}[q^{\alpha }{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7\\ \hfill +q^{\alpha }{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,y)+q^{\alpha }{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{2}x,y)],\end{array}\end{array}$$

(76)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbb{H}}_6(q^{\alpha +1};q^{\beta };q,x,y)={\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,y)+\frac{1-q}{1-q^{\alpha }}[q^{\alpha }{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,y)\hfill \\ & +q^{\alpha }{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,qy)+q^{\alpha }{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,qy)]\hfill \end{array}\end{array}$$

(77)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbb{H}}_6(q^{\alpha +1};q^{\beta };q,x,y)={\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,y)+\frac{1-q}{1-q^{\alpha }}[q^{\alpha }{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,y)\\ \hfill +q^{\alpha }{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,y)+q^{\alpha }{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{2}x,y)].\end{array}\end{array}$$

(78)

Proof. 

Using (14) and the relation

$$\begin{array}{c}\hfill \begin{array}{c}\hfill 1-q^{\alpha +2r+s}=1-q^{\alpha }+q^{\alpha }(1-q^s)+q^{\alpha +s}(1-q^r)+q^{\alpha +s+r}(1-q^r),\end{array}\end{array}$$

we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbb{H}}_7(q^{\alpha +1};q^{\beta },q^{\gamma };q,x,y)=\sum _{r,s=0}^{\infty }\left[\frac{1-q^{a+2r+s}}{1-q^{\alpha }}\right]\frac{{(q^{\alpha };q)}_{2r+s}}{{(q^{\beta };q)}_r{(q^{\gamma };q)}_s{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ & =\sum _{r,s=0}^{\infty }[1+q^{\alpha }\frac{1-q^s}{1-q^{\alpha }}+q^{\alpha +s}\frac{1-q^r}{1-q^{\alpha }}+q^{\alpha +s+r}\frac{1-q^r}{1-q^{\alpha }}]\frac{{(q^{\alpha };q)}_{2r+s}}{{(q^{\beta };q)}_r{(q^{\gamma };q)}_s{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ & ={\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,y)+\frac{1-q}{1-q^{\alpha }}[q^{\alpha }{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,y)\hfill \\ & +q^{\alpha }{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,qy)+q^{\alpha }{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,qy)].\hfill \end{array}\end{array}$$

Using the relation

$$\begin{array}{c}\hfill \begin{array}{c}\hfill 1-q^{\alpha +2r+s}=1-q^{\alpha }+q^{\alpha }(1-q^r)+q^{\alpha +r}(1-q^r)+q^{\alpha +2r}(1-q^s).\end{array}\end{array}$$

The proof of Equations (76)–(78) are similar lines to the proof of Equation (75). □

Theorem 15.

The following identity holds true for the functions ${\mathbb{H}}_6$ and ${\mathbb{H}}_7$

$$\begin{array}{c}\hfill \begin{array}{cc}& {[\alpha +1]}_q{\mathbb{H}}_7\left(q^{\alpha +2}\right)={\mathbb{H}}_7\left(q^{\alpha +1}\right)+qx{\left[\alpha \right]}_q{\left[\alpha \right]}_q{\mathbb{H}}_7+2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7\hfill \\ & +2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,qy)+2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,qy)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,qy)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,qy)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,q^{2}y)+q^{2\alpha +1}x{\left[{\theta }_{y,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,y)\hfill \\ & +q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,q^{2}y)+q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{2}x,q^{2}y),\hfill \end{array}\end{array}$$

(79)

$$\begin{array}{c}\hfill \begin{array}{cc}& {[\alpha +1]}_q{\mathbb{H}}_7\left(q^{\alpha +2}\right)={\mathbb{H}}_7\left(q^{\alpha +1}\right)+qx{\left[\alpha \right]}_q{\left[\alpha \right]}_q{\mathbb{H}}_7+2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7\hfill \\ & +2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,y)+2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{2}x,y)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,y)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{2}x,y)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{3}x,y)+q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,y)\hfill \\ & +q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{2}x,y)+q^{2\alpha +1}x{\left[{\theta }_{y,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{4}x,y),\hfill \end{array}\end{array}$$

(80)

$$\begin{array}{c}\hfill \begin{array}{cc}& {[\alpha +1]}_q{\mathbb{H}}_6\left(q^{\alpha +2}\right)={\mathbb{H}}_6\left(q^{\alpha +1}\right)+qx{\left[\alpha \right]}_q{\left[\alpha \right]}_q{\mathbb{H}}_6+2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6\hfill \\ & +2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,qy)+2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,qy)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,qy)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,qy)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,q^{2}y)+q^{2\alpha +1}x{\left[{\theta }_{y,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,y)\hfill \\ & +q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,q^{2}y)+q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{2}x,q^{2}y)\hfill \end{array}\end{array}$$

(81)

and

$$\begin{array}{c}\hfill \begin{array}{cc}& {[\alpha +1]}_q{\mathbb{H}}_6\left(q^{\alpha +2}\right)={\mathbb{H}}_6\left(q^{\alpha +1}\right)+qx{\left[\alpha \right]}_q{\left[\alpha \right]}_q{\mathbb{H}}_6+2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6\hfill \\ & +2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,y)+2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{2}x,y)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,y)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{2}x,y)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{3}x,y)+q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,y)\hfill \\ & +q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{2}x,y)+q^{2\alpha +1}x{\left[{\theta }_{y,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{4}x,y).\hfill \end{array}\end{array}$$

(82)

Proof. 

From (14), we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {[\alpha +1]}_q{\mathbb{H}}_7\left(q^{\alpha +2}\right)-{\mathbb{H}}_7\left(q^{\alpha +1}\right)=\sum _{r,s=0}^{\infty }\frac{1-q^{a+2r+s}}{1-q^{\alpha }}[\frac{1-q^{\alpha +2r+s+1}}{1-q}-1]\frac{{(q^{\alpha };q)}_{2r+s}}{{(q^{\beta };q)}_r{(q^{\gamma };q)}_s{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \\ & =\frac{q}{1-q^{\alpha }}\sum _{r,s=0}^{\infty }\left[\frac{(1-q^{\alpha +2r+s})(1-q^{\alpha +2r+s})}{1-q}\right]\frac{{(q^{\alpha };q)}_{2r+s}}{{(q^{\beta };q)}_r{(q^{\gamma };q)}_s{(q;q)}_r{(q;q)}_s}{x}^r{y}^s\hfill \end{array}\end{array}$$

and using

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{(1-q^{\alpha +2r+s})(1-q^{\alpha +2r+s})}{1-q}={\left[\alpha \right]}_q[1-q^{\alpha }+2q^{\alpha }(1-q^s)+2q^{\alpha +s}(1-q^r)+2q^{\alpha +s+r}(1-q^r)]\hfill \\ & +\frac{1}{1-q}[2q^{2\alpha +s}(1-q^r)(1-q^s)+2q^{2\alpha +s+r}(1-q^r)(1-q^s)+2q^{2\alpha +2s+r}{(1-q^r)}^2\hfill \\ & +q^{2\alpha }{(1-q^s)}^2+q^{2\alpha +2s}{(1-q^r)}^2+q^{2\alpha +2s+2r}{(1-q^r)}^2],\hfill \end{array}\end{array}$$

after simplification we obtain (79). The proof of Equations (80)–(82) are similar to the proof of Equation (79). □

Theorem 16.

The functions ${\mathbb{H}}_6$ and ${\mathbb{H}}_7$ satisfies the partial q-differential equations

$$\begin{array}{c}\hfill \begin{array}{cc}& (q^{\gamma -1}{\left[{\theta }_{y,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q-q^{\gamma -1}{\left[{\theta }_{y,q}\right]}_q+{\left[\gamma \right]}_q{\left[{\theta }_{y,q}\right]}_q-\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{y,q}\right]}_q-y{q}^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q-\frac{y{\left[\alpha \right]}_q}{1-q}){\mathbb{H}}_7\hfill \\ & =\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,qy)+\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,qy),\hfill \end{array}\end{array}$$

(83)

$$\begin{array}{c}\hfill \begin{array}{cc}& (q^{\gamma -1}{\left[{\theta }_{y,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q-q^{\gamma -1}{\left[{\theta }_{y,q}\right]}_q+{\left[c\right]}_q{\left[{\theta }_{y,q}\right]}_q-\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{x,q}\right]}_q-y{q}^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q-\frac{y{\left[\alpha \right]}_q}{1-q}){\mathbb{H}}_7\hfill \\ & =\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,y)+\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{2}x,y),\hfill \end{array}\end{array}$$

(84)

$$\begin{array}{c}\hfill \begin{array}{cc}& (q^{\beta -1}{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{x,q}\right]}_q-q^{\beta -1}{\left[{\theta }_{x,q}\right]}_q+{\left[\beta \right]}_q{\left[{\theta }_{y,q}\right]}_q-q^{2\alpha +1}x{\left[{\theta }_{y,q}^2\right]}_q-2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{y,q}\right]}_q\hfill \\ & -x{q}^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q-qx{\left[\alpha \right]}_q{\left[\alpha \right]}_q-x{\left[\alpha \right]}_q){\mathbb{H}}_7=2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,qy)\hfill \\ & +2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,qy)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,qy)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,qy)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,q^{2}y)\hfill \\ & +q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,q^{2}y)+q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{2}x,q^{2}y),\hfill \end{array}\end{array}$$

(85)

$$\begin{array}{c}\hfill \begin{array}{cc}& (q^{\beta -1}{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{x,q}\right]}_q-q^{\beta -1}{\left[{\theta }_{x,q}\right]}_q+{\left[\beta \right]}_q{\left[{\theta }_{y,q}\right]}_q-q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q-2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q-x{q}^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q\hfill \\ & -x{\left[\alpha \right]}_q-qx{\left[\alpha \right]}_q{\left[\alpha \right]}_q){\mathbb{H}}_7=2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,y)\hfill \\ & +2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{2}x,y)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,y)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{2}x,y)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{3}x,y)\hfill \\ & +q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{2}x,y)+q^{2\alpha +1}x{\left[{\theta }_{y,q}^2\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{4}x,y),\hfill \end{array}\end{array}$$

(86)

$$\begin{array}{c}\hfill \begin{array}{cc}& (q^{\beta -1}{\left[{\theta }_{y,q}\right]}_q{[{\theta }_{x,q}+{\theta }_{y,q}]}_q-q^{\beta -1}{\left[{\theta }_{y,q}\right]}_q+{\left[\beta \right]}_q{\left[{\theta }_{y,q}\right]}_q-\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{y,q}\right]}_q-y{q}^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q-\frac{y{\left[\alpha \right]}_q}{1-q}){\mathbb{H}}_6\hfill \\ & =\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,qy)+\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,qy),\hfill \end{array}\end{array}$$

(87)

$$\begin{array}{c}\hfill \begin{array}{cc}& (q^{\beta -1}{\left[{\theta }_{y,q}\right]}_q{[{\theta }_{x,q}+{\theta }_{y,q}]}_q-q^{\beta -1}{\left[{\theta }_{y,q}\right]}_q-\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{x,q}\right]}_q+{\left[\beta \right]}_q{\left[{\theta }_{y,q}\right]}_q-y{q}^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q-\frac{y{\left[\alpha \right]}_q}{1-q}){\mathbb{H}}_6\hfill \\ & =\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,y)+\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{2}x,y)],\hfill \end{array}\end{array}$$

(88)

$$\begin{array}{c}\hfill \begin{array}{cc}& (q^{\beta -1}{\left[{\theta }_{x,q}\right]}_q{[{\theta }_{x,q}+{\theta }_{y,q}]}_q-2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{y,q}\right]}_q-q^{\beta -1}\left[{\theta }_{x,q}\right]+{\left[\beta \right]}_q{\left[{\theta }_{y,q}\right]}_q-q^{2\alpha +1}x{\left[{\theta }_{y,q}^2\right]}_q-x{q}^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q\hfill \\ & -x{\left[\alpha \right]}_q-qx{\left[\alpha \right]}_q{\left[\alpha \right]}_q){\mathbb{H}}_6=2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,qy)\hfill \\ & +2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,qy)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,qy)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,qy)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,q^{2}y)\hfill \\ & +q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,x,q^{2}y)+q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{2}x,q^{2}y)\hfill \end{array}\end{array}$$

(89)

and

$$\begin{array}{c}\hfill \begin{array}{cc}& (q^{\beta -1}{\left[{\theta }_{x,q}\right]}_q{[{\theta }_{x,q}+{\theta }_{y,q}]}_q-q^{\beta -1}{\left[{\theta }_{x,q}\right]}_q+{\left[\beta \right]}_q{\left[{\theta }_{y,q}\right]}_q-2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q-x{q}^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q-q^{2\alpha +1}x\left[{\theta }_{x,q}^2\right]\hfill \\ & -x{\left[\alpha \right]}_q-qx{\left[\alpha \right]}_q{\left[\alpha \right]}_q){\mathbb{H}}_6=2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,y)\hfill \\ & +2q^{\alpha +1}x{\left[\alpha \right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{2}x,y)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,qx,y)\hfill \\ & +2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{2}x,y)+2q^{2\alpha +1}x{\left[{\theta }_{x,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,q^{3}x,y)\hfill \\ & +q^{2\alpha +1}x{\left[{\theta }_{x,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{2}x,y)+q^{2\alpha +1}x{\left[{\theta }_{y,q}^2\right]}_q{\mathbb{H}}_6(q^{\alpha };q^{\beta };q,q^{4}x,y).\hfill \end{array}\end{array}$$

(90)

Proof. 

From (68) and (70), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& ({\left[{\theta }_{y,q}\right]}_q{[{\theta }_{y,q}+\gamma -1]}_q-y{[2{\theta }_{x,q}+{\theta }_{y,q}+\alpha ]}_q){\mathbb{H}}_7={[\gamma -1]}_q{\left[{\theta }_{y,q}\right]}_q{\mathbb{H}}_7\left(q^{\gamma -1}\right)-y{\left[\alpha \right]}_q{\mathbb{H}}_7\left(q^{\alpha +1}\right)\hfill \\ & ={[\gamma -1]}_q\frac{(1-q^{\alpha })y}{(1-q^{\gamma -1})(1-q)}{\mathbb{H}}_7(q^{\alpha +1};q^{\beta },q^{\gamma };q,x,y)-y{\left[\alpha \right]}_q{\mathbb{H}}_7\left(q^{\alpha +1}\right)\hfill \\ & =({[\gamma -1]}_q\frac{(1-q^{\alpha })y}{(1-q^{\gamma -1})(1-q)}-y{\left[\alpha \right]}_q){\mathbb{H}}_7(q^{\alpha +1};q^{\beta },q^{\gamma };q,x,y)\hfill \\ & =(\frac{y}{1-q}{\left[\alpha \right]}_q-y{\left[\alpha \right]}_q){\mathbb{H}}_7(q^{\alpha +1};q^{\beta },q^{\gamma };q,x,y)=\left(\frac{qy}{1-q}\right){\left[\alpha \right]}_q{\mathbb{H}}_7(q^{\alpha +1};q^{\beta },q^{\gamma };q,x,y).\hfill \end{array}\end{array}$$

(91)

By using the relation

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {[n-k]}_q=q^{-k}({\left[n\right]}_q-{\left[k\right]}_q),\\ \hfill {}_q={\left[n\right]}_q+q^s{\left[k\right]}_q\end{array}\end{array}$$

and we write the q-differential operator

$$\begin{array}{c}\hfill \begin{array}{cc}& ({\left[{\theta }_{y,q}\right]}_q{[{\theta }_{y,q}+\gamma -1]}_q-y{[2{\theta }_{x,q}+{\theta }_{y,q}+\alpha ]}_q)\hfill \\ & =\left({\left[{\theta }_{y,q}\right]}_q\right(q^{\gamma }{[{\theta }_{y,q}-1]}_q+{\left[\gamma \right]}_q)-y(q^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q+{\left[\alpha \right]}_q\left)\right)\hfill \\ & =\left({\left[{\theta }_{y,q}\right]}_q\right(q^{\gamma -1}{\left[{\theta }_{y,q}\right]}_q-q^{\gamma -1}{\left[1\right]}_q+{\left[\gamma \right]}_q)-y(q^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q+{\left[\alpha \right]}_q\left)\right)\hfill \\ & =(q^{\gamma -1}{\left[{\theta }_{y,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q-q^{\gamma -1}{\left[{\theta }_{y,q}\right]}_q+{\left[\gamma \right]}_q{\left[{\theta }_{y,q}\right]}_q-y{q}^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q-y{\left[\alpha \right]}_q).\hfill \end{array}\end{array}$$

From the above relation and using (91), (71), we get

$$\begin{array}{c}\hfill \begin{array}{cc}& (q^{\gamma -1}{\left[{\theta }_{y,q}\right]}_q{\left[{\theta }_{y,q}\right]}_q-q^{\gamma -1}{\left[{\theta }_{y,q}\right]}_q+{\left[\gamma \right]}_q{\left[{\theta }_{y,q}\right]}_q-\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{y,q}\right]}_q-y{q}^{\alpha }{[2{\theta }_{x,q}+{\theta }_{y,q}]}_q-\frac{y{\left[\alpha \right]}_q}{1-q}){\mathbb{H}}_7\hfill \\ & =\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,x,qy)+\frac{q^{\alpha +1}y}{1-q}{\left[{\theta }_{x,q}\right]}_q{\mathbb{H}}_7(q^{\alpha };q^{\beta },q^{\gamma };q,qx,qy).\hfill \end{array}\end{array}$$

(92)

The proof Equations (84)–(90) are on the same lines as of Equation (83). We omit the proof of the given theorem. □

3. Concluding Remarks

This study is a continuation of the recent paper [31], we have investigated the q-analogues of hypergeometric Horn functions ${\mathbf{H}}_3$ and ${\mathbf{H}}_4$ and their various properties. In our present study, we have established several results, such as q-contiguous relations, q-differential relations and q-differential equations of the basic Horn functions ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$ under conditions on the numerator and denominator parameters. In addition, we have deeply discussed new properties of these extended basic Horn functions ${\mathbb{H}}_6$ and ${\mathbb{H}}_7$ such as the iq-contiguous relations, q-differential relations, and q-differential equations. Note that, by setting $q⟶1^{-}$, we obtain various known or unknown results for the Horn hypergeometric functions ${\mathbf{H}}_6$ and ${\mathbf{H}}_7$ established earlier in [28]. Therefore, other special types of these extensions are recommended for a parallel work of this study. More investigations will be carried out in the coming future results in other different fields of interest for quantum calculus on time scales and applications in the mathematical and physical sciences.