generic interval $\left[ a,b\right] $ is given by $\left[ 11\right] $ $$\int\nolimits_{b}^{a}f\left( x\right) d_{q}x=\int\nolimits_{0}^{b}f\left(
x\right) d_{q}x-\int\nolimits_{0}^{a}f\left( x\right) d_{q}x.  \tag{6}$$The improper integral is defined as $\left[ 5\right] $$$\int\nolimits_{0}^{\frac{\infty }{A}}f\left( x\right) d_{q}x=\left(
1-q\right) \sum_{n\in 
\mathbb{Z}
}\frac{q^{k}}{A}f\left( \frac{q^{k}}{A}\right) .  \tag{7}$$The $q$-analogues of the gamma function are defined by $\left[ 5\right] $$$\left. 
\begin{array}{l}
\Gamma _{q}\left( \alpha \right) =\dint\nolimits_{0}^{\frac{1}{1-q}}x^{\alpha -1}E_{q}\left( q\left( 1-q\right) x\right) d_{q}x,\text{ } \\ 
\\ 
_{q}\Gamma \left( \alpha \right) =K\left( A;\alpha \right)
\dint\nolimits_{0}^{\frac{\infty }{A\left( 1-q\right) }}x^{\alpha
-1}e_{q}\left( -\left( 1-q\right) x\right) d_{q}x,\end{array}\right.$$where $\alpha >0$ and that$$\left. K\left( A;t\right) =A^{t-1}\dfrac{\left( -q/A;q\right) _{\infty }}{\left( -q^{t}/A;q\right) _{\infty }}\dfrac{\left( -A;q\right) _{\infty }}{\left( -Aq^{1-t};q\right) _{\infty }},\right.  \tag{8}$$$$\left( a;q\right) _{n}=\prod\limits_{0}^{n-1}\left( 1-aq^{k}\right) ,\text{ }\left( a;q\right) _{\infty }=\underset{n\rightarrow \infty }{\lim }\left(
a;q\right) _{n}\text{ }.  \tag{9}$$for all $t\in 
\mathbb{R}
.$

The useful notations we need here are $\left[ 10\right] $$$\Gamma _{q}\left( x\right) =\dfrac{\left( q;q\right) _{\infty }}{\left(
q^{x};q\right) _{\infty }}\left( 1-q\right) ^{1-x}\text{ and }\left(
a;q\right) _{t}=\dfrac{\left( a;q\right) _{\infty }}{\left( aq^{t};q\right)
_{\infty }},\text{ }t\in 
\mathbb{R}
.  \tag{10}$$The $q$-analogues of the exponential function are given as$$\left. 
\begin{array}{l}
E_{q}\left( t\right) =\dsum\limits_{0}^{\infty }\left( -1\right) ^{n}\dfrac{q^{\frac{n\left( n-1\right) }{2}}}{\left( q;q\right) _{n}}t^{n}=\left(
t;q\right) _{\infty },\text{ }t\in 
\mathbb{C}
\\ 
\\ 
e_{q}\left( t\right) =\dsum\limits_{0}^{\infty }\dfrac{1}{\left( q;q\right)
_{n}}t^{n}=\dfrac{1}{\left( t,q\right) _{\infty }},\text{ }t<1.\end{array}\right\}  \tag{12}$$

The $q$-analogues of the hypergeometric function are defined in two ways 