$\left( 21\right) ,$ *we write* $$\begin{aligned}
N_{q}\left( x^{\alpha -1}f\left( x\right) \right) \left( u;v\right)  &=&\frac{\left( q;q\right) _{\infty }}{v}\sum\limits_{k\geq 0}q^{k}\frac{\left( 
\dfrac{u}{v}q^{k}\right) ^{\alpha -1}f\left( \dfrac{u}{v}q^{k}\right) }{\left( q;q\right) _{k}}  \notag \\
&=&\frac{u^{\alpha -1}}{v^{\alpha }}\left( q;q\right) _{\infty
}\sum\limits_{k\geq 0}\frac{q^{\alpha k}}{\left( q;q\right) _{k}}\sum\limits_{n\geq 0}A_{n}\left( \dfrac{u}{v}q^{k}\right) ^{n}  \notag \\
&=&\frac{u^{\alpha -1}}{v^{\alpha }}\left( q;q\right) _{\infty
}\sum\limits_{n\geq 0}A_{n}\left( \dfrac{u}{v}\right) ^{n}\sum\limits_{k\geq
0}\frac{q^{k\left( \alpha +n\right) }}{\left( q;q\right) _{k}}.  \TCItag{25}\end{aligned}$$Hence, taking into account $\left( 11\right) ,$ $\left( 25\right) $ simply reveals$$\begin{aligned}
N_{q}\left( x^{\alpha -1}f\left( x\right) \right) \left( u;v\right)  &=&\frac{u^{\alpha -1}}{v^{\alpha }}\left( q;q\right) _{\infty
}\sum\limits_{n\geq 0}A_{n}\left( \dfrac{u}{v}\right) ^{n}e_{q}\left(
q^{\alpha +n}\right)   \notag \\
&=&\frac{u^{\alpha -1}}{v^{\alpha }}\left( q;q\right) _{\infty
}\sum\limits_{n\geq 0}A_{n}\dfrac{u^{n}}{v^{n}}\frac{1}{\left( q^{\left(
\alpha +n\right) };q\right) _{\infty }}.  \TCItag{26}\end{aligned}$$Therefore, by aid of the Equations $\left( 10\right) $ and $\left( 26\right) 
$ we get that$$\begin{aligned}
N_{q}\left( x^{\alpha -1}f\left( x\right) \right) \left( u;v\right)  &=&\frac{u^{\alpha -1}}{v^{\alpha }}\left( q;q\right) _{\infty
}\sum\limits_{n\geq 0}A_{n}\left( \frac{u}{v}\right) ^{n}e_{q}\left(
q^{\alpha +n}\right)  \\
&=&\frac{\left( 1-q\right) ^{\alpha -1}}{v^{\alpha }}u^{\alpha
-1}\sum\limits_{n\geq 0}A_{n}\frac{u^{n}}{v^{n}}\frac{\Gamma _{q}\left(
\alpha +n\right) }{\left( 1-q\right) ^{-n}} \\
&=&\dfrac{\left( 1-q\right) ^{\alpha -1}}{v^{\alpha }}u^{\alpha
-1}\sum\limits_{n\geq 0}A_{n}\dfrac{u^{n}\left( 1-q\right) ^{n}}{v^{n}}\Gamma _{q}\left( \alpha +n\right) .\end{aligned}$$This completes the proof of the theorems.

<span style="font-variant:small-caps;">Theorem 2.</span> *Let* $\alpha $* be a positive real number. Then, the following hold.*

$\left( \mathtt{i}\right) $ *Let* $\Gamma _{q}\left( \alpha \right) $ *be the q-gamma function of the 