theorem (for $A_{0}=1,A_{n}=0,$ for $n\geq 1)$ gives $$_{q}N\left( x^{\alpha -1}\right) \left( u;v\right) =\dfrac{\left( \frac{u}{v}\right) ^{\alpha -1}\left( 1-q\right) ^{\alpha -1}\Gamma _{q}\left( \alpha
\right) }{K\left( \frac{u}{v};\alpha \right) }.  \tag{39}$$

Also, Equ. $\left( 39\right) $ reveals :

<span style="font-variant:small-caps;">Corrollary 10.</span> *The following hold true.*$$\begin{aligned}
\left( \mathtt{i}\right) \text{ \ }_{q}L\left( x^{\alpha -1}\right) \left(
v\right) &=&\dfrac{1}{v^{\alpha -1}}\frac{\left( 1-q\right) ^{\alpha
-1}\Gamma _{q}\left( \alpha \right) }{K\left( \frac{1}{v};\alpha \right) }.\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } 
\TCItag{40} \\
\left( \mathtt{ii}\right) \text{ }_{q}S\left( x^{\alpha -1}\right) \left(
u\right) &=&\frac{u^{\alpha -1}\left( 1-q\right) ^{\alpha -1}\Gamma
_{q}\left( \alpha \right) }{K\left( u;\alpha \right) }.  \TCItag{41}\end{aligned}$$Further, $\left( 39\right) $ and $\left( 41\right) $ jointly lead to the conclusion $\left( _{q}N\left( 1\right) \right) \left( u;v\right) =\dfrac{1}{K\left( \frac{u}{v};\alpha \right) }.$ Therefore, we are directed to the results$$_{q}L\left( 1\right) \left( v\right) =\dfrac{1}{K\left( v;1\right) }\text{
and }\left( _{q}S\left( 1\right) \right) \left( u\right) =\dfrac{1}{K\left(
u;1\right) }.$$

<span style="font-variant:small-caps;">Theorem 11.</span> *Let* $a$* be a real number and* $f\left( x\right) =$ $_{m-k}\Phi _{m-1}\left[ \left. 
\begin{array}{l}
a_{1},a_{2},...,a_{m-k},q^{\alpha } \\ 
b_{1},b_{2},...,b_{m-1}\end{array}\right\vert q,ax\right] .$ *Then, we have*$$\begin{aligned}
_{q}N\left( x^{\alpha -1}f\left( x\right) \right) \left( u;v\right) &=&\frac{\left( \frac{u}{v}\right) ^{\alpha }\left( 