$\left( 19\right) $ can similarly be performed in terms of that series as $$_{q}N\left( f\right) \left( u;v\right) =\sum_{k\in 
\mathbb{Z}
}^{\infty }\frac{\left( q;q\right) _{\infty }}{v}\frac{f\left( q^{k}\right) 
}{\left( -\dfrac{u}{v}q^{k};q\right) _{\infty }}.$$Hence, on parity of the fact $\left( a;q\right) _{k}=\dfrac{\left(
a;q\right) _{\infty }}{\left( aq^{k};q\right) _{\infty }}$ $($ for $a=\dfrac{-v}{u}),$ the previous equation has the series representation$$_{q}N\left( f\right) \left( u;v\right) =\frac{1}{\left( -\dfrac{v}{u};q\right) _{\infty }}\sum_{k\in 
\mathbb{Z}
}^{\infty }q^{k}f\left( q^{k}\right) \left( -\dfrac{u}{v};q\right) _{k} 
\tag{22}$$that we shall use in later investigations.

$N_{q}$ of Some Special Functions
=================================

In this section of this paper, we apply the analogue $N_{q}$ for certain functions of special type and extend the work to $q$-Laplace and $q$-Sumudu transforms. We assume the functions, unless otherwise stated, are of power series form,$$f\left( x\right) =\sum\limits_{n\geq 0}A_{n}x^{n},  \tag{23}$$where $A_{n}$ is some bounded sequence .

In what follows, we establish the following three main theorems of this section.

<span style="font-variant:small-caps;">Theorem 1.</span> *Let* $\alpha $* be a positive real number and* $f\left( x\right) =\sum\limits_{n\geq 0}A_{n}x^{n}$*. Then, we have*$$N_{q}\left( x^{\alpha -1}f\left( x\right) \right) \left( u;v\right) =\frac{\left( 1-q\right) ^{\alpha -1}}{v^{\alpha }}u^{\alpha -1}\sum\limits_{n\geq
0}A_{n}\frac{u^{n}}{v^{n}}\left( 1-q\right) ^{n}\Gamma _{q}\left( \alpha
+n\right) .  \tag{24}$$<span style="font-variant:small-caps;">Proof</span>* Let* $\alpha $ *be a positive real number and* $f\left( x\right) =\sum\limits_{n\geq 0}A_{n}x^{n}$ *be given. Then, on aid of* 