$a$* be a positive real number. Then, the following hold.*

$\left( \text{i}\right) $ $S_{q}\left( x^{\alpha -1}J_{2\mu }^{\left(
1\right) }\left( 2\sqrt{ax};q\right) \right) \left( u\right) =\left(
1-q\right) ^{\alpha -1}u^{\alpha -1}\Gamma _{q}\left( \alpha \right)
\dsum\limits_{n\geq 0}A_{n}u^{n}\left( q^{\alpha };q\right) _{n},$

$\left( \text{ii}\right) $ $L_{q}\left( x^{\alpha -1}J_{2\mu }^{\left(
1\right) }\left( 2\sqrt{ax};q\right) \right) \left( v\right) =\dfrac{\left(
1-q\right) ^{\alpha -1}}{v^{\alpha }}\Gamma _{q}\left( \alpha \right)
\dsum\limits_{n\geq 0}A_{n}\dfrac{1}{v^{n}}\left( q^{\alpha };q\right) _{n},$

*where* $A_{n}=\left( -1\right) ^{n}\dfrac{a^{\mu +n}}{\left( q;q\right) _{2\mu +n}\left( q;q\right) _{n}}.$

Corrollary 6 extends the results to the case of $q$-Sumudu and $q$-Laplace transforms as follows.

<span style="font-variant:small-caps;">Corrollary 8</span>. *Let* $a$* be a real number. Then, the following hold.*

$\left( \text{i}\right) S_{q}\left( J_{2\mu }^{\left( 1\right)
}\left( 2\sqrt{ax};q\right) \right) \left( u\right) =\sum\limits_{n\geq
0}A_{n}u^{n}\left( q^{\alpha };q\right) _{n};\ \left( \text{ii}\right)
L_{q}\left( J_{2\mu }^{\left( 1\right) }\left( 2\sqrt{ax};q\right) \right)
\left( v\right) =\dfrac{1}{v}\sum\limits_{n\geq 0}A_{n}\dfrac{\left(
q^{\alpha };q\right) _{n}}{v^{n}}.$

$_{q}N$ Transform of Special Functions
======================================

In this section of this paper, we are concerned with the study of the $q$-analogue $_{q}N$ of some special functions. We are precisely concerned with the series representation of the transform and getting some results related to $q$-Laplace and $q$-Sumudu transforms.

<span style="font-variant:small-caps;">Theorem 9.</span> *Let* $f\left( x\right)
=\sum\limits_{n\geq 0}A_{n}x^{n}$* and* $\alpha >0.$* Then, we have*$$_{q}N\left( x^{\alpha -1}f\left( x\right) \right) \left( u;v\right) =\left( 
