*

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

<span style="font-variant:small-caps;">Proof</span> Let $a$ be a real number, then by aid of $\left(
15\right) ,$ we consider to write $$J_{2\mu }^{\left( 1\right) }\left( 2\sqrt{ax};q\right) =x^{\mu
}\sum\limits_{n\geq 0}\frac{\left( -1\right) ^{n}a^{\mu +n}}{\left(
q;q\right) _{2\mu +n}\left( q;q\right) _{n}}x^{n}.$$By replacing $\alpha $ by $\alpha -\mu -1,$ and setting $A_{n}=\dfrac{\left(
-1\right) ^{n}a^{\mu +n}}{\left( q;q\right) _{2\mu +n}\left( q;q\right) _{n}},$ we, partially, get$$x^{\alpha -1}J_{2\mu }^{\left( 1\right) }\left( 2\sqrt{ax};q\right)
=x^{\alpha -1}\sum\limits_{n\geq 0}A_{n}x^{n}=x^{\alpha -1}f\left( x\right) .$$Hence, by Theorem 1, we obtain$$\begin{aligned}
N_{q}\left( x^{\alpha -1}J_{2\mu }^{\left( 1\right) }\left( 2\sqrt{ax};q\right) \right) \left( u;v\right) &=&\frac{\left( 1-q\right) ^{\alpha
-1}u^{\alpha -1}}{v^{\alpha }}\sum\limits_{n\geq 0}A_{n}\frac{u^{n}}{v^{n}}\left( q^{\alpha };q\right) _{n}\Gamma _{q}\left( \alpha \right) \\
&=&\frac{\left( 1-q\right) ^{\alpha -1}u^{\alpha -1}}{v^{\alpha }}\Gamma
_{q}\left( \alpha \right) \sum\limits_{n\geq 0}A_{n}\frac{u^{n}}{v^{n}}\left( q^{\alpha };q\right) _{n}.\end{aligned}$$This completes the proof of the corrollary.

In view of Corrollary 5, we have the following easy statement.

<span style="font-variant:small-caps;">Corrollary 6.</span> *Let* $a$* be a real number. Then, we have*

$N_{q}\left( J_{2\mu }^{\left( 1\right) }\left( 2\sqrt{ax};q\right) \right)
\left( u;v\right) =\dsum\limits_{n\geq 0}A_{n}u^{n}\left( q^{\alpha
};q\right) _{n}.$

Further, Corrollary 5 is expressed in terms of $q$-Sumudu and $q$-Laplace transforms as in the following results.

<span style="font-variant:small-caps;">Corrollary</span>**** ****<span style="font-variant:small-caps;">7</span>. *Let* 