\right\} ,\end{array}$

*respectively. Then, we have the following definitions.*

$\left( \mathtt{i}\right) $ *Over the set* $\hat{A},$ *we define the* $q$*-analogue of the Natural transform of first type as*$$N_{q}\left( f\right) \left( u;v\right) =\frac{1}{\left( 1-q\right) u}\dint\nolimits_{0}^{\frac{u}{v}}f\left( t\right) E_{q}\left( q\frac{v}{u}t\right) d_{q}t,  \tag{18}$$*provided the integral exists.*

$\left( \mathtt{ii}\right) $ *Over the set* $\check{A},$ *we define the* $q$*-analogue of the Natural transform of type two as*$$_{q}N\left( f\right) \left( u;v\right) =\frac{1}{\left( 1-q\right) }\dint\nolimits_{0}^{\infty }f\left( t\right) e_{q}\left( -\frac{v}{u}t\right) d_{q}t,  \tag{19}$$*when the integral exists.*

It seems very benificial to us to notice the following relations$$\left. 
\begin{array}{c}
N_{q}\left( f\right) \left( 1;v\right) =\left( L_{q}f\right) \left( v\right)
,\text{ \ \ }_{q}N\left( f\right) \left( 1;v\right) =\left( _{q}Lf\right)
\left( v\right) , \\ 
\\ 
N_{q}\left( f\right) \left( u;1\right) =\left( S_{q}f\right) \left( u\right)
,\text{ \ \ }_{q}N\left( f\right) \left( u;1\right) =\left( _{q}Sf\right)
\left( u\right) ,\end{array}\right.  \tag{20}$$where $L_{q}\left( S_{q}\right) $ and $_{q}L\left( _{q}S\right) $ are respectively the $q$-analogues of the Laplace $\left( \text{Sumudu}\right) $ transforms of first $\left( \text{second }\right) $ types; see, for example, $\left[ \text{4}\right] \left( \text{ resp.},\left[ 6\right] \right) $.

In terms of Jackson integral series representation, the $q$-analogue of $\left( 18\right) $ can be expressed as$$N_{q}\left( f\right) \left( u;v\right) =\frac{\left( q;q\right) _{\infty }}{v}\sum_{k\geq 0}\frac{q^{k}f\left( \dfrac{u}{v}q^{k}\right) }{\left(
q;q\right) _{k}},  \tag{21}$$whereas, the $q$-analogue of 