,$ $1\leq j\leq m,$ are its right and those of $G\left(
q^{1-a_{j}+\alpha _{j}s}\right) ,$ $1\leq j\leq n,$ are the left of $C.$ The integral converges if $\func{Re}\left( s\log x-\log \sin \pi s\right)
<0, $ for large values of $\left\vert s\right\vert $ on $C.$ Hence,$$\left\vert \arg \left( x\right) -w_{2}w_{1}^{-1}\log \left\vert x\right\vert
\right\vert <\pi ,\text{ }\left\vert q\right\vert <1,\text{ }\log
q=-w=-w_{1}-iw_{2},$$where $w_{1}$ and $w_{2}$ are real numbers.

Indeed, for $\alpha _{i}=\beta _{j}=1,$ for all $i,j,$ $\left(
45\right) $ gives the $q$-analogue of the Meijer’s $G$-function$$G_{M,N}^{m,n}\left[ x;q\left\vert 
\begin{array}{c}
a_{1},a_{1},...,a_{M} \\ 
b_{1},b_{2},...,b_{N},\end{array}\right. \right] =\frac{1}{2\pi i}\dint\nolimits_{C}\frac{\dprod\limits_{j=1}^{m}G\left( q^{b_{j}-s}\right) \dprod\limits_{j=1}^{n}G\left(
q^{1-a_{j}+s}\right) \pi x^{2}}{\dprod\limits_{j=m+1}^{N}G\left(
q^{1-b_{j}+s}\right) \dprod\limits_{j=n+1}^{M}G\left( q^{a_{j}-s}\right)
G\left( q^{1-s}\right) \sin \pi s}d_{q}s$$where $0\leq m\leq N;$ $0\leq n\leq M$ and $\func{Re}\left( s\log
x-\log \sin \pi s\right) <0$.

We have the following main result of this section.

<span style="font-variant:small-caps;">Theorem 15.</span> $\left( \mathtt{i}\right) $ *Let* $\lambda $* be any complex number and* $k\in \left( 0,\infty \right)
. $* The* $q$*-Natural transform* $N_{q}$* of the Fox’s* $H_{q}$*-Function is given as*

$N_{q}\left( x^{\lambda }H_{M,N}^{m,n}\left[ \lambda x^{k};q\left\vert 
\begin{array}{l}
\left( a_{1},\alpha _{1}\right) ,\left( a_{1},\alpha _{1}\right) ,...,\left(
a_{M},\alpha _{M}\right) \\ 
\left( b_{1},\beta _{1}\right) ,b_{2},...,\left( b_{N},\beta _{N}\right)\end{array}\right. \right] \right) \left( u;v\right) =\dfrac{u^{\lambda }}{v^{\lambda
+1}G\left( q\right) }$

$\ \ \ \ \ \ \ \ \ \ \ \ 