Assume $X$ a continuous non-negative random variable follow a Gamma distribution having shape parameter $k$ and scale parameter $\theta$ with the following CDF
\begin{align} F_{X}(x) =\frac{\gamma\left(k,\frac{x}{\theta}\right)}{\Gamma(k)}, \end{align}
and accordingly I suppose it have the following MGF (not sure of the MGF expression because I saw different expressions)
\begin{align} M_{X}(t) =\left(\frac{1}{1-\theta t}\right)^k, \end{align}
Let random variable $Y = X^2$.
How do I find the MGF of $Y$? I know that there is a variable transformation on the PDF $f_X(x)$ to obtain $f_Y(y)$, for example:
$f_Y(y)=f_X(g^{-1}(y)) \vert\frac{d}{d y}g^{-1}(y)\vert.$
My question is there also a variable transformation that can be applied on $M_{X}(t)$ to have $M_{Y}(t)$?