A random variable $Y$ with support $R_Y=(-\infty, \infty)$ has moment generating function $$M_Y(t)=\left( \frac{\theta^t \Gamma(k+t)}{\Gamma(k)} \right)^n$$ where $\theta\in(0,\infty), k\in(0,\infty), n\in\mathbb{N},$ and $\Gamma(\cdot)$ represents the gamma function. What is the probability density function of $Y$?
Truthfully, I'm unfamiliar with the characteristic function and inverse Fourier transforms, which seems to be the typical approach to solving these types of problems. Further, this seems like a particularly challenging problem since $t$ appears as both the argument to the Gamma function and in $\theta^t$.
Is it possible to actually derive this PDF? If so, can you do it?