I'm suppose to obtain the density function of this mgf for a discrete random variable $Y$ taking values in range $[0, \infty)$, where $E(e^{ty}) = e^{-l}[1-e^t(1-p)]^{-ld},c\in \mathbb{R}^+,p \in [0,1]$.
My intuition was to find the taylor polynomial of this function but it quickly blew into a incomprehensible mess.
By using Binomial series expansion, we have
\begin{align} e^{-\lambda}[1-e^t(1-p)]^{-\lambda c} &= e^{-\lambda }\left[1+\sum_{k=1}^\infty\frac{\prod_{j=0}^{k-1}(-\lambda c - j)}{k!} (-e^t(1-p))^k\right] \end{align}
Now, we just have to read off the coefficient of $e^{tk}$:
$$Pr(Y=k) = e^{-\lambda}\frac{(1-p)^k}{k!}\prod_{j=0}^{k-1}(\lambda c + j)$$