For $k\ge3$, using the fact that the probability generating function for $X$ extends to a $k$-th complex root of unity, find a closed form formula for $P(X\in k\mathbb{Z})$, for $X\in\text{Po}(\gamma)$.
Using the fact that the probability generating function for $X$, $E[s^X]=e^{\gamma(s-1)}$, extends to a $k$-th complex root of unity, we get that \begin{align*} e^{\gamma(\eta-1)}&=\sum_{n\ge0}\eta^n\cdot\frac{\gamma^n}{n!}e^{-\gamma}\\ &=\sum_{m\in k\mathbb{Z}}\frac{\gamma^m}{m!}e^{-\gamma}+\sum_{m\notin k\mathbb{Z}}\eta^m\cdot\frac{\gamma^m}{m!}e^{-\gamma} \end{align*} Which implies that \begin{align*} P(X\in k\mathbb{Z})&=\sum_{m\in k\mathbb{Z}}\frac{\gamma^m}{m!}e^{-\gamma}\\ &=e^{\gamma(\eta-1)}-\sum_{m\notin k\mathbb{Z}}\eta^m\cdot\frac{\gamma^m}{m!}e^{-\gamma} \end{align*} Now from here I do not know how to write this sum in a closed form, any hints or help would be greatly appreciated.