I have the following question:
Consider a random variable $M \sim \mathrm{Poisson} (\lambda)$. I would like to evaluate the following expectation:
\begin{align*} \mathbb{E} \left[ 1 / \Gamma (M + a) \right] \end{align*} where $a$ is a fixed positive integer and $\Gamma (\cdot)$ is just the gamma function.
I would like to obtain an analytical expression for the above quantity. If some further restriction helps, we can consider the case where both $a$ and $\lambda$ are large.