Could anyone help to derive a closed form of the following recurrence sequence? (or any comments?)
$N, M$ are positive integers and $\sigma$ is a positive real number.
\begin{align} c_0 &= \frac{1}{M^N}, \\ c_m &= \frac{M^N}{m} \sum_{k=1}^{m} \frac{(-1)^k (kN-m+k)}{k!(M+k) \sigma^{2k}} c_{m-k}. \end{align}