 Solution

The additional transition is $n\to n+k$ at rate $ϵ(n)$. Its gain term at state $n$ comes from state $n-k$, so the new master equation iswhere the gain term is zero when $n<k$. In the generating-function equation the new contribution isFor a general state-dependent $ϵ$, this is not closed in $\varphi$. Even when it is closed, $s^k-1$ is not proportional to $s-1$ for $k>1$, so the Poisson ansatz used in part (a) is not preserved.