I have a Poisson process where new elements arrive to a set with Poisson intensity $\lambda$. Initially, there are $N_0$ elements in the set. The probability that there are $N_0 + M$ elements in the set at time $t$ is $Pr[N(t) = N_0+M] = \frac{(\lambda t)^M}{M!} e^{-\lambda t}$.
I'm interested in the expected number of subsets of size $k$. That is, I want to compute
$$f(k,t) = \sum_{M=0}^{\infty} \frac{(\lambda t)^M}{M!} e^{-\lambda t} \binom{N_0+M}{k}$$
Is there some formula or approximation for this expectation when $N_0$ is large?
