Let $k∈N, \lambda \in \Bbb R$ and $X∼\operatorname{Poi}(λ)$. Is there an explicit function $f$ such that $E[f(X)]=λ(λ−1)...(λ−k+1)$?
I can prove that there exists a linear combination of the moments that works, because they contain a polynomial of every degree, but the coefficients of the moments are Stirling numbers so this linear combination is not explicit. I'd like to have a more concrete $f$ is possible.
Such a function is termed an unbiased estimator for $k! \binom {\lambda} {k}$.