Sufficiency, completeness, UMVUE, Cramér–Rao

An unbiased estimator g(X) must satisfy $\sum g(k){\lambda }^k{e}^{-\lambda }/k$! $=h(\lambda )$; the left side is $e^{-\lambda }$ times a power series in $\lambda$, which can never equal $\lambda$⁻${}^1($unbounded at $0).\lambda ,{\lambda }^2$ and $e^{-\lambda }$ = P(X = 0) are all estimable.