Complete statistic: Poisson Distribution

If $s(\lambda) = \sum_{k=0}^\infty h(k)\frac{(n\lambda)^k}{k!}$ and $s(\lambda) =0$ for all $\lambda$, then clearly $h(0)=0$ since $s(0)=h(0)$.

Similarly if you find the $m$th derivative of $s(\lambda)$ at $\lambda=0$, which must also be $0$, you will have $h(m)=0$ for all $m$.