I'm interested in a closed-form (special functions included) for the integer moments of the Poisson distribution. Let $X\sim\operatorname{Poisson}(\lambda)$. Then we have for the moment generating function $$ M_X(t)=\exp(\lambda(e^t-1))=\sum_{k=0}^\infty(\mathsf EX^k)\frac{t^k}{k!}. $$ So if we can write the moment generating function as a Taylor series in $t$ we can extract the closed-form I'm looking for from the coefficients of said series. Write $$ M_X(t)=e^{-\lambda}\exp(\lambda e^t)=e^{-\lambda}\sum_{n=0}^\infty\frac{\lambda^n e^{nt}}{n!}. $$ Expanding the remining exponential and interchanging order of summation gives $$ M_X(t)=e^{-\lambda}\sum_{n=0}^\infty\frac{\lambda^n}{n!}\sum_{k=0}^\infty\frac{n^kt^k}{k!} =\sum_{k=0}^\infty \left(e^{-\lambda}\sum_{n=0}^\infty\frac{\lambda^n n^k}{n!}\right)\frac{t^k}{k!}. $$ So we have $$ \mathsf EX^k=e^{-\lambda}\sum_{n=0}^\infty\frac{\lambda^n n^k}{n!}. $$ Can we write this series in terms of known special functions? I'm pretty sure this can be written as a higher-order hypergeometric function (something like $_{k+1}F_{k}$) but not sure if there are other simpler closed-forms.
Edit:
We can write a hypergeometric form. For $k=1,2,\dots$ the series evaluates to $$ \mathsf EX^k=\lambda e^{-\lambda} {_{k-1}F_{k-1}}\left({2,\dots,2 \atop1,\dots,1};\lambda\right). $$ But can this be reduced to simpler special functions?
For completeness, @Kurt.G posted a link to the Wiki article on the Poisson distribution. It turns out that for $X\sim\operatorname{Poisson}(\lambda)$ $$ \mathsf EX^k=T_n(\lambda)=B_n(\lambda,\dots,\lambda), $$ where $T_n(x)$ is the $n$th Touchard Polynomial, which can also be written in terms of the the complete Bell polynomial $B_n(x,\dots,x)$.
For a simple Mathematica implementation we have the one-liner