TITLE: What is $\sum_{k=0}^\infty \frac{k^nz^k}{k!}$?
QUESTION [5 upvotes]: Let $n\in\mathbb{N}$. Can you find a simple expression for the function
$f_n(z)=\sum_{k=0}^\infty \frac{k^nz^k}{k!}$.
By simple expression I just mean a finite linear combination of powers of $z$, exponential functions, (or something else?)

For example in the case $n=1$, we have
$f_1(z)=ze^z$.
I am really struggling to see an expression for the general $n\in\mathbb{N}$ case. Any help would be much appreciated.

REPLY [5 votes]: In terms of Stirling numbers of the second kind for any $n\geq 1$ we have
$$ k^n = \sum_{j=1}^{n}{n \brace j}\binom{k}{j}j! \tag{1}$$
and since
$$ \sum_{k\geq 0}\frac{z^k}{k!}\binom{k}{j}j! = z^j e^z \tag{2}$$
it follows that
$$ \sum_{k\geq 0}\frac{k^n z^j}{k!} = e^z\sum_{j=1}^{n}{n\brace j}z^j\tag{3} $$
so the wanted series is the product between $e^z$ and a Bell polynomial.