Evaluting sum $\sum \limits_{n=0}^\infty\frac{n^k}{n!}$

Inspired by this question,I was interested if the following sum has a closed form.Looking for $k$ integer I found the Dobinski's formula so that the sum when $k$ is natural number is $e\cdot B_k$ where $B_k$ is the $k$-th Bell number.I am interested if it's known whether the sum $$\sum_{n=0}^\infty\frac{n^k}{n!}$$ has a closed form for some other values of $k$.

Solution 1:

I am interested if it's known whether the sum $\displaystyle\sum_{n=0}^\infty\frac{n^k}{n!}$ has a closed form for some other values of k.

No, not really. I guess it depends on what you are willing to accept as such. For fractional values, there are no known closed forms, other than extensions of Bell numbers to fractional indices: but this is a trivial observation. Also, if you are willing to omit the first term, corresponding to $n=0$, for $k=-1$ you'll get $-\gamma+\text{Ei}(1)$, and for other negative integer values various hypergeometric functions: but one can argue that these are mere rewritings which ultimately tell us nothing about the sum's true value.