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 $k$.
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@Nirbhay You really shouldn't be editing old questions just to add \limits to the title. It doesn't really improve the question. It's just a stylistic choice. More importantly, it bumps the question to the front page, thereby pushing new questions off the front page. And while not as bad as using \displaystyle, it can sometimes cause alignment issues due to the fact that more vertical space is being used. – user232456 Feb 18 '17 at 15:20
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@user232456 Okay. I'll take care next time .... :-) – Feb 19 '17 at 03:54
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See also Compute $\frac{1}{e}\sum\limits_{n=0}^{\infty}\frac{n^{k}}{n!}$ for $k=0, 1, 2 ... $ and Value of $\sum_{n=0}^{\infty}\frac{n^k}{n!}$ for $k \in \mathbb{N}$. – Martin Sleziak Nov 02 '19 at 06:19
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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.
Lucian
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