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This is a generalization of Prove the following series $\sum\limits_{s=0}^\infty \frac{1}{(sn)!}$ that I have no idea how to solve.

Multisection of series allows us to show that $\sum\limits_{s=0}^\infty \frac{1}{(sn)!} =\frac{1}{n}\sum\limits_{r=0}^{n-1}\exp\left(\cos\frac{2r\pi}{n}\right)\cos\left(\sin\frac{2r\pi}{n}\right) $.

Is there a closed form for $\sum\limits_{s=0}^\infty \frac{1}{(sn)!} $ for non-integer $n$?

This would obviously have to use the Gamma function and probably involve integrals (this last guess based on just a feeling).

If this non-integral multisection could be done for $e^x$, would it work for general power series?

Inquiring minds want to know.

Note: Doing a Google search for "non-integral multisection" does not turn up anything applicable.

Any ideas out there?

marty cohen
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  • you mean for $s \in \mathbb{N}^*$ : $\sum_{n \ge 0} \frac{1}{(ns)!} = \sum_{n \ge 0}\frac{1}{n!} \frac{1}{s}\sum_{k=0}^{s-1} e^{2i \pi n k/s} = \frac{1}{s}\sum_{k=0}^{s-1} e^{e^{2i \pi nk/s}}$ – reuns Oct 02 '16 at 06:07
  • and $\Gamma(sn+1)$ instead of $(sn)!$ ? – reuns Oct 02 '16 at 06:22
  • No. I used the notation of the original question: s is the index of summation and n is the increment. The original question assumed that n was an integer, which leads to standard multisection. My question asks what happens if n is not an integer, such as 3/2 or pi. – marty cohen Oct 02 '16 at 06:24
  • Yes to the Gamma question. – marty cohen Oct 02 '16 at 06:25
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    ok but use $e^{2i \pi ks/n}$ not $\cos,\sin$. And I don't think there is a closed-form, because it leads to things of the kind $\int_a^\infty \frac{1}{\Gamma(nx+1)}dx$ – reuns Oct 02 '16 at 06:26
  • The term "transseries" comes to mind ... There is an explanation/introduction online by Gerald Edgar (It is not that I knew how to actually do this problem, but transseries-concept seems to be pretty general and possibly expressive enough for such generalizations to fractional series-indexes) – Gottfried Helms Dec 19 '16 at 07:44

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