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?