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An identity of the Bell numbers is given by $$B_n=\frac{1}{e}\sum_{x=1}^\infty \frac{x^n}{x!}$$ and I was wondering if it would be valid to define fractional Bell numbers in the same way, to preserve this identity. So, to calculate what I would take as $B_{1/2}$, I would have to calculate $$\frac{1}{e}\sum_{x=1}^\infty \frac{\sqrt x}{x!}$$ However, I have no idea how to calculate this sum, and I am afraid that it may not have a closed form.

Can somebody show me how to evaluate this, or a better way to define fractional Bell numbers?

NOTE: I am thinking that perhaps I can express $$\frac{\sqrt x}{x!}$$ in the form $$\int_a^b f(x,n)\,dn$$ for some $a$, $b$, $f$, so that I can convert my sum into an integral. Any ideas how to do this?

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    $\begingroup$ if there is an integral representation probably it can be extended by analytic continuation. Try to see for what values of $w$ this integral make sense $$B_w = \frac{\Gamma(w+1)}{2 \pi i e} \int_{\gamma} \frac{e^{e^z}}{z^{w+1}} \, dz. $$ $\endgroup$ – Masacroso Jul 24 '17 at 18:34
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    $\begingroup$ searching for "generalized Bell numbers" I found some proposals (not sure if equivalent or different). In particular this paper seems interesting by it geometrical interpretation. I will add also this paper that seems interesting because it is an understandable extension of the Bell numbers. $\endgroup$ – Masacroso Jul 24 '17 at 19:00
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    $\begingroup$ the extension of Bell Numbers to complex arguments through the integral cited by Masacroso is well discussed in this article $\endgroup$ – G Cab Aug 2 '17 at 15:48
  • $\begingroup$ I studied generalized Bell-numbers as an exercise for myself. Maybe you like go.helms-net.de/math/binomial_new/04_5_SummingBellStirling.pdf $\endgroup$ – Gottfried Helms Oct 15 '17 at 10:08

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