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I'm interested in power series of form $$f(z)=\sum_{k=0}^\infty \frac{z^k}{(k!)^\alpha}.$$ When $\alpha=1$, this becomes $\exp(z)$. For $\alpha=2$ this is a Bessel function and for larger integer $\alpha$ we get a hypergeometric series. These special functions ($\alpha>1$) have integral expressions in form of some integral of an elementary function.

Can anything be said for non-integral values of $\alpha$? If $\alpha>1$ is an arbitrary real number, is there a hope to write an integral expression for the sum? In general, has this form of series been studied anywhere in the literature? Any useful techniques to work with them?

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  • $\begingroup$ even something simple as $\sum_{k=0}^\infty 1/\sqrt{k!}$ doesn't seem to be a known special function... $\endgroup$ – Carlo Beenakker Feb 21 '20 at 19:59
  • $\begingroup$ It may not have a name but I was wondering if it can be written in some form (such as integral expression) to turn the sum into a continuous integral so that one can work with the function in nontrivial ways. It seems like Barnes integrals can be used to produce such expressions, so that may be a potential approach. $\endgroup$ – MCH Feb 21 '20 at 21:56
  • $\begingroup$ Related: mathoverflow.net/questions/84958/… [for $\alpha = 1/2$, but fedja's accepted answer applies to all $\alpha \in (0,1)$]; mathoverflow.net/questions/85013 [about positivity of $\sum_{r=0}^n (-1)^r {n \choose r}^{1/2}$, again with answers that generalize to $\sum_{r=0}^n (-1)^r {n \choose r}^\alpha$ with $\alpha < 1$]. $\endgroup$ – Noam D. Elkies Feb 27 at 16:37
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This function is really known and has its name: Le Roy function, cf. https://www.tandfonline.com/doi/abs/10.1080/10652469.2018.1472592?journalCode=gitr20

and references therein.

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F. Olver, in 'Asymptotics and Special Functions,' chapter 8, has shown that $$ F_\rho(x):=\sum_{j=0}^\infty \Big( \frac{x^j}{j!} \Big)^\rho \sim \frac{\exp(\rho \, x)}{\sqrt{\rho}(2\,\pi\,x)^{(\rho-1)/2}} \Big(1+O(1/x)\Big) $$ for $ 0<\rho\le 4$ and $x \to \infty.$ The OP might get some hints from that analysis.

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