# Integral expressions for Bessel-like power series

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?

• even something simple as $\sum_{k=0}^\infty 1/\sqrt{k!}$ doesn't seem to be a known special function... – Carlo Beenakker Feb 21 '20 at 19:59
• 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. – MCH Feb 21 '20 at 21:56
• 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 $\alpha < 1$]. – Noam D. Elkies Feb 27 at 16:37

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.