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?

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