I want to understand the function $$f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n, \ \ \ \forall x\in\mathbb{R},$$ for any possible real $\alpha\geq0$. I know that for $\alpha$ integer, $f_\alpha(x) = P_\alpha(x)\times e^x$ with $P_\alpha$ a polynomial of degree $\alpha$. The coefficients of the polynomial can be obtained with some combinatorics. Do you have any idea on how to understand the general case?
Thanks for attention.
I'm not entirely sure how useful this partial answer is for you, but we could probably use the ideas from this paper by Flajolet and Prodinger on generalizing Dobiński's formula to complex arguments.
To wit, since
$$\mathscr{B}_k(z)\exp\,z=\sum_{n=0}^\infty \frac{n^k z^n}{n!}$$
where $\mathscr{B}_k(z)$ is the Bell polynomial, and we have the exponential generating function
$$\sum_{k=0}^\infty \frac{\mathscr{B}_k(z)}{k!}u^k=\exp((\exp\,u-1)z)$$,
we can then consider applying Cauchy's differentiation formula,
$$f^{(n)}(0)=\frac{n!}{2\pi i}\oint_\gamma \frac{f(z)}{z^{n+1}}\mathrm dz$$
to the exponential generating function. (Note that $\gamma$ can be an anticlockwise closed contour, or a Hankel contour encircling the negative real axis.)
I'll probably try out a few numerical experiments on this when I get to my own computer.