Recognize this sum I have strong feeling that the function, 
$$
f_\alpha (x) = \sum_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)},
$$
is a known special function (here $\Gamma(x)$ is the usual extension of the factorial). Is this the case?   
 A: This is so-called generalized Mittag-Leffler function, more exactly the Wright function (as series ) or the Fox function (as inverse Mellin transform). A lot is known about them
Start with
1. http://en.wikipedia.org/wiki/Fox%E2%80%93Wright_function
You may use inet search with these names.
Other useful references are:


*

*A.Kilbas, M.Saigo. H-transforms: theory and applications.
http://books.google.ru/books?id=SL-HqdvUYzEC&pg=PA352&lpg=PA352&dq=Kilbas+saigo&source=bl&ots=xuQ-J79z6c&sig=RrYNqEKUIYv64RuK6urTemqUC28&hl=ru&sa=X&ei=eG5LU9bKEoa7ygOt4oCACQ&ved=0CD8Q6AEwAg#v=onepage&q=Kilbas%20saigo&f=false

*A.M. Mathai, Ram Kishore Saxena, Hans J. Haubold. The H-Function:
Theory and Applications. http://www.springer.com/physics/theoretical,+mathematical+%26+computational+physics/book/978-1-4419-0915-2

*Papers of V.Kiryakova, e.g. Multiple (multiindex) Mittag–Leffler functions and relations to generalized fractional calculus:
http://www.sciencedirect.com/science/article/pii/S0377042700002922
(and much more her papers on the subject).

*MULTI-PARAMETRIC MITTAG-LEFFLER FUNCTIONS AND THEIR EXTENSION.
Anatoly A. Kilbas , Anna A. Koroleva, Sergei V. Rogosin:
http://link.springer.com/article/10.2478/s13540-013-0024-9
and so on... For sure you will find enough in these references, hope it will be useful! 
A: This is an entire function of order $1/\alpha$ when $\alpha>1$. So for irrational $\alpha$
it cannot satisfy any linear differential equation with polynomial coefficients.
If $0<\alpha<1$, the order is $1$ but the type is minimal, so again it cannot
satisfy any such equation. This excludes most special functions.
(But does not exclude their compositions with some irrational power inside).
Entire solutions of linear differential equations with polynomial coefficients have
rational order and normal type. 
One can obtain an integral representation of this function by taking the integral representation of the Mittag-Leffler function and then a sort of Laplace transform of it. 
Edit. If $\alpha=1$ it is expressed in terms of a Bessel function as the comment below shows.
