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.