Probably the answer is negative. Your series is a restriction of the analytic function in two variables:
$$F(\zeta,q)=\sum_{n=0}^\infty\frac{q^{n^2}}{n!}\zeta^n,\quad |q|<1,$$
obtained by setting $q=\exp(-ia)$ and $\zeta=|z|^2>0$.

This important function has been studied much, see 
<a href="http://www.maths.qmul.ac.uk/~pjc/csgnotes/sokal/">lectures of Alan Sokal</a> for a survey of known results, and there is no indication of its expression in terms of other special functions.