Series with product of different power

Question

I would like to know the analytical expression for the following series (if it exists):

$\sum_{n=0}^{\infty} \frac{1}{n!} a^n b^{n^2}$

Does anybody have a clue on how to proceed?

Answer 1

There have been questions on $\sum_{n \ge 0} x^{n^2}$ and similar lacunary series here. Such functions have singularities that are dense on the boundary of the circle of convergence, and can't be extended outside.