Is there anything known about ordinary generating functions of the form $$ S(z;a) = \sum_{n=1}^\infty \Gamma(ani)z^n, $$ for $z \in \mathbb{C}$ and $a \in \mathbb{R}$, $a \neq 0$. Here $i$ is the imaginary unit. Is there a nice expression for it, or good bounds, in terms of $z,a$? Thanks.
Edit 1: An application of the root test and the fact that $|\Gamma(\sigma + it)| \sim \sqrt{2\pi}|t|^{\sigma - 1/2}e^{-\pi |t|/2}$ as $|t| \to \infty$, for $\sigma$ fixed (see for instance Corollary 16 here) shows that $S(z;a)$ has radius of convergence $e^{\pi|a|/2}$.
Edit 2: I have also posted this question here