Is there any information on the asymptotics of $J_n(z)$ as $n\to \pm\infty$ for fixed $z$ (real or imaginary)? I originally wanted to ask about the modified Bessel functions $I_n(z)$, but found out that this was equivalent for changing $z$ from real <--> imaginary.

I am looking at quantum wave packets for quantised solitons and have to examine the convergence of sums of the form $$ \sum_{n\in\mathbb{Z}} I_n(z)\, a^n\, b^{n^2}\ . $$ Any help or references would be greately appreciated.