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.
From https://functions.wolfram.com/Bessel-TypeFunctions/BesselI/11/ $$ \sum_{k=-\infty}^\infty I_k(x)t^k = \exp\left(\frac{x}{2}\left(t+\frac{1}{t}\right)\right) \\ \sum_{k=-\infty}^\infty J_k(x)t^k = \exp\left(\frac{x}{2}\left(t-\frac{1}{t}\right)\right) $$ In particular, $I_k(x) \to 0$ as $k\to\infty$ faster than any power $t^k$.
The most comprehensive general reference is G. N. Watson, Treatise on the theory of Bessel functions, multiple editions, the latest: Cambridge 1995. It contains the asymptotics you are looking for.
