Is there something similar to Jacobi's anger expansion for powers of modified Bessel functions?
What I mean is an evaluation of the series $$ \sum_{n\in\mathbb{Z}}\mathrm{e}^{\mathrm{i}n\theta}\left(I_n(x)\right)^m $$ where $x>0,\theta\in\mathbb{R}$ and $m\in\mathbb{N}$ and large?
I would like to prove the conjectured identity $$ \lim_{m\to\infty}\frac{\sum_{n\in\mathbb{Z}}\mathrm{e}^{\mathrm{i}n\theta}\left(I_n(x)\right)^m}{\sum_{n\in\mathbb{Z}}\left(I_n(x)\right)^m} = 1 \qquad(\theta\in[0,2\pi))\,.$$
It seems to me like this problem is somewhat related to the following intution:
One could think of $\mathbb{Z}\ni n \mapsto \exp(-x)I_n(x)$ as a discrete Gaussian, i.e., \begin{align} \sum_{n\in\mathbb{Z}}\mathrm{e}^{\mathrm{i}n\theta}\left(I_n(x)\right)^m &= \exp(+mx)\sum_{n\in\mathbb{Z}}\mathrm{e}^{\mathrm{i}n\theta}\left(\exp(-x)I_n(x)\right)^m\\&\approx \exp(+mx)\sum_{n\in\mathbb{Z}}\mathrm{e}^{\mathrm{i}n\theta}\exp(-xmn^2)\\&=\exp(mx)\Theta_3(\theta/2,\exp(-mx/2)) \end{align}where $\Theta_3$ is the Jacobi elliptic theta function, and then Mathematica says $$ \lim_{m\to\infty} \frac{\Theta_3(\theta/2,\exp(-mx/2)) }{\Theta_3(0,\exp(-mx/2)) } = 1\,. $$
However it is not clear how to make this rigorous...