It is well known that the generating function for the Bessel function is $$f(z) = \exp \left (\frac12 \left (z - \frac1z \right ) w \right ).$$
So, we have $$f(z) = \sum_{\nu = -\infty}^{\infty} J_\nu(w) z^\nu.$$
Okay excellent! It is quite easy for those that pay attention well to the details to derive our friend the Bessel function from this (series and integral representations).
Now my question is actually: What is the (physical) interpretation of this $f(z)$? I know that for Hermite polynomials, the similar generating function is something that has to do with the random walk. This makes lots of sense thanks to our friend the Ornstein-Uhlenbeck operator!
What is it here? I have plotted $f$ for $w = 1$ under the image of a circle. That gives me some kickass animation if I let the radius grow. But what the heck is it?
The Bessel functions are intimately connected to the wave equation, so an interpretation in that direction would be nice.
I am not sure if this is what you are after, but notice that $$ f(i \mathrm{e}^{i \theta}, r) = \mathrm{e}^{i \cos(\theta) r} = \mathrm{e}^{i z} $$ That is, it is a plane wave (and solves the wave equation), and the expansion of the plane wave in a series of Bessel function is the celebrated Jacobi-Anger expansion.
Added: You seem to have plotted $\Re(f(i \mathrm{e}^{i \theta}, r)) = \cos(r \cos(\theta)$ for different values of radius $r$: