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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.

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I'm interested in the interpretation you mention for the Hermite polynomials' generating function. Do you have a reference? –  Antonio Vargas May 4 '12 at 16:04
    
@AntonioVargas Not at this moment. I will ask my advisor, he had a reference. –  Jonas Teuwen May 4 '12 at 16:05
    
I originally had an answer identifying the generating function as a velocity potential for fluid flow around a long circular cylinder but I was completely wrong! –  Antonio Vargas May 13 '12 at 2:22
    
@AntonioVargas What was completely wrong about it? –  Jonas Teuwen May 13 '12 at 8:26
    
The potential for that fluid flow was $(w/2)(z+r^2/z)$, which doesn't have the exponential and has a different sign on the $1/z$ term. I can't think of a physical reason for exponentiating a potential or for using a complex "radius" $r$, so I don't think the interpretation can be fixed. –  Antonio Vargas May 13 '12 at 15:10

1 Answer 1

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 $r$: enter image description here

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Well, this might be what I am after. Would it be possible to elaborate maybe a bit on the pattern I have seen? (The image of circles under $f$)? Also I am interested why exactly these arguments for $f$ solve the wave equation, what does this mean? –  Jonas Teuwen May 4 '12 at 16:29
    
@JonasTeuwen I have updated the plot with some images. Are these like what you were getting? –  Sasha May 4 '12 at 16:36
    
No, I plotted $\exp (\frac12 (r e^{it} - \frac1r e^{-it}))$ which gave me something else for different $r$ :-). Maybe I messed up. These look nice too. –  Jonas Teuwen May 4 '12 at 16:40
    
Actually I plotted this in the complex plane. –  Jonas Teuwen May 4 '12 at 16:52

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