7
$\begingroup$

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.

  • 1
    I'm interested in the interpretation you mention for the Hermite polynomials' generating function. Do you have a reference?2012-05-04
  • 0
    @AntonioVargas Not at this moment. I will ask my advisor, he had a reference.2012-05-04
  • 0
    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!2012-05-13
  • 0
    @AntonioVargas What was *completely* wrong about it?2012-05-13
  • 0
    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.2012-05-13

1 Answers 1