Bessel Polynomial Linearization Coefficients

Linearization coefficients for Bessel polynomials, and certain rational polynomial spectra.

Three simple functions generate linearization coefficients for Bessel polynomials. Three additional functions apply Bessel polynomial expansions to calculate rational polynomial spectra of certain correlation functions.

Applications center on the modified Bessel correlation function. Its Fourier transform is a simple rational polynomial spectrum that the physicist or engineer will immediately associate with cascade single-pole filters.

Expansions in Bessel polynomials facilitate analysis and numerical evaluation of more complicated product forms of these correlation functions. What was a Fourier transform to a hypergeometric function, instead expands into a sum of elementary rational polynomial spectra weighted by linearization coefficients. Best of all, this can be done in an efficient and numerically stable way!

These results are applied to calculate the spectrum of the correlation function of a Gamma-Gamma fading process describing laser beam propagation through turbulent media.