A probabilistic proof of product formulas for spherical Bessel functions and their matrix analogues
Abstract
We write, for geometric index values, a probabilistic proof of the product formula for spherical Bessel functions. Our proof has the merit to carry over without any further effort to Bessel-type hypergeometric functions of one matrix argument. Moreover, the representative probability distribution involved in the matrix setting is shown to be closely related to matrix-variate normal distributions and to the symmetrization of upper-left corners of Haar distributed orthogonal matrices. Once we did, we use the latter relation to perform a detailed analysis of this probability distribution. In case it is absolutely continuous with respect to Lebesgue measure on the space of real symmetric matrices, the product formula for Bessel-type hypergeometric functions of two matrix arguments is obtained from Weyl integration formula.
Keywords
Cite
@article{arxiv.1202.5165,
title = {A probabilistic proof of product formulas for spherical Bessel functions and their matrix analogues},
author = {Luc Deleaval and Nizar Demni},
journal= {arXiv preprint arXiv:1202.5165},
year = {2012}
}