Probabilistic proof of product formulas for Bessel functions

We write, for geometric index values, a probabilistic proof of the product formula for spherical Bessel functions. Though our proof looks elementary in the real variable setting, it has the merit to carry over without any further effort to Bessel-type hypergeometric functions of one matrix argument, thereby avoid complicated arguments from differential geometry. Moreover, the representative probability distribution involved in the last setting is shown to be closely related to the symmetrization of upper-left corners of Haar-distributed orthogonal matrices. Analysis of this probability distribution is then performed and in case it is absolutely continuous with respect to Lebesgue measure on the space of real symmetric matrices, we derive an invariance-property of its density. As a by-product, Weyl integration formula leads to the product formula for Bessel-type hypergeometric functions of two matrix arguments.