Open Access
November 2015 Probabilistic proof of product formulas for Bessel functions
Luc Deleaval, Nizar Demni
Bernoulli 21(4): 2419-2429 (November 2015). DOI: 10.3150/14-BEJ649


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.


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Luc Deleaval. Nizar Demni. "Probabilistic proof of product formulas for Bessel functions." Bernoulli 21 (4) 2419 - 2429, November 2015.


Received: 1 January 2013; Revised: 1 January 2014; Published: November 2015
First available in Project Euclid: 5 August 2015

zbMATH: 1364.33009
MathSciNet: MR3378472
Digital Object Identifier: 10.3150/14-BEJ649

Keywords: Conditional independence , hypergeometric functions , Matrix-variate normal distribution , product formula

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

Vol.21 • No. 4 • November 2015
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