## Bernoulli

• Bernoulli
• Volume 21, Number 4 (2015), 2419-2429.

### Probabilistic proof of product formulas for Bessel functions

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 21, Number 4 (2015), 2419-2429.

Dates
Revised: January 2014
First available in Project Euclid: 5 August 2015

https://projecteuclid.org/euclid.bj/1438777599

Digital Object Identifier
doi:10.3150/14-BEJ649

Mathematical Reviews number (MathSciNet)
MR3378472

Zentralblatt MATH identifier
1364.33009

#### Citation

Deleaval, Luc; Demni, Nizar. Probabilistic proof of product formulas for Bessel functions. Bernoulli 21 (2015), no. 4, 2419--2429. doi:10.3150/14-BEJ649. https://projecteuclid.org/euclid.bj/1438777599

#### References

• [1] Biane, P., Bougerol, P. and O’Connell, N. (2009). Continuous crystal and Duistermaat–Heckman measure for Coxeter groups. Adv. Math. 221 1522–1583.
• [2] Chikuse, Y. (2003). Statistics on Special Manifolds. Lecture Notes in Statistics 174. New York: Springer.
• [3] Chybiryakov, O., Demni, N., Gallardo, L., Rösler, M., Voit, M. and Yor, M. (2008). Harmonic and Stochastic Analysis of Dunkl Processes (P. Graczyk, M. Rösler and M. Yor, eds.). Travaux en Cours 71. Paris: Hermann.
• [4] Collins, B. (2003). Intégrales matricielles et probabilités non commutatives. Ph.D. thesis, Paris VI.
• [5] Dunkl, C.F. and Xu, Y. (2001). Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications 81. Cambridge: Cambridge Univ. Press.
• [6] Faraut, J. (2008). Analysis on Lie Groups. An Introduction. Cambridge Studies in Advanced Mathematics 110. Cambridge: Cambridge Univ. Press.
• [7] Faraut, J. and Korányi, A. (1994). Analysis on Symmetric Cones. Oxford Mathematical Monographs. New York: The Clarendon Press, Oxford Univ. Press.
• [8] Federer, H. (1969). Geometric Measure Theory. Die Grundlehren der Mathematischen Wissenschaften 153. New York: Springer.
• [9] Herz, C.S. (1955). Bessel functions of matrix argument. Ann. of Math. (2) 61 474–523.
• [10] Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.
• [11] Ragozin, D.L. (1973/74). Rotation invariant measure algebras on Euclidean space. Indiana Univ. Math. J. 23 1139–1154.
• [12] Revuz, D. (1975). Markov Chains. North-Holland Mathematical Library 11. Amsterdam: North-Holland.
• [13] Revuz, D. (1997). Probabilités, Editeurs des Sciences et des Arts. Paris: Hermann.
• [14] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press. Translated from the 1990 Japanese original. Revised by the author.
• [15] Watson, G.N. (1995). A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library. Cambridge: Cambridge Univ. Press. Reprint of the second (1944) edition.