Beta distributions and Sonine integrals for Bessel functions on symmetric cones

TL;DR

This paper extends Sonine integral formulas for Bessel functions on symmetric cones to a broader range of parameters by analytically continuing Beta distributions as tempered distributions, revealing new admissible indices.

Contribution

It introduces an analytic extension of Beta measures as tempered distributions for Bessel functions on symmetric cones, expanding the known parameter range for Sonine formulas.

Findings

Extended Sonine formulas to new parameter ranges.

Identified conditions when Beta distributions remain measures.

Characterized indices where Sonine formulas exist, revealing gaps in admissible ranges.

Abstract

There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index $\mu+ \nu$ with respect to such functions of lower index $\mu.$ For Bessel functions on matrix cones, Sonine formulas involve beta densities $\beta_{\mu,\nu}$ on the cone and trace already back to Herz. The Sonine representations known so far on symmetric cones are restricted to continuous ranges $\Re\mu, \Re \nu > \mu_0$, where the involved Beta densities are probability measures and the limiting index $\mu_0\geq 0$ depends on the rank of the cone. It is zero only in the one-dimensional case, but larger than zero in all multivariate cases. In this paper, we study the extension of Sonine formulas for Bessel functions on symmetric cones to values of $\nu$ below the critical limit $\mu_0$. This is achieved by an analytic extension of the involved Beta measures as tempered distributions. Following recent ideas by A. Sokal for Riesz distributions on symmetric cones, we analyze for which indices the obtained Beta distributions are still measures. At the same time, we characterize the indices for which a Sonine formula between the related Bessel functions exists. As for Riesz distributions, there occur gaps in the admissible range of indices which are determined by the so-called Wallach set.