Beta Distributions and Sonine Integrals for Bessel Functions on Symmetric Cones

Abstract

There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index μ+ν with respect to such functions of lower index μ. For Bessel functions on matrix cones, Sonine formulas involve beta densities βμ,ν on the cone and trace already back to Herz. The Sonine representations known so far on symmetric cones are restricted to continuous ranges ℜμ,ℜν>μ0, where the involved beta densities are probability measures and the limiting index μ0≥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 ν below the critical limit μ0. 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.

Fractional Calculus in Russia at the End of XIX Century

“…There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index µ + ν in terms of such functions of lower index µ (see, e.g., Reference [30]). For Bessel functions on matrix cones, Sonine formulas involve beta densities β µ,nu on the cone and go already back to Herz.…”

Sonine Formulas and Intertwining Operators in Dunkl Theory

“…(2) Corollary 3.7 should be compared with the results of [RV2, Section 4] for Bessel functions on symmetric cones, which are closely related to the Bessel functions J B k . In [RV2,Theorem 4] also a sufficient condition for the existence of Sonine formulas between Bessel functions on a symmetric cone is given. It is based on the knowledge of the parameters for which Riesz distributions are actually measures.…”

“…We still conjecture that V k ′ ,k is positive for k ′ > k > 0 in the A n−1 -case. We also mention that for Bessel functions on symmetric cones, Sonine formulas were recently studied in [RV2].…”

Preface: Orthogonal polynomials, special functions, and applications

“…This special issue features papers by plenary speakers at OPSFA14. These reflect the varied nature of talks at the conference with papers discussing aspects of Askey–Wilson polynomials, Sonine formulas for Bessel functions, cubatures, exceptional orthogonal polynomials, rational solutions of Painlevé equations, and multipeakon solutions of partial differential equations …”

“…Rösler and Voit study the extension of Sonine formulas for Bessel functions on symmetric cones, which is achieved by an analytic extension of the involved beta measures as tempered distributions.…”