Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC

Abstract

In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type $BC$. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds $G/K$ over one of the (skew) fields $\mathbb F= \mathbb R, \mathbb C, \mathbb H.$ We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hypergroup algebras which include the commutative convolution algebras of $K$-biinvariant functions on $G$.