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

TL;DR

This paper develops explicit product formulas and positive convolution structures for certain Heckman-Opdam hypergeometric functions of type BC, linking them to spherical functions on Grassmann manifolds and hypergroup algebras.

Contribution

It introduces new explicit product formulas and positive convolution structures for Heckman-Opdam hypergeometric functions of type BC, extending their analytic and algebraic understanding.

Findings

Derived explicit product formulas for hypergeometric functions of type BC.

Established positive convolution structures and hypergroup algebras.

Connected hypergeometric functions to spherical functions on Grassmann manifolds.

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 $G$.