Opdam's hypergeometric functions: product formula and convolution structure in dimension 1

TL;DR

This paper derives a product formula for Opdam's hypergeometric functions related to Dunkl-Cherednik operators, explores their convolution structure, and connects to R"osler's Dunkl kernel via a rational limit.

Contribution

It provides an explicit integral product formula for Opdam's hypergeometric functions and introduces a convolution structure, extending Dunkl analysis in one dimension.

Findings

Derived an explicit integral product formula for $G_{\lambda}^{(\alpha,eta)}$

Established a convolution structure associated with these functions

Connected the product formula to R"osler's Dunkl kernel via a rational limit

Abstract

Let $G_{\lambda}^{(\alpha,\beta)}$ be the eigenfunctions of the Dunkl-Cherednik operator $T^{(\alpha,\beta)}$ on $\mathbb{R}$. In this paper we express the product $G_{\lambda}^{(\alpha,\beta)}(x)G_{\lambda}^{(\alpha,\beta)}(y)$ as an integral in terms of $G_{\lambda}^{(\alpha,\beta)}(z)$ with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula of M. R\"osler for the Dunkl kernel. We then define and study a convolution structure associated to $G_{\lambda}^{(\alpha,\beta)}$.