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

By Jean-Philippe Anker, Fatma Ayadi, Mohamed Sifi. 2010

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Abstract. Let G (,) $\operatorname{G}_\lambda ^{(\alpha ,\beta )}$ be the eigenfunctions of the Dunkl–Cherednik operator T (,) $\operatorname{T}^{(\alpha ,\beta )}$ on $\mathbb {R}$. In this paper we express the product G (,) (x)G (,) (y)$\operatorname{G}_\lambda ^{(\alpha ,\beta )}(x)\,\operatorname{G}_\lambda ^{(\alpha ,\beta )}(y)$ as an integral in terms of G (,) (z)$\operatorname{G}_\lambda ^{(\alpha ,\beta )}(z)$ 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. Rösler for the Dunkl kernel. We then define and study a convolution structure associated to G (,) $\operatorname{G}_\lambda ^{(\alpha ,\beta )}$.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e37446241b18e8c3ec75cc429db2ea81fb0c895c