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arXiv · 1310.3075

Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type BC

Michael Voit
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Abstract

In this paper we present explicit product formulas for a continuous two-parameter family of Heckman-Opdam hypergeometric functions of type BC on Weyl chambers $C_q\subset \mathbb R^q$ of type $B$. These formulas are related to continuous one-parameter families of probability-preserving convolution structures on $C_q\times\mathbb R$. These convolutions on $C_q\times\mathbb R$ are constructed via product formulas for the spherical functions of the symmetric spaces $U(p,q)/ (U(p)\times SU(q))$ and associated double coset convolutions on $C_q\times\mathbb T$ with the torus $\mathbb T$. We shall obtain positive product formulas for a restricted parameter set only, while the associated convolutions are always norm-decreasing. Our paper is related to recent positive product formulas of Rösler for three series of Heckman-Opdam hypergeometric functions of type BC as well as to classical product formulas for Jacobi functions of Koornwinder and Trimeche for rank $q=1$.

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Michael Voit. 2013-10-11. Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type BC. https://arxiv.org/abs/1310.3075

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