
Journal of Lie Theory 25 (2015), No. 1, 009036 Copyright Heldermann Verlag 2015 Product Formulas for a TwoParameter Family of HeckmanOpdam Hypergeometric Functions of Type BC Michael Voit Fakultät Mathematik, Technische Universität, Vogelpothsweg 87, 44221 Dortmund, Germany michael.voit@math.tudortmund.de [Abstractpdf] \def\R{{\Bbb R}} \def\T{{\Bbb T}} We present explicit product formulas for a continuous twoparameter family of HeckmanOpdam hypergeometric functions of type $BC$ on Weyl chambers $C_q\subset \mathbb R^q$ of type $B$. These formulas are related to continuous oneparameter families of probabilitypreserving convolution structures on $C_q\times\R$. These convolutions on $C_q\times\R$ 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 $C_q\times\T$ with the torus $\T$. We shall obtain positive product formulas for a restricted parameter set only, while the associated convolutions are always normdecreasing. \endgraf Our paper is related to recent positive product formulas of R\"osler for three series of HeckmanOpdam hypergeometric functions of type $BC$ as well as to classical product formulas for Jacobi functions of Koornwinder and Trimeche for rank $q=1$. Keywords: Hypergeometric functions associated with root systems, HeckmanOpdam theory, hypergroups, product formulas, Grassmann manifolds, spherical functions, signed hypergroups, Haar measure. MSC: 33C67, 43A90, 43A62, 33C80 [ Fulltextpdf (392 KB)] for subscribers only. 