Product Formulas for a Two-Parameter Family of Heckman-Opdam Hypergeometric Functions of Type BC

Michael Voit¹

¹Fakultät Mathematik, Technische Universität, Vogelpothsweg 87, 44221 Dortmund, Germany

Journal of Lie Theory, Tome 25 (2015) no. 1, pp. 9-36

Résumé

\def\R{{\Bbb R}} \def\T{{\Bbb T}} 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 $C_q\times\R$. 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 norm-decreasing. \endgraf Our paper is related to recent positive product formulas of R\"osler 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 $q=1$.

Détail

Classification : 33C67, 43A90, 43A62, 33C80
Mots-clés : Hypergeometric functions associated with root systems, Heckman-Opdam theory, hypergroups, product formulas, Grassmann manifolds, spherical functions, signed hypergroups, Haar measure