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In [M. R\"osler and M. Voit. Integral Representation and Uniform Limits forSome Heckman-Opdam Hypergeometric Functions of type BC, Transactions of theAmerican Mathematical Society, Vol. 368, No. 8, 6005-6032, 2016.], R\"osler andVoit give a formula for generalized spherical functions of type $BC$ in termsof the spherical functions of type A. We use this formula to describe preciselythe support of the associated generalized Abel transform. Furthermore, wederive a similar formula for the generalized spherical functions in therational Dunkl setting. The support of the intertwining operator V is alsodeduced.  We also show, as a consequence, that a Laplace-type expression exists for thegeneralized spherical functions both in the trigonometric Dunkl setting and inthe rational Dunkl setting.

Laplace-type representation for some generalized spherical functions of type BC