A Laplace-type representation for some generalized spherical functions of type $BC$
Rösler and Voit (2016) gave a formula for the generalized spherical function of type $BC$ in terms of the spherical functions of type $A$. We use this formula to describe precisely the support of the associated generalized Abel transform. Furthermore, we derive a similar formula for the generalized spherical functions in the rational Dunkl setting. The support of the intertwining operator $V$ is also deduced.
As a consequence, we show that a Laplace-type expression exists for the generalized spherical functions both in the trigonometric Dunkl setting and in the rational Dunkl setting.