  In this paper, we extend the iterative expression for the generalized
spherical functions associated to the root systems of type $A$ previously
obtained beyond regular elements. We also provide the corresponding expression
in the flat case. From there, we derive a Laplace-type representation for the
generalized spherical functions associated to the root systems of type $A$ in
the Dunkl setting as well as in the trigonometric Dunkl setting. This
representation leads us to describe precisely the support of the generalized
Abel transform. Thanks to a recent result of Rejeb, this allows us to give the
support for the Dunkl intertwining operator.
