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

A Laplace-type representation of the generalized spherical functions associated to the root systems of type A