Dunkl intertwining operator for symmetric groups

In this note, we express explicitly the Dunkl kernel and generalized Bessel functions of type A(n-1) by the Humbert's function Phi((n))(2), with one variable specified. The obtained formulas lead to a new proof of Xu's integral expression for the intertwining operator associated to symmetric groups, which was recently reported by Y. Xu [Intertwining operator associated to symmetric groups and summability on the unit sphere, arXiv:2004.08727].