Dunkl intertwining operator for symmetric groups

TL;DR

This paper provides explicit formulas for the Dunkl kernel and generalized Bessel functions of type A_{n-1} using Humbert's function, offering a new proof of Xu's integral expression for the symmetric group intertwining operator.

Contribution

It introduces explicit formulas for key functions in Dunkl theory related to symmetric groups and offers a new proof of an existing integral expression for the intertwining operator.

Findings

Explicit formulas for Dunkl kernel and Bessel functions of type A_{n-1}

New proof of Xu's integral expression for the intertwining operator

Connection established between Dunkl functions and Humbert's function

Abstract

In this note, we express explicitly the Dunkl kernel and generalized Bessel functions of type $A_{n-1}$ by the Humbert's function $\Phi_{2}^{(n)}$, 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 in [21].