An integral expression for the Dunkl kernel in the dihedral setting

Abstract

Abstract Using the results in [12] where a construction of the Dunkl intertwining operator for a large set of regular parameter functions is provided, we establish an integral expression for the Dunkl kernel in the context of the dihedral group 𝒟 n {\mathcal{D}_{n}} with constant parameter function k ∈ ℂ {k\in\mathbb{C}} and arbitrary order n ≥ 2 {n\geq 2} . Our main tool is a differential system that leads to the explicit expression of the Dunkl kernel whenever an appropriate solution of it is obtained. In particular, an explicit expression of the Dunkl kernel E k ⁢ ( x , y ) {E_{k}(x,y)} is given when one of its argument x or y is invariant under the action of any reflection in the dihedral group. We obtain also a generating series for the homogeneous components K m ⁢ ( x , y ) {K_{m}(x,y)} , m ∈ ℤ + {m\in\mathbb{Z}^{+}} , of the Dunkl kernel and provide new sharp estimates for the Dunkl kernel in the large context k ∈ ℂ {k\in\mathbb{C}} , n ≥ 2 {n\geq 2} and - 2 ⁢ n ⁢ k ≠ 1 , 2 , 3 , … {-2nk\neq 1,2,3,\dots\,} .