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An integral expression for the Dunkl kernel in the dihedral setting

Maslouhi M.

Journal of Applied Analysis

2017

Vol. 23, nr 2

73--84

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 Dn with constant parameter function k ∈ ℂ and arbitrary order n ≥ 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 Ek(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 Km(x, y), m ∈ ℤ+, of the Dunkl kernel and provide new sharp estimates for the Dunkl kernel in the large context k ∈ ℂ, n ≥ 2 and −2nk ≠ 1, 2, 3, . . . .