First hitting time of the boundary of a wedge of angle pi/4 by a radial Dunkl process

Abstract

In this paper, we establish an integral representation for the density of the reciprocal of the first hitting time of the boundary of even dihedral wedges by a radial Dunkl process having equal multiplicity values. Doing so provides another proof and extends to all even dihedral groups the main result proved in [11]. We also express the weighted Laplace transform of this density through the fourth Lauricella function and establish similar results for odd dihedral wedges.

On a Neumann-type series for modified Bessel functions of the first kind

“…Consequently, one gets in both cases analogous representations of (C (3) In [7], the following representation of Gegenbauer polynomials was derived for integer k ≥ 1: led to the interesting identity (9). The latter has been recently used by the secondly named author in order to solve important problems related to the first hitting time of the boundary of a dihedral wedge by a radial Dunkl process ( [6]). On the other hand, the obtained Horn's series is a major step toward the conjectured Laplace-type representation of the generalized Bessel function associated with dihedral groups.…”

“…This expansion is an instance of a more general one (see for instance [11, page 213] or [14]) and is also the inverse (in the sense of composition) of (6). In this direction, we can extend it to even integers n = 2q, q ≥ 1, as follows.…”