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

Abstract

In this paper, we are interested in a Neumann-type series for modified Bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of Bessel functions while the second one is based on a quadratic transformation for the Gauss hypergeometric function and opens the way to derive further expressions when the orders of the underlying dihedral groups are powers of two. More generally, we give another proof of De Bie & al formula expressing this series as a Φ 2 -Horn confluent hypergeometric function. In the course of proving, we shed the light on the occurrence of multiple angles in their formula through elementary symmetric functions, and get a new representation of Gegenbauer polynomials.2010 Mathematics Subject Classification. 33C45; 33C52; 33C65.

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

“…In this paper, we shall rather extend the aforementioned integral representation to all even dihedral groups with equal multiplicity values. Our proof of this extension is different from the one written in [11] and appeal to an identity proved in [7] and satisfied by ultraspherical polynomials (see eq. (15) below).…”

“…Now, recall from [7], Proposition 1, the following identity: for any integers M ≥ 0, q ≥ 1, any real numbers k > 0, ξ ∈ [0, π], we have: (15) m,j≥0 M=2m+qj…”

The Dunkl kernel and intertwining operator for dihedral groups

“…R m e −∆y/2 p(−iy) e −∆y/2 e i y,z e −|y| 2 /2 dy. (12) Applying V κ on both sides of ( 12) with respect to z, it follows that…”

Generalized Bessel functions of dihedral-type: expression as a series of confluent Horn functions and Laplace-type integral representation

“…Proof. Recall formula (5) (this is Proposition 1 in [8]): for two integers n, M such that n ≥ 1, M ≥ 0 and for ξ ∈ [0, π], we have (9) m,j≥0 M=2m+nj…”

“…Results towards this goal have been recently obtained in a series of papers ([2, 6, 7, 8, 10, 18]). In particular, the identity recalled below in (5) and proved in [8] shows that, if one of the two variables of the generalized Bessel function lies on the boundary of the dihedral wedge and if the multiplicity function is constant, then it may be expressed through the confluent Horn function Φ 2 .…”