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

Abstract

In the first part of this paper, we express the generalized Bessel function associated with dihedral systems and a constant multiplicity function as a infinite series of confluent Horn functions. The key ingredient leading to this expression is an extension of an identity involving Gegenbauer polynomials proved in a previous paper by the authors, together with the use of the Poisson kernel for these polynomials. In particular, we derive an integral representation of this generalized Bessel function over the standard simplex. The second part of this paper is concerned with even dihedral systems and boundary values of one of the variables. Still assuming that the multiplicity function is constant, we obtain a Laplace-type integral representation of the corresponding generalized Bessel function, which extends to all even dihedral systems a special instance of the Laplace-type integral representation proved in [2].

Dunkl intertwining operator for symmetric groups

“…In 2020, for a restricted class of functions, an elegant integral expression was obtained for the intertwining operator in [27], using an integral over the simplex. This result was further explored in the general case in [7] and [11]. In particular, a new integral expression for the intertwining operator was given in [7].…”

The Dunkl kernel and intertwining operator for dihedral groups

“…An explicit expression in the Laplace domain for the generalized Bessel function was obtained in [7], using the same Laplace transform technique as for Dunkl dihedral kernel. Recently, a Laplace type expression for the generalized Bessel function for even dihedral groups with one variable specified is given in [14].…”