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

In the first part of this paper, we express the generalized Bessel function associated with dihedral systems and a constant multiplicity function as an 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 Amri and Demni (Moscow Math J 17(2):1–15, 2017).