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).

在本文的第一部分中，我们将与二面体系统相关的广义贝塞尔函数和恒定的多重性函数表示为汇合的Horn函数的无穷级数。导致此表达式的关键因素是作者在以前的论文中证明的涉及Gegenbauer多项式的恒等式的扩展，以及将Poisson核用于这些多项式。特别是，我们在标准单纯形上得出了此广义Bessel函数的积分表示。本文的第二部分涉及偶二面体系统和变量之一的边界值。仍然假设多重性函数是常数，我们获得了对应的广义贝塞尔函数的拉普拉斯型积分表示，