The Dunkl operators associated to a dihedral group are a pair of differential-difference operators that generate a commutative algebra acting on differentiable functions in $\mathbb{R}^2$. The intertwining operator intertwines between this algebra and the algebra of differential operators. The main result of this paper is an integral representation of the intertwining operator on a class of functions. As an application, closed formulas for the Poisson kernels of $h$-harmonics and sieved Gegenbauer polynomials are deduced when one of the variables is at vertices of a regular polygon, and similar formulas are also derived for several other related families of orthogonal polynomials.

中文翻译：

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与二面体群相关的交织算子

与二面体群相关联的 Dunkl 算子是一对微分差分算子，它们生成作用于 $\mathbb{R}^2$ 中可微函数的交换代数。交织算子交织在这个代数和微分算子的代数之间。本文的主要结果是对一类函数的交织算子的积分表示。作为应用，当变量之一位于正多边形的顶点时，推导出$h$-谐波和筛分Gegenbauer多项式的泊松核的闭合公式，并且还推导出其他几个相关的正交多项式族的类似公式。