与二面体群有关的多项式

与二面体群相关的向量多项式和矩阵权

Vector Polynomials and a Matrix Weight Associated to Dihedral Groups

研究了值在二面体群的二维不可约模中的两实变量多项式空间作为Dunkl算子的标准模。考虑了单参数情形（省略了偶二面体群的双参数情形）。通过求解一个边值问题，然后计算归一化常数，得到了高斯形式的矩阵权函数。构造了齐次调和多项式的一个正交基。这些多项式的系数被发现是平衡终止$_4F_3$系列。

Intertwining operators associated to dihedral groups

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 simila…

球面上的Radon变换和与二面体群相关的广义Bessel函数

Radon Transform on spheres and generalized Bessel function associated with dihedral groups

受Dunkl算子理论的启发，我们考虑一个包含修正Bessel函数和Gegenbauer多项式的生成级数，它推广了L.盖根鲍尔。当Gegenbauer多项式的参数为严格的正整数时，我们实际上使用Fourier和Radon变换的反演公式来推导这个级数的闭公式。作为副产品，我们得到了偶二面体群的广义Bessel函数在重数和为整数时的一个相对简单的整表示。特别是，我们恢复以前的结果得到的平方对称性保持组，我们给一个特殊的兴趣的六边形。本文是关闭适应我们的方法，以奇数二面体群，从而耗尽列表Weyl二面体群。

Dunkl kernel associated with dihedral group

In this paper, we pursue the investigations started in \cite{Mas-You} where the authors provide a construction of the Dunkl intertwining operator for a large subset of the set of regular multiplicity values. More precisely, we make concrete the action of this operator on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function. In particular, we obtain a formula for the corresponding Dunkl kernel and another representation of the generalized Bessel function already derived in \cite{Demni0}. When the multiplicity function is every…

The Dunkl kernel and intertwining operator for dihedral groups

Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists a unique linear operator called intertwining operator which intertwines between this algebra and the algebra of standard differential operators. There also exists a generalization of the Fourier transform in this context called Dunkl transform. In this paper, we determine an integral expression for the Dunkl kernel, which is the integral kernel of the Dunkl transform, for all dihedral groups. We also determine an integral expression for the intertwining ope…

Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order

Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying `octahedral' polynomial, indexed by the degree and order and having a non-classical kind of orthogonality, is identified, and recurrences for it a…