Vector-Valued Polynomials and a Matrix Weight Function with $B_2$-Action

TL;DR

This paper develops a theory of vector-valued orthogonal polynomials associated with the $B_2$ reflection group, constructing an orthogonal basis, an exponential kernel, and analyzing the inner product structure with matrix weights.

Contribution

It introduces a new framework for vector-valued polynomials with $B_2$ symmetry, including explicit operators, basis construction, and matrix kernel representation.

Findings

Orthogonal basis for harmonic vector-valued polynomials constructed.

Inner product positive only under specific parameter conditions.

Matrix kernel expressed via hypergeometric functions.

Abstract

The structure of orthogonal polynomials on $\mathbb{R}^{2}$ with the weight function $| x_{1}^{2}-x_{2}^{2}|^{2k_{0}}| x_{1}x_{2}|^{2k_{1}}e^{-(x_{1}^{2}+x_{2}^{2})/2}$ is based on the Dunkl operators of type $B_{2}$. This refers to the full symmetry group of the square, generated by reflections in the lines $x_{1}=0$ and $x_{1}-x_{2}=0$. The weight function is integrable if $k_{0},k_{1},k_{0}+k_{1}>-\frac{1}{2}$. Dunkl operators can be defined for polynomials taking values in a module of the associated reflection group, that is, a vector space on which the group has an irreducible representation. The unique 2-dimensional representation of the group $B_{2}$ is used here. The specific operators for this group and an analysis of the inner products on the harmonic vector-valued polynomials are presented in this paper. An orthogonal basis for the harmonic polynomials is constructed, and is used to define an exponential-type kernel. In contrast to the ordinary scalar case the inner product structure is positive only when $(k_{0},k_{1})$ satisfy $-\frac{1}{2}<k_{0}\pm k_{1}<\frac{1}{2}$. For vector polynomials $(f_{i})_{i=1}^{2}$, $(g_{i})_{i=1}^{2}$ the inner product has the form $\iint_{\mathbb{R}^{2}}f(x) K(x) g(x)^{T}e^{-(x_{1}^{2}+x_{2}^{2})/2}dx_{1}dx_{2}$ where the matrix function $K(x)$ has to satisfy various transformation and boundary conditions. The matrix $K$ is expressed in terms of hypergeometric functions.