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.
Categories: math.CA