
SIGMA 9 (2013), 007, 23 pages arXiv:1210.1177
https://doi.org/10.3842/SIGMA.2013.007
VectorValued Polynomials and a Matrix Weight Function with B_{2}Action
Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 229044137, USA
Received October 16, 2012, in final form January 23, 2013; Published online January 30, 2013
Abstract
The structure of orthogonal polynomials on $\mathbb{R}^{2}$ with the weight
function $\vert x_{1}^{2}x_{2}^{2}\vert ^{2k_{0}}\vert
x_{1}x_{2}\vert ^{2k_{1}}e^{( x_{1}^{2}+x_{2}^{2}) /2}$ 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
vectorvalued polynomials are presented in this paper. An orthogonal basis for
the harmonic polynomials is constructed, and is used to define an
exponentialtype 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.
Key words:
matrix Gaussian weight function; harmonic polynomials.
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