Vector-Valued Polynomials and a Matrix Weight Function with B2-Action

The structure of orthogonal polynomials on R with the weight function |x1 − x2| |x1x2|1e 2 1+x 2 2)/2 is based on the Dunkl operators of type B2. This refers to the full symmetry group of the square, generated by reflections in the lines x1 = 0 and x1− x2 = 0. The weight function is integrable if k0, k1, k0 +k1 > − 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 B2 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 (k0, k1) satisfy − 12 < k0 ± k1 < 1 2 . For vector polynomials (fi) 2 i=1, (gi) 2 i=1 the inner product has the form ∫∫ R2 f(x)K(x)g(x) T e−(x 2 1+x 2 2dx1dx2 where the matrix function K(x) has to satisfy various transformation and boundary conditions. The matrix K is expressed in terms of hypergeometric functions.