SIGMA 10 (2014), 044, 23 pages      arXiv:1306.6599      https://doi.org/10.3842/SIGMA.2014.044

Vector Polynomials and a Matrix Weight Associated to Dihedral Groups

Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA

Received January 22, 2014, in final form April 10, 2014; Published online April 15, 2014

Abstract
The space of polynomials in two real variables with values in a 2-dimensional irreducible module of a dihedral group is studied as a standard module for Dunkl operators. The one-parameter case is considered (omitting the two-parameter case for even dihedral groups). The matrix weight function for the Gaussian form is found explicitly by solving a boundary value problem, and then computing the normalizing constant. An orthogonal basis for the homogeneous harmonic polynomials is constructed. The coefficients of these polynomials are found to be balanced terminating ₄F₃-series.

Key words: standard module; Gaussian weight.

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