Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 043, 11 pages      arXiv:1302.3632

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

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

Received February 15, 2013, in final form June 07, 2013; Published online June 12, 2013

This is a sequel to [SIGMA 9 (2013), 007, 23 pages], in which there is a construction of a 2×2 positive-definite matrix function K(x) on R2. The entries of K(x) are expressed in terms of hypergeometric functions. This matrix is used in the formula for a Gaussian inner product related to the standard module of the rational Cherednik algebra for the group W(B2) (symmetry group of the square) associated to the (2-dimensional) reflection representation. The algebra has two parameters: k0, k1. In the previous paper K is determined up to a scalar, namely, the normalization constant. The conjecture stated there is proven in this note. An asymptotic formula for a sum of 3F2-type is derived and used for the proof.

Key words: matrix Gaussian weight function.

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  1. Dunkl C.F., Vector-valued polynomials and a matrix weight function with B2-action, SIGMA 9 (2013), 007, 23 pages, arXiv:1210.1177.

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