Vector-Valued Polynomials and a Matrix Weight Function with $B_2$-Action. II

TL;DR

This paper completes the construction of a specific matrix function related to the rational Cherednik algebra for the group B2, proving a previous conjecture and providing asymptotic formulas for hypergeometric sums.

Contribution

It proves a conjecture from a prior work by explicitly determining the normalization of a matrix function associated with B2 and derives asymptotic formulas for hypergeometric series.

Findings

Conjecture on the matrix function normalization is proven.

Explicit formulas for the matrix entries are established.

Asymptotic behavior of hypergeometric sums is derived.

Abstract

This is a sequel to [SIGMA 9 (2013), 007, 23 pages, arXiv:1210.1177], in which there is a construction of a $2\times2$ positive-definite matrix function $K (x)$ on $\mathbb{R}^{2}$. 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 $W(B_2)$ (symmetry group of the square) associated to the (2-dimensional) reflection representation. The algebra has two parameters: $k_{0}$, $k_{1}$. 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 $_{3}F_{2}$-type is derived and used for the proof.