Orthogonal Laurent polynomials of two real variables

TL;DR

This paper develops a framework for orthogonal Laurent polynomials in two variables using a specific monomial ordering, leading to recurrence relations, kernel formulas, and connections to one-variable cases, with potential for future applications.

Contribution

It introduces a novel ordering of Laurent monomials for two variables, enabling the derivation of recurrence relations and kernel formulas for orthogonal Laurent polynomials.

Findings

Derived five-term recurrence relations for the polynomials.

Established Christoffel-Darboux and kernel formulas.

Connected the two-variable case to the one-variable scenario.

Abstract

In this paper we consider an appropriate ordering of the Laurent monomials $x^{i}y^{j}$, $i,j \in \mathbb{Z}$ that allows us to study sequences of orthogonal Laurent polynomials of the real variables $x$ and $y$ with respect to a positive Borel measure $\mu$ defined on $\mathbb{R}^2$ such that $\{ x=0 \}\cup \{ y=0 \} \not\in \textrm{supp}(\mu)$. This ordering is suitable for considering the {\em multiplication plus inverse multiplication operator} on each varibale $\left( x+\frac{1}{x}\right.$ and $\left. y+\frac{1}{y}\right)$, and as a result we obtain five-term recurrence relations, Christoffel-Darboux and confluent formulas for the reproducing kernel and a related Favard's theorem. A connection with the one variable case is also presented, along with some applications for future research.