Orthogonal Laurent polynomials of two real variables

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 $\hat{I}¼$ defined on ${\mathbb{R}}^2$ such that $\{x=0\}\cup \{y=0\}\notin \text{supp}(\hat{I}¼)$. This ordering is suitable for considering the {\em multiplication plus inverse multiplication operator} on each varibale $(x+\frac{1}{x}$ and $y+\frac{1}{y})$, 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.