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.
Categories: math.NA cs.NA