Mixed Multiple Orthogonal Laurent Polynomials on the Unit Circle

TL;DR

This paper introduces mixed multiple orthogonal Laurent polynomials on the unit circle, analyzing their properties, recurrence relations, and perturbations, expanding the theoretical framework of orthogonal polynomials in complex analysis.

Contribution

It develops a systematic theory for mixed multiple orthogonal Laurent polynomials on the unit circle, including their recurrence relations, Christoffel-Darboux kernels, and measure perturbations.

Findings

Derived recurrence relations in banded matrix form

Established Christoffel-Darboux kernels and relations

Formulated Christoffel and Geronimus perturbation formulas

Abstract

Mixed orthogonal Laurent polynomials on the unit circle of CMV type are constructed utilizing a matrix of moments and its Gauss--Borel factorization and employing a multiple extension of the CMV ordering. A systematic analysis of the associated multiple orthogonality and biorthogonality relations, and an examination of the degrees of the Laurent polynomials is given. Recurrence relations, expressed in terms of banded matrices, are found. These recurrence relations lay the groundwork for corresponding Christoffel-Darboux kernels and relations, as well as for elucidating the ABC theorem. The paper also develops the theory of diagonal Christoffel and Geronimus perturbations of the matrix of measures. Christoffel formulas are found for both perturbations.