CMV biorthogonal Laurent polynomials. II: Christoffel formulas for Geronimus-Uvarov perturbations

TL;DR

This paper extends the theory of CMV biorthogonal Laurent polynomials by deriving Christoffel formulas for Geronimus-Uvarov perturbations, providing explicit quasideterminantal expressions for perturbed polynomials and kernels.

Contribution

It introduces new Christoffel formulas for Laurent polynomial perturbations involving Geronimus-Uvarov transformations, expanding the analytical tools for biorthogonal Laurent polynomial families.

Findings

Derived connection formulas for perturbed polynomials and kernels.

Established quasideterminantal Christoffel formulas for general perturbations.

Provided explicit expressions for perturbations supported on the unit circle.

Abstract

This paper is a continuation of the recent paper "CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus transformations" by the same authors. The behavior of quasidefinite sesquilinear forms for Laurent polynomials in the complex plane, characterized by bivariate linear functionals, and corresponding CMV biorthogonal Laurent polynomial families --including Sobolev and discrete Sobolev orthogonalities--- under two type of Geronimus--Uvarov transformations is studied. Either the linear functionals are multiplied by a Laurent polynomial and divided by the complex conjugate of a Laurent polynomial, with the addition of appropriate masses (linear functionals supported on the zeros of the perturbing Laurent polynomial in the denominator) or vice-versa, multiplied by the complex conjugate of a Laurent polynomial and divided by a Laurent polynomial. The connection formulas for the CMV biorthogonal Laurent polynomials, their norms, and Christoffel--Darboux kernels are given. For prepared Laurent polynomials these connection formulas lead to quasideterminantal (quotient of determinants) Christoffel formulas expressing an arbitrary degree perturbed biorthogonal Laurent polynomial in terms of $2N_{\mathfrak C}+2N_\Gamma$ unperturbed biorthogonal Laurent polynomials, their second kind functions or Christoffel--Darboux kernel and its mixed versions. When the linear functionals are supported on the unit circle, a particularly relevant role is played by the reciprocal polynomial, and the Christoffel formulas provide now with two possible ways of expressing the same perturbed quantities in terms of the original ones, one using only the nonperturbed biorthogonal family of Laurent polynomials, and the other using the Christoffel--Darboux kernel and its mixed versions.