Orthogonal Laurent polynomials in unit circle, extended CMV ordering and 2D Toda type integrable hierarchies

Abstract

Orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss--Borel factorization of a Cantero-Moral-Velazquez moment matrix, which is constructed in terms of a complex quasi-definite measure supported in the unit circle. The factorization of the moment matrix leads to orthogonal Laurent polynomials in the unit circle and the corresponding second kind functions. Jacobi operators, 5-term recursion relations and Christoffel-Darboux kernels, projecting to particular spaces of truncated Laurent polynomials, and corresponding Christoffel-Darboux formulae are obtained within this point of view in a completely algebraic way. Cantero-Moral-Velazquez sequence of Laurent monomials is generalized and recursion relations, Christoffel-Darboux kernels, projecting to general spaces of truncated Laurent polynomials and corresponding Christoffel-Darboux formulae are found in this extended context. Continuous deformations of the moment matrix are introduced and is shown how they induce a time dependant orthogonality problem related to a Toda-type integrable system, which is connected with the well known Toeplitz lattice. Using the classical integrability theory tools the Lax and Zakharov-Shabat equations are obtained. The dynamical system associated with the coefficients of the orthogonal Laurent polynomials is explicitly derived and compared with the classical Toeplitz lattice dynamical system for the Verblunsky coefficients of Szegő polynomials for a positive measure. Discrete flows are introduced and related to Darboux transformations. Finally, the representation of the orthogonal Laurent polynomials (and its second kind functions), using the formalism of Miwa shifts, in terms of $\tau$-functions is presented and bilinear equations are derived.

References in corpus (3)

• Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy
• The multicomponent 2D Toda hierarchy: Discrete flows and string equations
• Matrix biorthogonal polynomials on the unit circle and non-Abelian Ablowitz-Ladik hierarchy

Cited by in corpus (13)

• Matrix Orthogonal Laurent Polynomials on the Unit Circle and Toda Type Integrable Systems
• Multivariate orthogonal polynomial and integrable systems
• Darboux transformations for multivariate orthogonal polynomials
• On the Christoffel--Darboux formula for generalized matrix orthogonal polynomials of multigraded-Hankel type
• Another Christoffel--Darboux Formula for Multiple Orthogonal Polynomials of Mixed Type
• Isospectral flows related to Frobenius-Stickelberger-Thiele polynomials
• Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies
• The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type
• CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus perturbations
• Matrix biorthogonal polynomials in the unit circle: Riemann-Hilbert problem and matrix discrete Painleve II system
• Darboux transformations for CMV matrices
• CMV biorthogonal Laurent polynomials. II: Christoffel formulas for Geronimus-Uvarov perturbations
• Spectral quantization of discrete random walks on half-line, and orthogonal polynomials on the unit circle