Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies
We connect the theory of orthogonal Laurent polynomials on the unit circle and the theory of Toda-like integrable systems using the Gauss–Borel factorization of a Cantero–Moral–Velázquez moment matrix, that we construct in terms of a complex quasi-definite measure supported on the unit circle. The factorization of the moment matrix leads to orthogonal Laurent polynomials on the unit circle and the corresponding second kind functions. We obtain Jacobi operators, 5-term recursion relations, Christoffel–Darboux kernels, and corresponding Christoffel–Darboux formulas from this point of view in a completely algebraic way. We generalize the Cantero–Moral–Velázquez sequence of Laurent monomials, recursion relations, Christoffel–Darboux kernels, and corresponding Christoffel–Darboux formulas in this extended context. We introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a Toda-type integrable system, which is connected with the well known Toeplitz lattice. We obtain the Lax and Zakharov–Shabat equations using the classical integrability theory tools. We explicitly derive the dynamical system associated with the coefficients of the orthogonal Laurent polynomials and we compare it 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, we obtain the representation of the orthogonal Laurent polynomials (and their second kind functions), using the formalism of Miwa shifts in terms of τ -functions and the subsequent bilinear equations.