Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II

TL;DR

This paper explores transformations of bi-orthogonal systems on the unit circle, linking them to integrable systems and isomonodromic deformations, with explicit derivations of compatibility conditions and Hirota-Miwa equations.

Contribution

It introduces Christoffel-Geronimus-Uvarov transformations for bi-orthogonal polynomials on the unit circle within semi-classical weights, connecting them to integrable systems and Schlesinger transformations.

Findings

Derived Christoffel-Geronimus-Uvarov transformations for bi-orthogonal systems.

Established compatibility of Schlesinger transformations with system structures.

Formulated Hirota-Miwa equations for tau-functions and Toeplitz determinants.

Abstract

We derive the Christoffel-Geronimus-Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities $ \{z_1, ..., z_M \} $ the bi-orthogonal system is known to be isomonodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel-Geronimus-Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system - the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota-Miwa equations are derived for the $ \tau $-functions or equivalently Toeplitz determinants of the system.