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

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.

to appear J. Approx. Theory

References in corpus (4)

• Painlevé transcendent evaluations of finite system density matrices for 1d impenetrable Bosons
• Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N)
• Biorthogonal Laurent polynomials, Toeplitz determinants, minimal Toda orbits and isomonodromic tau functions
• Fermionic approach to the evaluation of integrals of rational symmetric functions