Bi-orthogonal Polynomials on the Unit Circle, regular semi-classical Weights and Integrable Systems

Abstract

The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of bi-orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle $w(z)={\prod }_{j=1}^m(z-z_j(t){)}^{{\rho }_j}$, consisting of $m\in {\mathbb{Z}}_{>0}$ finite singularities, difference equations with respect to the bi-orthogonal polynomial degree $n$ (Laguerre-Freud equations or discrete analogs of the Schlesinger equations) and differential equations with respect to the deformation variables $z_j(t)$ (Schlesinger equations) are derived completely characterising the system.

This extends and supersedes math-ph/0308036

References in corpus (1)

• Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles