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

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

2004-12-20

arxiv.org/abs/math/0412394v1

Mathematics

Classical Analysis and ODEs

This extends and supersedes math-ph/0308036

Scientific paper

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^m_{j=1}(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.

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