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

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.