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) = ∏m j=1(z − zj(t)) ρj , consisting of m ∈ 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 zj(t) (Schlesinger equations) are derived completely characterising the system.