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) = πm j=1(z-z j(t)))Pj, consisting of m ∈ ℤ> 0 finite singularities, difference equations with respect to the bi-orthogonal polynomial degree..