Asymptotics of polynomials orthogonal over the unit disk with respect to a polynomial weight without zeros on the unit circle

Abstract

We establish asymptotic formulas for polynomials that are orthogonal over the unit disk with respect to a weight of the form |w(z)|2, where w(z) is a polynomial without zeros on the unit circle |z|=1. The formulas put in evidence a strong connection between the behavior of the polynomials and the reproducing kernel of an associated weighted Bergman space, which produces interesting new features in the presence of a weight w with interior zeros.