Asymptotic behaviour of Verblunsky coefficients

Let V (z) = m j=1(z − ζj ), ζh = ζk, h = k and |ζj| = 1, j = 1, . . . , m, and consider the polynomials orthogonal with respect to |V |2 dμ, ϕn(|V |2 dμ;z), where μ is a finite positive Borel measure on the unit circle with infinite points in its support, such that the reciprocal of its Szeg˝o function has an analytic extension beyond |z| < 1. In this paper we deduce the asymptotic behaviour of their Verblunsky coefficients. By means of this result, an asymptotic representation for these polynomials inside the unit circle is also obtained. © 2006 Elsevier Inc. All rights reserved.