Asymptotics for orthogonal polynomials beyond the analytic boundary (Q1915758)

Let \(\mu\) be a finite positive measure on the unit circle such that Szegö's condition is satisfied and Szegö's exterior function \(D(z)\) can be extended analytically to all \([|z|> \rho]\) with \(\rho< 1\). It is well-known, that in this case \(\lim_n {\varphi_n(z)\over z^n}= D(z)\), locally uniformly on all \([|z|> \rho]\). The author studies what occurs with this strong asymptotics formula when the orthogonality measure is of the form \(|\omega(z)|^2 d\mu(z)\), where \(\mu\) is as described above and \(\omega(z)\) is an algebraic polynomial whose zeros are in \([1< |z|< \rho^{- 1}]\).