Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle

Strong asymptotics of orthogonal polynomials on the unit circle with respect to a weight of the form
$W{(z)}=w{(z)}\prod\nolimits_{k=1}^m{\vert z-a_k\vert}^{2\beta_k}$
, |z| = 1, |a
k
| = 1, β
k
> −1/2, k = 1, …, m, where w(z) > 0 for |z| = 1 can be extended as a holomorphic and nonvanishing function to an annulus containing the unit circle. The formulas obtained are valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials, the behavior of their leading and Verblunsky coefficients, and we give an alternative proof of the Fisher-Hartwig conjecture about the asymptotics of Toeplitz determinants for such type of weights. The main technique is the steepest descent analysis of Deift and Zhou, based on the matrix Riemann-Hilbert characterization proposed by Fokas, Its, and Kitaev.