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) Π k=1 m | z-ak | 2βk, |z|=1, | ak |=1, βk >-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.