Orthogonal polynomials with respect of a class of Fisher-Hartwig symbols

Abstract

In this paper we give an asymptotic of the coefficients of the orthogonal polynomials on the unit circle, with respect of a weight of type $\displaystyle{ f : \theta \mapsto \prod_{1\le j \le M} \vert 1 - e^{i(\theta_{j}-\theta)}\vert ^{2\alpha_{j}} c}$ with $\theta_{j}\in ]-\pi,\pi]$, $-\frac{1}{2} < \alpha_{j}<\frac{1}{2}$ and $c$ a sufficiently smooth function.