  Asymptotic behavior of orthogonal polynomials on the circle, with respect to
a weight having a fractional zero on the torus. Applications to the eigenvalues
of certain unitary random matrices. This paper is devoted to the orthogonal
polynomial on the circle, with respect to a weight of type $ f=(1-\cos \theta
)^\alpha c$ where $c$ is a sufficiently smooth function and $\alpha \in
]-{1/2}, {1/2}[$. We obtain an asymptotic expansion of the coefficients of this
polynomial and of $\Phi^{(p)}_{N}(1)$ for all integer $p$. These results allow
us to obtain an asymptotic expansion of the associated Christofel-Darboux
kernel, and to compute the distribution of the eigenvalues of a family of
random unitary matrices. The proof of the resuts related with the orthogonal
polynomials are essentialy based on the inversion of Toeplitz matice associated
to the symbol $f$.
