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Asymptotic behavior of orthogonal polynomials on the circle, with respect toa weight having a fractional zero on the torus. Applications to the eigenvaluesof certain unitary random matrices. This paper is devoted to the orthogonalpolynomial 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 thispolynomial and of $\Phi^{(p)}_{N}(1)$ for all integer $p$. These results allowus to obtain an asymptotic expansion of the associated Christofel-Darbouxkernel, and to compute the distribution of the eigenvalues of a family ofrandom unitary matrices. The proof of the resuts related with the orthogonalpolynomials are essentialy based on the inversion of Toeplitz matice associatedto the symbol $f$.

Comportement asymptotique des polyn\^omes orthogonaux associ\'es \`a un  poids ayant un z\'ero d'ordre fractionnaire sur le cercle. Applications aux  valeurs propres d'une classe de matrices al\'eatoires unitaires