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The first part of this paper is devoted to the study of the orthogonalpolynomial on the circle, with respect of a weight of type $f_\alpha (\theta) =(2\cos \theta- 2\cos \theta_0)^{2\alpha} c_1$ with $\theta_0 \in ]0,\pi[$, -1/2<\alpha<1/2 and c_1 a sufficiently smooth function. In a second part of thepaper we obtain an asymptotic of the entries $(T_N f_\alpha)^{-1}_{k+1,l+1}$for sufficiently large values of $k,l$, that provides a lower bound on theeigenvalues of this matrix.

Asymptotic of the terms of the Gegenbauer polynomial on the unit circle and applications to the inverse of Toeplitz matrices