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Let   be a nontrivial probability measure on the unit circle   the density of its absolutely continuous part,   its Verblunsky coefficients, and   its monic orthogonal polynomials. In this paper we compute the coefficients of   in terms of the   . If the function   is in   , we do the same for its Fourier coefficients. As an application we prove that if   and if   is a polynomial, then with   and S the left-shift operator on sequences we have             We also study relative ratio asymptotics of the reversed polynomials   and provide a necessary and sufficient condition in terms of the Verblunsky coefficients of the measures   and   for this difference to converge to zero uniformly on compact subsets of   .

Coefficients of Orthogonal Polynomials on the Unit Circle and Higher-Order Szego Theorems