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

Let $\mu$ be a non-trivial probability measure on the unit circle $\mathrm{\partial }\text{bbD}$, $w$ the density of its absolutely continuous part, ${\alpha }_n$ its Verblunsky coefficients, and ${\mathrm{\Phi }}_n$ its monic orthogonal polynomials. In this paper we compute the coefficients of ${\mathrm{\Phi }}_n$ in terms of the ${\alpha }_n$. If the function $\log w$ is in $L^1(d\theta )$, we do the same for its Fourier coefficients. As an application we prove that if ${\alpha }_n\in {\mathrm{\ell }}^4$ and $Q(z)={\sum }_{m=0}^N{q}_m{z}^m$ is a polynomial, then with $\bar Q(z) = \sum_{m=0}^N \bar q_m z^m$and$S$theleftshiftoperatoronsequenceswehave$|Q(e^{i\theta})|^2 \log w(\theta) \in L^1(d\theta)$ifandonlyif${\bar Q(S)\alpha}_n \in \ell^2$. We also study relative ratio asymptotics of the reversed polynomials ${\mathrm{\Phi }}_{n+1}^{\ast }(\mu )\mathrm{/}{\mathrm{\Phi }}_n^{\ast }(\mu )-{\mathrm{\Phi }}_{n+1}^{\ast }(\nu )\mathrm{/}{\mathrm{\Phi }}_n^{\ast }(\nu )$ and provide a necessary and sufficient condition in terms of the Verblunsky coefficients of the measures $\mu$ and $\nu$ for this difference to converge to zero uniformly on compact subsets of $\text{bbD}$.