Leonid Golinskii, Andrej Zlatos
Coefficients of Orthogonal Polynomials on the Unit Circle and 
Higher Order Szego Theorems
(72K, Latex 2e)

ABSTRACT.  Let $\mu$ be a non-trivial probability measure on the unit circle 
$\partial\bbD$, $w$ the density of its absolutely continuous part, 
$\alpha_n$ its Verblunsky coefficients, and $\Phi_n$ its monic 
orthogonal polynomials. In this paper we compute the coefficients 
of $\Phi_n$ 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 $\alpha_n \in \ell^4$ and $Q(z) \equiv 
\sum_{m=0}^N q_m z^m$ is a polynomial, then with $\bar Q(z) \equiv 
\sum_{m=0}^N \bar q_m z^m$ and $S$ the left shift operator on 
sequences we have $|Q(e^{i\theta})|^2 \log w(\theta) \in L^1(d\theta)$ if and only if $\{\bar Q(S)\alpha\}_n \in \ell^2$. We also study relative ratio asymptotics of the reversed polynomials 
$\Phi_{n+1}^*(\mu)/\Phi_n^*(\mu)-\Phi_{n+1}^*(\nu)/\Phi_n^*(\nu)$ 
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 
$\bbD$.