Meromorphic Szego functions and asymptotic series for Verblunsky coefficients

Abstract

We prove that the Szeg\H{o} function, $D(z)$, of a measure on the unit circle is entire meromorphic if and only if the Verblunsky coefficients have an asymptotic expansion in exponentials. We relate the positions of the poles of $D(z)^{-1}$ to the exponential rates in the asymptotic expansion. Basically, either set is contained in the sets generated from the other by considering products of the form, $z_1 ... z_\ell \bar z_{\ell-1}... \bar z_{2\ell-1}$ with $z_j$ in the set. The proofs use nothing more than iterated Szeg\H{o} recursion at $z$ and $1/\bar z$.