  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 $1/\bar z$.
