  We show that the parameters $a_n, b_n$ of a Jacobi matrix have a complete
asymptotic series $ a_n^2 -1 &= \sum_{k=1}^{K(R)} p_k(n) \mu_k^{-2n} +
O(R^{-2n}) b_n &= \sum_{k=1}^{K(R)} p_k(n) \mu_k^{-2n+1} + O(R^{-2n}) $ where
$1 < |\mu_j| < R$ for $j\leq K(R)$ and all $R$ if and only if the Jost
function, $u$, written in terms of $z$ (where $E=z+z^{-1}$) is an entire
meromorphic function. We relate the poles of $u$ to the $\mu_j$'s.
