Meromorphic Jost Functions and Asymptotic Expansions for Jacobi Parameters

We show that the parameters $a_n$, $b_n$ of a Jacobi matrix have a complete asymptotic expansion
$$ a_n^2-1=\sum_{k=1}^{K(R)} p_k(n) \mu_k^{-2n}+ O(R^{-2n}),\qquad b_n=\sum_{k=1}^{K(R)} p_k(n)\mu_k^{-2n+1}+O(R^{-2n}), $$
where $1<|\mu_j|<R$ for $j\le K(R)$ 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.