Barry Simon 
Jost Functions and Jost Solutions for Jacobi Matrices, III. Asymptotic Series for Decay and Meromorphicity
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ABSTRACT.  We show that the parameters $a_n, b_n$ of a Jacobi matrix have a complete asymptotic series 
\begin{align*} 
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}) 
\end{align*} 
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.