11.8]). It would be interesting to extend the results of the current paper to that case.

It is a pleasure to thank Chuck Newman and Percy Deift for the hospitality of the Courant Institute where some of this work was done.

OPRL With Periodic Jacobi Coefficients
======================================

[s2]{} In this section, we analyze the zeros of OPRL with Jacobi coefficients obeying . We begin with a summary of the theory of transfer matrices, discriminants, and Abelian functions associated to this situation. A reference for much of this theory is von Moerbeke [@vMoer]; a discussion of the discriminant can be found in Hochstadt [@Hoch7], von Moerbeke [@vMoer], Toda [@Toda], and Last [@Last92]. The theory is close to the OPUC theory developed in Chapter 11 of [@OPUC2].

Define the $2\times 2$ matrix, $${\label}{2.1} 
A_k(z) = {\frac}{1}{a_{k+1}}\, \begin{pmatrix} 
z-b_{k+1} & -a_k \\
a_{k+1} & 0 \end{pmatrix}$$ where $${\label}{2.2} 
a_0 \equiv a_p$$ Thus $${\label}{2.3} 
\det (A_k) = {\frac}{a_k}{a_{k+1}}$$ and the abstract form of $${\label}{2.4} 
zu_n = a_{n+1} u_{n+1} + b_{n+1} u_n + a_n u_{n-1}$$ is equivalent to $${\label}{2.5} 
\binom{u_{n+1}}{u_n} = A_n \binom{u_n}{u_{n-1}}$$ So, in particular, $${\label}{2.6} 
\binom{p_{n+1}(z)}{p_n(z)} = A_n A_{n-1} \dots A_0 \binom{1}{0}$$

This motivates the definition of the transfer matrix, $${\label}{2.7} 
T_n(z) = 