System of recursive equations for the partition functions of 1D models

In this note we consider several kind of partition functions of one-dimensional models with nearest - neighbor interactions $I_n, n\in \mathbf{Z}$ and spin values $\pm 1$. We derive systems of recursive equations for each kind of such functions. These systems depend on parameters $I_n, n\in \mathbf{Z}$. Under some conditions on the parameters we describe solutions of the systems of recursive equations.