  We perform the spectral analysis of a family of Jacobi operators $J(\alpha)$
depending on a complex parameter $\alpha$. If $|\alpha|\neq1$ the spectrum of
$J(\alpha)$ is discrete and formulas for eigenvalues and eigenvectors are
established in terms of elliptic integrals and Jacobian elliptic functions. If
$|\alpha|=1$, $\alpha \neq \pm 1$, the essential spectrum of $J(\alpha)$ covers
the entire complex plane. In addition, a formula for the Weyl $m$-function as
well as the asymptotic expansions of solutions of the difference equation
corresponding to $J(\alpha)$ are obtained. Finally, the completeness of
eigenvectors and Rodriguez-like formulas for orthogonal polynomials, studied
previously by Carlitz, are proved.
