We perform the spectral analysis of a family of Jacobi operators J(alpha) depending on a complex parameter alpha. If |alpha| not equal 1 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 not equal perpendicular to 1, the essential spectrum of J(alpha) covers the entire complex plane. In addition, a formula for theWeyl 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.