We study connections between diagonal Padé approximants and spectral properties of second order difference operators with complex coefficients. In a first part, we identify the diagonal of the Padé table with a particular difference operator which is shown to have maximal resolvent set. The spectrum of an asymptotically periodic complex difference operator is given, and we prove convergence on the resolvent set for the corresponding sequence of Padé approximants to the associated Weyl function.

In the second part, we give convergence results for the diagonal of the Padé table in the general case where the recurrence coefficients are uniformly bounded. The grow of Padé denominators is related to the Green function of the spectrum of the associated difference operator, and the connection to orthogonality on the real line is studied. In order to approximate the Weyl function locally uniformly on the whole resolvent set, we finally propose the concept of smoothed Padé approximants where we remedy the undesired phenomenon of spurious zeros of Padé denominators.