The diagonal of the Pad e table andthe approximation of the Weyl functionof second order di erence

We study connections between diagonal Pad e approximants and spectral properties of second order diierence operators with complex coeecients. In a rst part, we identify the diagonal of the Pad e table with a particular diierence operator which is shown to have maximal resolvent set. The spectrum of an asymptotically periodic complex diierence operator is given, and we prove convergence on the resolvent set for the corresponding sequence of Pad e approximants to the associated Weyl function. In the second part, we give convergence results for the diagonal of the Pad e table in the general case where the recurrence coeecients are uniformly bounded. The growth of Pad e denominators is related to the Green function of the spectrum of the associated diierence operator, and the connection with orthogonality on the real line is studied. In order to approximate the Weyl function locally uniformly on the whole resolvent set, we nally propose the concept of smoothed Pad e approximants where we remedy the undesired phenomenon of spurious zeros of Pad e denominators.