It is possible to generalize the fruitful interaction between (real or
complex) Jacobi matrices, orthogonal polynomials and Pade approximants at
infinity by considering rational interpolants, (bi-)orthogonal rational
functions and linear pencils zB-A of two tridiagonal matrices A, B, following
Spiridonov and Zhedanov.

In the present paper, beside revisiting the underlying generalized Favard
theorem, we suggest a new criterion for the resolvent set of this linear pencil
in terms of the underlying associated rational functions. This enables us to
generalize several convergence results for Pade approximants in terms of
complex Jacobi matrices to the more general case of convergence of rational
interpolants in terms of the linear pencil. We also study generalizations of
the Darboux transformations and the link to biorthogonal rational functions.
Finally, for a Markov function and for pairwise conjugate interpolation points
tending to infinity, we compute explicitly the spectrum and the numerical range
of the underlying linear pencil.