  We present a class of algorithms based on rational Krylov methods to compute
the action of a generalized matrix function on a vector. These algorithms
incorporate existing methods based on the Golub-Kahan bidiagonalization as a
special case. By exploiting the quasiseparable structure of the projected
matrices, we show that the basis vectors can be updated using a short
recurrence, which can be seen as a generalization to the rational case of the
Golub-Kahan bidiagonalization. We also prove error bounds that relate the error
of these methods to uniform rational approximation. The effectiveness of the
algorithms and the accuracy of the bounds is illustrated with numerical
experiments.
