  We derive frequency moment sum rules for the retarded electronic Green's
function and self-energy of the Holstein model for both equilibrium and
nonequilibrium cases. We also derive sum rules for the phonon propagator in
equilibrium and nonequilibrium. These sum rules allow one to benchmark
nonequilibrium calculations and help with interpreting the behavior of
electrons driven out of equilibrium by an applied electric field. We exactly
evaluate the sum rules when the system is in the atomic limit. We also discuss
the application of these sum rules to pump/probe experiments like time-resolved
angle-resolved photoemission spectroscopy.
