  We derive sum rules for the phonon self-energy and the electron-phonon
contribution to the electron self-energy of the Holstein-Hubbard model in the
limit of large Coulomb interaction U. Their relevance for finite U is
investigated using exact diagonalization and dynamical mean-field theory. Based
on these sum rules, we study the importance of vertex corrections to the
electron-phonon interactionin a diagrammatic approach. We show that they are
crucial for a sum rule for the electron self-energy in the undoped system while
a sum rule related to the phonon self-energy of doped systems is satisfied even
if vertex corrections are neglected. We provide explicit results for the vertex
function of a two-site model.
