  We investigate the interplay between the electron-electron and the
electron-phonon interaction in the Hubbard-Holstein model. We implement the
flow-equation method to investigate within this model the effect of correlation
on the electron-phonon effective coupling and, conversely, the effect of
phonons in the effective electron-electron interaction. Using this technique we
obtain analytical momentum-dependent expressions for the effective couplings
and we study their behavior for different physical regimes. In agreement with
other works on this subject, we find that the electron-electron attraction
mediated by phonons in the presence of Hubbard repulsion is peaked at low
transferred momenta. The role of the characteristic energies involved is also
analyzed.
