  We discuss interaction of strongly correlated electrons (described within the
Hubbard model solved by dynamical mean-field theory (DMFT)) with Debye and
Einstein phonons using recently developed DMFT+Sigma computational scheme.
Electron-phonon interaction (EPI) is analyzed in adiabatic approximation
(assuming the validity of Migdal theorem), allowing the neglect of EPI vertex
corrections. This approach is valid for EPI coupling constant L < e_F/w_ph~10,
where e_F is Fermi energy and w_ph is Debye or Einstein frequency. For moderate
values of L only small changes in the electronic density of states are observed
in DMFT+Sigma approximation for both weakly and strongly correlated metallic
regimes. Metal-insulator (Mott) transition due to the increase of Hubbard
interaction U is slightly inhibited by EPI. Our main aim is to discuss the
interplay of "kinks" in electronic dispersion due to EPI and recently
discovered kinks of electronic origin. For the certain region of model
parameters coexistence of phonon "kinks" in electronic dispersion with purely
electronic "kinks" is readily observed and we formulate some simple criteria of
such coexistence. However, for most general combinations of model parameters
phonon "kinks" make electronic "kinks" hardly observable. In the general case
an increase of Hubbard interaction U rapidly suppresses the slope of electronic
dispersion within the phonon "kink". These results are important for deeper
understanding of the shape and evolution of electronic dispersions in strongly
correlated systems such as copper oxides, where different kinds of "kinks" were
recently observed in ARPES experiments.
