We describe an approach for calculations of phonon contributions to the
electron spectral function, including both quasiparticle properties and
satellites. The method is based on a cumulant expansion for the retarded
one-electron Green's function and a many-pole model for the electron
self-energy. The electron-phonon couplings are calculated from the Eliashberg
functions, and the phonon density of states is obtained from a Lanczos
representation of the phonon Green's function. Our calculations incorporate ab
initio dynamical matrices and electron-phonon couplings from the density
functional theory code ABINIT. Illustrative results are presented for several
elemental metals and for Einstein and Debye models with a range of coupling
constants. These are compared with experiment and other theoretical models.
Estimates of corrections to Migdal's theorem are obtained by comparing with
leading order contributions to the self-energy, and are found to be significant
only for large electron-phonon couplings at low temperatures