  Mono- and multilayer FeSe thin films grown on SrTiO$_\mathrm{3}$ and
BiTiO$_\mathrm{3}$ substrates exhibit a greatly enhanced superconductivity over
that found in bulk FeSe. A number of proposals have been advanced for the
mechanism of this enhancement. One possibility is the introduction of a
cross-interface electron-phonon ($e$-$ph$) interaction between the FeSe
electrons and oxygen phonons in the substrates that is peaked in the forward
scattering (small ${\bf q}$) direction due to the two-dimensional nature of the
interface system. Motivated by this, we explore the consequences of such an
interaction on the superconducting state and electronic structure of a
two-dimensional system using Migdal-Eliashberg theory. This interaction
produces not only deviations from the expectations of conventional
phonon-mediated pairing but also replica structures in the spectral function
and density of states, as probed by angle-resolved photoemission spectroscopy,
scanning tunneling microscopy/spectroscopy, and quasi-particle interference
imaging. We also discuss the applicability of Migdal-Eliashberg theory for a
situation where the \ep interaction is peaked at small momentum transfer and in
the FeSe/STO system.
