The coupled electron-phonon system is considered for phonon spectra of Einstein and Debye forms. The single-particle electron Green's function G is calculated in a nonperturbative manner in both models, and its spectral weight function is examined to determine the validity of a quasiparticle picture. The weight function and the poles of G both lead to several branches of excitations rather than a single "dressed" electron. The asymptotic time dependence of the G is found, and the effect of multiphonon processes on the electron decay rate is discussed. The electronic polarizability, P of the interacting system is calculated with the aid of a generalized Ward's identity for the electron-phonon vertex. This identity, which is a consequence of electronic charge conservation, is derived in an Appendix. The calculation of P is carried out in the limit that the Fermi velocity is small compared with the phase velocity of the polarization field. An Appendix on the formal development of the Green's function equations is included.