  We use the dynamical mean field theory to develop a systematic and
computationally tractable method for studying electron-phonon interactions in
systems with arbitrary electronic correlations. The method is formulated as an
adiabatic expansion around the limit of static phonons. No specific electronic
groundstate is assumed. We derive an effective low-frequency phonon action
whose coefficients are static local correlation functions of the underlying
electron system. We identify the correct expansion parameters. At a critical
electron-phonon interaction strength the system undergoes a transition to a
polaronic state. We determine the location of this polaronic instability in the
presence of electron-electron interactions, doping, and quantum lattice
fluctuations and present the formalism needed for study of the electron
self-energy and effective mass.
