We present a generalized Momentum Average method for solving electron-phonon problems in the single-carrier limit on infinite lattices. This approach, which we call the Generalized Green's function Cluster Expansion, expands the Green's function in orders of phonon cloud size and then solves the expansion (a system of coupled linear equations). To demonstrate the broad applicability of the method, we solve for ground and excited-state properties of canonical electron-phonon polaron lattice models that have applicability to modeling real-world experiments, such as the Holstein, Peierls and two-phonon-mode Holstein+Peierls models, and show that, where applicable, our results agree with other numerically exact calculations but with reduced computational cost. Finally, we highlight that our method is easily generalized to finite-temperature, higher-dimensions, multi-carrier-band and multi-carrier (e.g. bipolaron) models and note that it can reach the adiabatic limit of small phonon frequencies, a regime where other methods have historically struggled.
