  We generalize the family of approximate Momentum Average (MA) methods to
formulate a numerically exact, convergent hierarchy of equations whose solution
provides an efficient algorithm to compute the Green's function of a particle
dressed by bosons, suitable in the entire parameter regime. We use this
approach to extract ground state properties and spectral functions. Our
approximation-free framework, dubbed the Generalized Green's function Cluster
Expansion (GCCE), allows access to exact numerical results in the extreme
adiabatic limit, where many standard methods struggle or completely fail. We
showcase the performance of the method, specializing to three important models
of charge-boson coupling in solids and molecular complexes: the molecular
Holstein model, which describes coupling between charge density and local
distortions, the Peierls model, which describes modulation of charge hopping
due to inter-site distortions, and a more complex Holstein+Peierls system with
couplings to two different phonon modes, paradigmatic of charge-lattice
interactions in organic crystals. The GGCE serves as an efficient approach that
can be systematically extended to different physical scenarios, thus providing
a tool to model the frequency-dependence of dressed particles in realistic
settings.
