We generalize the family of approximate momentum average methods to formulate a numerically exact, convergent hierarchy of equations whose solution provides an efficient algorithm to compute the Greens 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 Greens function cluster expansion (GGCE), 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 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 intersite 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.
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