  In the context of a single electron two orbital Holstein system coupled to
dispersionless bosons, we develop a general method to correct single particle
Green's function using a power series correction(PSC) scheme. We then outline
the derivations of various flavors of cumulant approximation through the PSC
scheme and explain the assumptions and approximations behind them. Finally, we
compute and compare PSC spectral function with cumulant and exact diagonalized
spectral functions and elucidate three regimes of this problem - two that
cumulant explains and one where cumulant fails. We find that the exact and the
PSC spectral functions match within spectral broadening across all three
regimes.
