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A series of weak-coupling perturbation theories which include thelowest-order vertex corrections are applied to the attractive Holstein model ininfinite dimensions. The approximations are chosen to reproduce the iteratedperturbation theory in the limit of half-filling and large phonon frequency(where the Holstein model maps onto the Hubbard model). Comparison is made withquantum Monte Carlo solutions to test the accuracy of different approximationschemes.Comment: 31 pages, 15 figures, typeset in ReVTe

Vertex-corrected perturbation theory for the electron-phonon problem with nonconstant density of states