  I examine electron-phonon mediated superconductivity in the intermediate
coupling and phonon frequency regime of the quasi-2D Holstein model. I use an
extended Migdal-Eliashberg theory which includes vertex corrections and spatial
fluctuations. I find a d-wave superconducting state that is unique close to
half-filling. The order parameter undergoes a transition to s-wave
superconductivity on increasing filling. I explain how the inclusion of both
vertex corrections and spatial fluctuations is essential for the prediction of
a d-wave order parameter. I then discuss the effects of a large Coulomb
pseudopotential on the superconductivity (such as is found in contemporary
superconducting materials like the cuprates), which results in the destruction
of the s-wave states, while leaving the d-wave states unmodified.
