  I present results from an approach that extends the Eliashberg theory by
systematic expansion in the vertex function; an essential extension at large
phonon frequencies, even for weak coupling. In order to deal with
computationally expensive double sums over momenta, a dynamical cluster
approximation (DCA) approach is used to incorporate momentum dependence into
the Eliashberg equations. First, I consider the effects of introducing partial
momentum dependence on the standard Eliashberg theory using a quasi-local
approximation; which I use to demonstrate that it is essential to include
corrections beyond the standard theory when investigating d-wave states. Using
the extended theory with vertex corrections, I compute electron and phonon
spectral functions. A kink in the electronic dispersion is found in the normal
state along the major symmetry directions, similar to that found in
photo-emission from cuprates. The phonon spectral function shows that for weak
coupling $W\lambda < \omega_0$, the dispersion for phonons has weak momentum
dependence, with consequences for the theory of optical phonon mediated d-wave
superconductivity, which is shown to be 2nd order in $\lambda$. In particular,
examination of the order parameter vs. filling shows that vertex corrections
lead to d-wave superconductivity mediated via simple optical phonons. I map out
the order parameters in detail, showing that there is significant induced
anisotropy in the superconducting pairing in quasi-2D systems.
