We review the Dynamical Mean Field Theory of the Holstein Polaron
Problem in order to compute the small polaron Green's functions.
The Renormalized Perturbation Expansion (RPE) play a central role
and allows to compute all the local and non local Green's functions
for the electron and the small polaron, for finite or infinite size
systems, with periodic or non periodic boundary conditions. We
introduce a restricted basis for the phonons to study the
decoupling scheme of the Green's functions in a Local Approximation
via exact diagonalizations. As a bonus, we furnish all the C/C++
programs built step by step, from a pedagogical point of view.
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