  We give a detailed discussion of the recently developed Generalized Dynamical
Mean-Field Theory (GDMFT) for a mixture of bosonic and fermionic particles. We
show that this method is non-perturbative and exact in infinite dimensions and
reliably describes the full range from weak to strong coupling. Like in
conventional Dynamical Mean-Field Theory, the small parameter is 1/z, where z
is the lattice coordination number. We apply the GDMFT scheme to a mixture of
spinless fermions and bosons in an optical lattice. We investigate the
ossibility of a supersolid phase, focussing on the case of 1/2 filling for the
fermions and 3/2 filling for the bosons.
