  We formulate a bosonic dynamical mean-field theory (B-DMFT) which provides a
comprehensive, thermodynamically consistent framework for the theoretical
investigation of correlated lattice bosons. The B-DMFT is applicable for
arbitrary values of the coupling parameters and temperature and becomes exact
in the limit of high spatial dimensions d or coordination number Z of the
lattice. In contrast to its fermionic counterpart the construction of the
B-DMFT requires different scalings of the hopping amplitudes with Z depending
on whether the bosons are in their normal state or in the Bose-Einstein
condensate. A detailed discussion of how this conceptual problem can be
overcome by performing the scaling in the action rather than in the Hamiltonian
itself is presented. The B-DMFT treats normal and condensed bosons on equal
footing and thus includes the effects caused by their dynamic coupling. It
reproduces all previously investigated limits in parameter space such as the
Beliaev-Popov and Hartree-Fock-Bogoliubov approximations and generalizes the
existing mean-field theories of interacting bosons. The self-consistency
equations of the B-DMFT are those of a bosonic single-impurity coupled to two
reservoirs corresponding to bosons in the condensate and in the normal state,
respectively. We employ the B-DMFT to solve a model of itinerant and localized,
interacting bosons analytically. The local correlations are found to enhance
the condensate density and the Bose-Einstein condensate (BEC) transition
temperature T_{BEC}. This effect may be used experimentally to increase T_{BEC}
of bosonic atoms in optical lattices.
