  Local n-particle vertex functions represent the crucial ingredient for all
diagrammatic extensions of dynamical mean field theory (DMFT). Hitherto their
application has been restricted -with a few exceptions- to the n=2-particle
vertex while higher-order vertices have been neglected. In this paper we derive
a general analytical expression for the n-particle (one-particle reducible)
vertex of the Falicov-Kimball model (FKM). We observe that the magnitude of
such vertex functions itself strongly increases with the particle-number n. On
the other hand, their effect on generic Feynman diagrams remains rather
moderate due to the damping effect of the Green's functions present in such
diagrams. Nevertheless, they yield important contributions to the self-energy
corrections calculated in diagrammatic extensions of DMFT as we explicitly
demonstrate using the example of dual fermion (DF) calculations for the
two-dimensional FKM at quarter filling of stationary f-electrons where
corrections to the self-energy obtained from the three-particle vertex are
indeed comparable in magnitude to corresponding corrections stemming from the
two-particle vertex.
