  We use a novel approach to analyze the one dimensional spinless
Falicov-Kimball model. We derive a simple effective model for the occupation of
the localized orbitals which clearly reveals the origin of the known ordering.
Our study is extended to a quantum model with hybridization between the
localized and itinerant states; we find a crossover between the well-known
weak- and strong-coupling behaviour. The existence of electronic polarons at
intermediate coupling is confirmed. A phase diagram is presented and discussed
in detail.
