In this paper we extend the Falicov-Kimball model (FKM) to the case where the quasi-particles entering the FKM are not ordinary fermions. As an example we first discuss how the FKM can be generalized to the case with spin-dependent hopping. Afterwards we discuss several cases where the quasi-particles entering the FKM are Majorana fermions (extended Majorana-Falicov-Kimball Model (MFKM). Two examples of extended MFKM are discussed in detail: (i) a $p$-wave BCS superconductor on a bipartite lattice and (ii) a BCS-Anderson model. We also discuss the most general forms of extended MFKM, including a brief discussion on the case where the Majorana fermions represent spins, but not real fermion particles.
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