Cλ​-extended oscillator algebras generalizing the Calogero-Vasiliev
algebra, where Cλ​ is the cyclic group of order λ, are
studied both from mathematical and applied viewpoints. Casimir operators of the
algebras are obtained, and used to provide a complete classification of their
unitary irreducible representations under the assumption that the number
operator spectrum is nondegenerate. Deformed algebras admitting Casimir
operators analogous to those of their undeformed counterparts are looked for,
yielding three new algebraic structures. One of them includes the Brzezi\'nski
{\em et al.} deformation of the Calogero-Vasiliev algebra as a special case. In
its bosonic Fock-space representation, the realization of
Cλ​-extended oscillator algebras as generalized deformed oscillator
ones is shown to provide a bosonization of several variants of supersymmetric
quantum mechanics: parasupersymmetric quantum mechanics of order p=λ−1 for any λ, as well as pseudosupersymmetric and
orthosupersymmetric quantum mechanics of order two for λ=3.Comment: 48 pages, LaTeX with amssym, no figures, to be published in Int. J.
Theor. Phy