  We explore a model for the one-dimensional quantum oscillator based upon the
Lie superalgebra sl(2|1). For this purpose, a class of discrete series
representations of sl(2|1) is constructed, each representation characterized by
a real number beta>0. In this model, the position and momentum operators of the
oscillator are odd elements of sl(2|1) and their expressions involve an
arbitrary parameter gamma. In each representation, the spectrum of the
Hamiltonian is the same as that of the canonical oscillator. The spectrum of
the momentum operator can be continuous or infinite discrete, depending on the
value of gamma. We determine the position wavefunctions both in the continuous
and discrete case, and discuss their properties. In the discrete case, these
wavefunctions are given in terms of Meixner polynomials. From the embedding
osp(1|2)\subset sl(2|1), it can be seen why the case gamma=1 corresponds to the
paraboson oscillator. Consequently, taking the values (beta,gamma)=(1/2,1) in
the sl(2|1) model yields the canonical oscillator.
