  A new model for the finite one-dimensional harmonic oscillator is proposed
based upon the algebra u(2)_{\alpha}. This algebra is a deformation of the Lie
algebra u(2) extended by a parity operator, with deformation parameter
{\alpha}. A class of irreducible unitary representations of u(2)_{\alpha} is
constructed. In the finite oscillator model, the (discrete) spectrum of the
position operator is determined, and the position wave functions are shown to
be dual Hahn polynomials. Plots of these discrete wave functions display
interesting properties, similar to those of the parabose oscillator. We show
indeed that in the limit, when the dimension of the representations goes to
infinity, the discrete wave functions tend to the continuous wave functions of
the parabose oscillator.
