A finite oscillator model with equidistant position spectrum based on an extension of $\mathfrak{s}\mathfrak{u}(2)$

We consider an extension of the real Lie algebra $\mathfrak{s}\mathfrak{u}(2)$ by introducing a parity operator $P$ and a parameter $c$. This extended algebra is isomorphic to the Bannai-Ito algebra with two parameters equal to zero. For this algebra we classify all unitary finite-dimensional representations and show their relation with known representations of $\mathfrak{s}\mathfrak{u}(2)$. Moreover, we present a model for a one-dimensional finite oscillator based on the odd-dimensional representations of this algebra. For this model, the spectrum of the position operator is equidistant and coincides with the spectrum of the known $\mathfrak{s}\mathfrak{u}(2)$ oscillator. In particular the spectrum is independent of the parameter $c$ while the discrete position wavefunctions, which are given in terms of certain dual Hahn polynomials, do depend on this parameter.