In the context of a two-parameter $(\alpha, \beta)$ deformation of the
canonical commutation relation leading to nonzero minimal uncertainties in both
position and momentum, the harmonic oscillator spectrum and eigenvectors are
determined by using techniques of supersymmetric quantum mechanics combined
with shape invariance under parameter scaling. The resulting supersymmetric
partner Hamiltonians correspond to different masses and frequencies. The
exponential spectrum is proved to reduce to a previously found quadratic
spectrum whenever one of the parameters $\alpha$, $\beta$ vanishes, in which
case shape invariance under parameter translation occurs. In the special case
where $\alpha = \beta \ne 0$, the oscillator Hamiltonian is shown to coincide
with that of the q-deformed oscillator with $q > 1$ and its eigenvectors are
therefore $n$-$q$-boson states. In the general case where $0 \ne \alpha \ne
\beta \ne 0$, the eigenvectors are constructed as linear combinations of
$n$-$q$-boson states by resorting to a Bargmann representation of the latter
and to $q$-differential calculus. They are finally expressed in terms of a
$q$-exponential and little $q$-Jacobi polynomials.