This paper has been motivated by a recent paper by Dey [Phys. Rev. D 91, 044024 (2015)] on the known Arik-Coon
q oscillator. We construct
q coherent, even and odd
q-cat states in Fock representation for the Biedenharn-Macfarlane
q oscillator with
q>1 and study their nonclassical properties. The
q-coherent states minimize the Heisenberg uncertainty relation between the generalized position and momentum operators as well as the
x and
y components of a
q-deformed
su(1,1) algebra in the Schwinger boson representation. The latter is also minimized by the even and odd
q-cat states. We show that, contrary to the undeformed harmonic oscillator, the squeezing effect in both position and momentum operators can be exhibited by odd
q-cat states. It is also violated by even
q-cat states. Furthermore, it is shown that the antibunching effect and sub-Poissonian or super-Poissonian statistics can simultaneously appear by each of the even or odd
q-cat states. Finally, a unitary Fock representation of the
q-deformed
su(1,1) algebra is obtained by the
q-deformed Bargmann-Fock realization.
We present the truncated (or [Formula: see text]-deformed) para-Bose oscillator algebra of order [Formula: see text], with [Formula: see text] being a half-integer, along with its [Formula: see text]-dimensional Fock representation. The model reduces to the truncated simple harmonic oscillator and the conventional para-Bose oscillator algebras in the limiting cases where the parameters [Formula: see text] and [Formula: see text] approach zero and infinity, respectively. A Fock–Bargmann realization is constructed in such a way that the creation operator is expressed as a function of both deformation parameters [Formula: see text] and [Formula: see text]. Two-component orthogonal cat states are formulated through suitable superpositions of even and odd parity states within the Fock representation space. The corresponding positive-definite measures over the entire complex plane are derived in terms of Meijer’s [Formula: see text]-functions parameterized by [Formula: see text] and [Formula: see text]. In addition, the nonclassical properties of the even and odd cat states are analyzed, including the manifestation of sub-Poissonian statistics, photon antibunching effect, and quadrature squeezing. We further investigate how the parameters [Formula: see text] and [Formula: see text] individually influence the enhancement or suppression of entanglement in the associated quasi-Bell states.