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.
In a previous paper (2007 J. Phys. A: Math. Theor. 40 11105), we constructed a class of coherent states for a polynomially deformed su (2) algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed su (1, 1) algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su (1, 1) algebra which contains the Barut–Girardello set and the Perelomov set of the SU (1, 1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.