  We introduce an alternative factorization of the Hamiltonian of the quantum
harmonic oscillator which leads to a two-parameter self-adjoint operator from
which the standard harmonic oscillator, the one-parameter oscillators
introduced by Mielnik, and the Hermite operator are obtained in certain limits
of the parameters. In addition, a single Bernoulli-type parameter factorization
which is different of the one introduced by M. A. Reyes, H. C. Rosu, and M. R.
Gutierrez, Phys. Lett. A 375 (2011) 2145 is briefly discussed in the final part
of this work
