2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/199410By factorization of the Hamiltonian describing the quantum mechanics of the continuous q-Hermite polynomial, the creation and annihilation operators of the q-oscillator are obtained. They satisfy a q-oscillator algebra as a consequence of the shape-invariance of the Hamiltonian. A second set of q-oscillator is derived from the exact Heisenberg operator solution. Now the q-oscillator stands on the equal footing to the ordinary harmonic oscillator.12pages, no figures. Document-class changed; q->1 limit added; refs.[9] and [10] updated. To appear in Phys. Lett. BHigh Energy Physics - TheoryMathematical PhysicsClassical Analysis and ODEsQuantum AlgebraExactly Solvable and Integrable SystemsQuantum Physicstext