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The operator L , the weight factor ω ( x ) and the eigenfunctions H δ $_{n}$( x ), all depend on the factorization parameter δ . In the case δ = 0, we have that α ( x ) ≡ 1, and B and B ∗ become the original anihilation/creation operators (4). Also, the eigenvalue equation (21) becomes Schr¨ odinger’s equation for the SHO, and H δ $_{n}$( x ) become the quantum eigenfunctions. On the other hand, in the limit δ → ∞ , L , ω , and H δ n become 

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and the eigenvalue equation (21) becomes Hermite’s differential equation. Therefore, we can call eq.(21) a new generalized Hermite equation for the SHO, which includes Schr¨ odinger’s and Hermite’s equations as particular cases for limiting values of the parameter δ , linked by a continuos parametric transformation from one to the other. This also allows us to call the 

