FIG. 1: The first four generalized Hermite functions H δ $_{n}$( x ) (arbitrary scale) for different values of the parameter δ . When δ = 0 we get the quantum eigenfunctions ψ$_{n}$ ( x ), and for increasing δ , the eigenfunctions H δ n begin to acquire the form of the Hermite polynomials, H$_{n}$( x ), modulated at x → ±∞ by the vanishing exponential tails e − x $^{2}$/ $^{2}$. 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 L → δe − x 2 d 2 dx 2 − 2 δxe − x 2 d dx − δe − x 2 , ω → 2 δe − x 2 , H δ n → 1 √$_{δ}$ H$_{n}$( x ) , 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 6