eigenfunctions H δ $_{n}$( x ) new generalized Hermite functions . A plot of the first generalized Hermite functions is shown in Fig.(1). Notice the change in the functions behaviour as δ goes from zero to infinity, showing the continuos passage from the quantum mechanics eigenfunctions to the Hermite polynomials. The corresponding raising and lowering operators, c ∗ , c , are easily deduced from the action of a ∗ and a over ψ$_{n}$ ( x ). The non-mutually adjoint operators IV. RAISING AND LOWERING OPERATORS c ∗ = α a $_{∗}$1 α = 1 √$_{2}$ ( − d dx + 1 + 2 δe − x 2 1 + δe − x 2 x ) (25) c = α a 1 α = 1 √$_{2}$ ( d dx + x 1 + δe − x 2 ) (26) become the creation/anihilation operators of the quantum mechanics problem as δ → 0, and the raising and lowering operators for the Hermite polynomials as δ → ∞ . They satisfy the relations which are the quantum eigenfunctions creation/anihilation operations when δ = 1, and become the raising/lowering operations for the Hermite polynomials in the limit δ → ∞ . They also satisfy that and, therefore, Finally, due to the relation a ∗ = (1 /α ) B ∗ and eq.(19), c ∗ and c correspond to the cre- ation/anihilation operators introduced by Mielnik[2] as triple operator products. [ c, c $^{∗}$] = 1 . (28) cc $^{∗}$H δ n = α ( H + 1 2 ) 1 α H δ n = ( n + 1) H δ n , c $^{∗}$c H δ n = α H − 1 2 1 α H δ n = n H δ n , c $^{∗}$H δ n = √$_{n}$$_{+ 1 H}$ δ n +1 , c H δ n = √$_{n}$$_{H}$ δ n − 1 , (27) 7