V. CONCLUSION In this article we have introduced an alternative factorization for the SHO hamiltonian that leads to the new self adjoint equation (21), a parametric equation that becomes the SHO Schr¨ odinger’s equation in the limit δ → 0, and Hermite’s equation in the limit δ → ∞ . We therefore call this equation the new generalized Hermite equation and the functions H δ $_{n}$( x ) (23) the new generalized Hermite functions . We have also shown that the raising and lowering operators (25-26) correspond to the appropriate operators in these limits. We would like to acknowledge Erika Roldan for interesting and useful conversations. We also acknowledge support from CONACYT of Mexico, through a scholarship for MRG. Acknowledgments [1] Witten, E., Nucl. Phys. B (1981), 513-554. [2] Mienik, B., J. Math. Phys. 25 (1984), 3387-3389. [3] Cooper, F., Khare, A. and Sukhatme, U. P., Supersymmetry in quantum mechanics , World Sci., 2001. [4] Fern´ andez, D. J., Int. Jour. Mod. Phys. A12 (1997), 171-176. [5] Chang, S. J., and Chung, H. S., J. Korean Math. Soc. 46 (2009), 327-345 [6] Gould, H. W. and Hopper, A. T., Duke Math. J. 29 (1962), 51-64; Bao, W. and Shen, J., Jour. Comp. Phys. 227 (2008) 9778-9793. [7] Aldaya, V., Bisquert, J., and Navarro-Salas, J., Phys. Lett. A 156 (1991) 381-385. [8] Arfken, G. B. and Weber, H. J., Mathematical methods for physicists , Academic Press, 2001. 8