Factorization procedure and new generalized Hermite functions

We propose an alternative factorization for the simple harmonic oscillator hamiltonian which includes Mielnik's isospectral factorization as a particular case. This factorization is realized in two non-mutually adjoint operators whose inverse product, in the simplest case, lead to a new Sturm-Liouville eigenvalue equation which includes Schrodinger's equation for the oscillator and Hermite's equation as particular cases for limiting values of the factorization's parameter, and whose eigenfunctions allow us to define new generalized Hermite functions.