The Weber-Hermite differential equation, obtained as the dimensionless form of the stationary Schroedinger equation for a linear harmonic oscillator in quantum mechanics, has been expressed in a generalized form through introduction of a constant
conjugation parameter according to the transformation
, where the conjugation parameter is set to unity (
) at the end of the evaluations. Factorization in normal order form yields
composite eigenfunctions, Hermite polynomials and corresponding
positive eigenvalues, while factorization in the anti-normal order form yields the partner composite anti-eigenfunctions, anti-Hermite polynomials and negative eigenvalues. The two sets of solutions are related by an
reversal conjugation rule
. Setting
provides the standard Hermite polynomials and their partner anti-Hermite polynomials. The anti-Hermite polynomials satisfy a new differential equation, which is interpreted as the conjugate of the standard Hermite differential equation.