%0 Journal Article
%T Composite Hermite and Anti-Hermite  Polynomials
%A Joseph Akeyo Omolo
%J Advances in Pure Mathematics
%P 817-827
%@ 2160-0384
%D 2015
%I Scientific Research Publishing
%R 10.4236/apm.2015.514076
%X 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 <em>conjugation parameter <img alt=\"\" src=\"Edit_6de6be9a-3041-474d-be4d-21c283efd926.bmp\" /></em>according to the transformation <img alt=\"\" src=\"Edit_621f64a8-91d7-4291-8d34-863b37696392.jpg\" />, where the conjugation parameter is set to unity (<img alt=\"\" src=\"Edit_d2827724-84b1-4700-9861-8eab631a596b.jpg\" />) at the end of the evaluations. Factorization in normal order form yields <img alt=\"\" src=\"Edit_5c406e4f-fd43-4009-a3e5-23dbdf7b2804.jpg\" /> <em></em><em>composite </em>eigenfunctions, Hermite polynomials and corresponding <em>positive </em>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 <img alt=\"\" src=\"Edit_c4d880d0-a717-40fb-8108-fb10adffeb9a.jpg\" /> <em>reversal </em>conjugation rule <img alt=\"\" src=\"Edit_cf03d53e-4f44-43a7-97f4-dc583053f128.jpg\" />. Setting <img alt=\"\" src=\"Edit_f1a4a391-40cb-4701-bcdf-190a83f29c2f.jpg\" /> 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.
%K Weber-Hermite Differential Equation
%K Eigenfunctions
%K Anti-Eigenfunctions
%K Hermite
%K Anti-Hermite
%K Positive-Negative Eigenvalues
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=61821