Generalized Hermite polynomials and supergaussian forms (Q1125832)

Generalized Hermite polynomials \(H_{n,m}(x)\) were studied. Their generating function was given, association with Humbert and Gould- Hopper type polynomials, differential equations satisfied by \(H_{n,m}(x)\), recurrence properties, introduction and harmonic oscillator-like functions, nonorthogonal properties, relevant creation -- annihilation operator algebra, generalized forms of harmonic oscillator eigenfunctions were discussed. Extension of generalized Hermite polynomials leading to the class of Bell polynomials were dealt with. Super-Gauss-Hermite functions with pseudo-orthogonal properties were also discussed. They are shown for \(m=4\).