Generalized Hermite polynomials associated with functions of parabolic cylinder

Authors:
A. Wünsche
Affiliations:
Humboldt-Universität, Institut für Physik, Invalidenstrasse 110, D-10115 Berlin, Germany
Venue:
Applied Mathematics and Computation - Special issue: Advanced special functions and related topics in differential equations, third Melfi workshop, proceedings of the Melfi school on advanced topics in mathematics and physics
Year:
2003

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Abstract

Associated Hermite polynomials He n v (z) which generalize the usual (scaled) Hermite polynomials He n (z) corresponding to v = 0 are introduced for the purpose to represent the raising and lowering of the indices of functions of the parabolic cylinder D v (z) in finite integer steps. Properties of these polynomials such as recursion relations, explicit representations and the differential equation are derived. The generation of the associated Hermite polynomials from the usual Hermite polynomials by differential operators representable by means of the confluent hypergeometric function is given. An application for the explicit calculation of the functions of the parabolic cylinder for negative integer indices is discussed. Other applications are visible for the investigation of the zeros of the functions of the parabolic cylinder.