Generalized Hermite polynomials associated with functions of parabolic cylinder

摘要

Associated Hermite polynomials Heνn(z) which generalize the usual (scaled) Hermite polynomials Hen(z) corresponding to ν=0 are introduced for the purpose to represent the raising and lowering of the indices of functions of the parabolic cylinder Dν(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.