Associated Hermite Polynomials Related to Parabolic Cylinder Functions

Abstract

In \nanalogy to the role of Lommel polynomials \xa0in relation to Bessel \nfunctions Jv(z) the theory of \nAssociated Hermite polynomials in the scaled form \xa0with parmeter v to Parabolic Cylinder \nfunctions Dv(z) is developed. The \ngroup-theoretical background with the 3-parameter group of motions M(2) in the plane for \nBessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder \nfunctions is discussed and compared with formulae, in particular, for the \nlowering and raising operators and the eigenvalue equations. Recurrence \nrelations for the Associated Hermite polynomials and for their derivative and \nthe differential equation for them are derived in detail. Explicit expressions \nfor the Associated Hermite polynomials with involved Jacobi polynomials at \nargument zero are given and by means of them the Parabolic Cylinder functions \nare represented by two such basic functions.