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Associated Hermite Polynomials Related to Parabolic Cylinder Functions

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Document pages: 28 pages

Abstract: Inanalogy to the role of Lommel polynomials  in relation to Besselfunctions Jv(z) the theory ofAssociated Hermite polynomials in the scaled form  with parmeter v to Parabolic Cylinderfunctions Dv(z) is developed. Thegroup-theoretical background with the 3-parameter group of motions M(2) in the plane forBessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinderfunctions is discussed and compared with formulae, in particular, for thelowering and raising operators and the eigenvalue equations. Recurrencerelations for the Associated Hermite polynomials and for their derivative andthe differential equation for them are derived in detail. Explicit expressionsfor the Associated Hermite polynomials with involved Jacobi polynomials atargument zero are given and by means of them the Parabolic Cylinder functionsare represented by two such basic functions.

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