Associated Hermite Polynomials Related to Parabolic Cylinder Functions - Advances in Pure Mathematics

APM > Vol.9 No.1, January 2019

Advances in Pure Mathematics

Volume 9, Issue 1 (January 2019)

ISSN Print: 2160-0368 ISSN Online: 2160-0384

Google-based Impact Factor: 0.50 Citations h5-index & Ranking

Associated Hermite Polynomials Related to Parabolic Cylinder Functions ()

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DOI: 10.4236/apm.2019.91002 851 Downloads 2,155 Views Citations

Author(s)

Alfred Wünsche

Affiliation(s)

Max-Planck-Group “Nonclassical Radiation” at Institut für Physik, Humboldt-Universität Berlin, Berlin, Germany.

ABSTRACT

In analogy to the role of Lommel polynomials

in relation to Bessel functions J_v(z) the theory of Associated Hermite polynomials in the scaled form

with parmeter v to Parabolic Cylinder functions D_v(z) is developed. The group-theoretical background with the 3-parameter group of motions M(2) in the plane for Bessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.

KEYWORDS

Bessel Functions, Lommel Polynomials, Parabolic Cylinder Functions, Associated Hermite Polynomials, Jacobi polynomials, Recurrence Relations, Lowering and Raising Operators, Heisenberg-Weyl Group, Motion Group of Plane, Irreducible Representations

Share and Cite:

Wünsche, A. (2019) Associated Hermite Polynomials Related to Parabolic Cylinder Functions. Advances in Pure Mathematics, 9, 15-42. doi: 10.4236/apm.2019.91002.

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