TITLE:
Associated Hermite Polynomials Related to Parabolic Cylinder Functions
AUTHORS:
Alfred Wünsche
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
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.9 No.1,
January
23,
2019
ABSTRACT: In
analogy to the role of Lommel polynomials in relation to Bessel
functions Jv(z) the theory of
Associated Hermite polynomials in the scaled form with parmeter v to Parabolic Cylinder
functions Dv(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.