Temme, N.M. (2010) Parabolic Cylinder Functions, Chapter. 12. In: NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, New York, 303-319.

has been cited by the following article:

• 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.