Orthogonal polynomials

Complex plane

Pure mathematics

Classical orthogonal polynomials

Mathematics

Bessel function

Laguerre polynomials

Mathematical analysis

Hermite polynomials

Cylinder

Parabolic partial differential equation

Geometry

Wilson polynomials

Differential equation

Parabolic cylinder function

Jacobi polynomials

Bessel polynomials

Advances in Pure Mathematics

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.

Associated Hermite Polynomials Related to Parabolic Cylinder Functions