Lie algebra representations and 1-parameter 2D-Hermite polynomials

Abstract

ABSTRACTThe representations of the Lie algebras generate in a natural way all known classical special polynomials. This allows one to greatly simplify the theory of orthogonal polynomials by expressing them in terms of the corresponding Lie algebra or Lie group. In this article, the problem of framing the 1-parameter 2D-Hermite polynomials (1P2DHP) (which are 2D orthogonal polynomials) into the context of the irreducible representations and of the four-dimensional Lie algebra is considered. This approach stress the mathematical relevance of 2D-orthogonal polynomials and Lie algebras. Certain relations involving the 1P2DHP are obtained using the approach adopted by Miller. The linear differential operators serve as useful tools towards obtaining these relations. The analysis has been carried out by generalizing the formalism relevant to 1P2DHP . Certain examples involving 2D-Hermite polynomials and Laguerre polynomials are obtained as special cases.