Differential Operator Representation of sl (2, R) as Modules over Univariate and Bivariate Hermite Polynomials

TL;DR

This paper explores how the Lie algebra sl(2,R) can be represented through differential operators acting on univariate and bivariate Hermite polynomials, revealing new relations and general representations involving other polynomial bases.

Contribution

It introduces a general form of sl(2,R) representation using differential operators on Hermite, Laguerre, and Legendre polynomials, and derives new relations via the Baker-Campbell-Hausdorff formula.

Findings

Derived new relations for Hermite polynomials using Lie algebra representations.

Established a general framework for sl(2,R) representations with various polynomial bases.

Connected differential equations with Lie algebra structures in polynomial spaces.

Abstract

This paper presents the connections between univariate and bivariate Hermite polynomials and associated differential equations with specific representations of Lie algebra sl(2,R) whose Cartan sub-algebras coincide the associated differential operators of these differential equations . Applying the Baker-Campbell-Hausdorff formula to generators of these algebras, results in new relation for one-variable and Bivariate Hermite polynomials. A general form of sl(2,R) representation by differential operators and arbitrary polynomial basis such as Laguerre and Legendre polynomials is introduced.