Hermite, Higher order Hermite, Laguerre type polynomials and Burgers like equations

TL;DR

This paper explores the connection between multivariable Hermite and Laguerre polynomials and their associated Burgers-like equations, extending classical solutions to higher order and generalized cases.

Contribution

It introduces new Burgers-type equations linked to multivariable Hermite and Laguerre polynomials, broadening the understanding of their nonlinear PDE relationships.

Findings

Derived Burgers equations for Hermite polynomials

Extended equations to Laguerre and generalized polynomials

Established connections between polynomials and nonlinear PDEs

Abstract

The multivariable version of ordinary and generalized Hermite polynomials are the natural solutions of the classical heat equation and of its higher order versions. We derive the associated Burgers equations and show that analogous non-linear partial differential equations can be derived for Laguerre polynomials and for the relevant generalizations.