DJKM algebras and non-classical orthogonal polynomials

TL;DR

This paper introduces new families of orthogonal polynomials linked to the universal central extensions of specific Lie algebras, highlighting their differential equations and non-classical nature.

Contribution

It identifies and characterizes non-classical orthogonal polynomials associated with DJKM algebras, expanding understanding of their algebraic and differential properties.

Findings

Polynomials satisfy fourth-order linear differential equations

They are orthogonal but not of classical type

New connections between Lie algebra extensions and orthogonal polynomials

Abstract

We describe families of polynomials arising in the study of the universal central extensions of Lie algebras introduced by Date, Jimbo, Kashiwara, and Miwa in their work on the Landau-Lifshitz equations. Two of the families of polynomials we show satisfy certain forth order linear differential equations, are orthogonal and are not of classical type.