Families of orthogonal Laurent polynomials, hyperelliptic Lie algebras and elliptic integrals

TL;DR

This paper introduces families of orthogonal Laurent polynomials linked to hyperelliptic Lie algebras and elliptic integrals, revealing their properties, differential equations, and orthogonality relations, with applications in number theory and conformal field theory.

Contribution

It presents new families of Laurent polynomials connected to hyperelliptic Lie algebras, detailing their properties, differential equations, and orthogonality, and deriving new elliptic integral identities.

Findings

Polynomials satisfy specific second order linear differential equations.

Families are orthogonal with respect to explicit kernels.

New identities of elliptic integrals are derived.

Abstract

We describe a family of polynomials discovered via a particular recursion relation, which have connections to Chebyshev polynomials of the first and the second kind, and the polynomial version of Pell's equation. Many of their properties are listed in Section 3. We show that these families of polynomials in the variable $t$ satisfy certain second order linear differential equations that may be of interest to mathematicians in conformal field theory and number theory. We also prove that these families of polynomials in the setting of Date-Jimbo-Kashiwara-Miwa algebras when multiplied by a suitable power of $t$ are orthogonal with respect to explicitly-described kernels. Particular cases lead to new identities of elliptic integrals (see Section 5).