Superelliptic Affine Lie algebras and orthogonal polynomials II

TL;DR

This paper studies superelliptic affine Lie algebras, computes recursion relations for a natural cocycle basis, and links polynomial solutions of associated differential equations to ultraspherical polynomials.

Contribution

It introduces a classification of initial conditions, derives differential equations for polynomial families, and connects these to ultraspherical polynomials in the context of superelliptic Lie algebras.

Findings

Classified four structural types of initial conditions.

Derived fourth-order differential equations for polynomial families.

Connected polynomial solutions to ultraspherical polynomials.

Abstract

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $r,m\ge 2$. The universal central extension of the superelliptic current algebra $\mathfrak{g}\otimes A$ is $\widehat{\mathfrak{g}\otimes A}\cong\mathfrak{g}\otimes A \oplus(\Omega^1_A/dA)$, where $A=\mathbb{C}[t,t^{-1},u]/\langle u^m-(1-2ct^r+t^{2r})\rangle$. We compute the recursion relations governing a natural cocycle basis in $\Omega^1_A/dA$ and encode them by generating functions admitting closed integral expressions of superelliptic type. The $2r$ possible choices of initial conditions are classified into four structural types; two canonical choices (types~1 and~2) produce two distinguished polynomial families. We prove that these polynomials satisfy fourth-order linear ordinary differential equations in~$c$, valid for all integers $r,m\ge 2$. For the type~2 family the proof combines the Picard-Fuchs theory of the superelliptic curve $u^m=1-2ct^r+t^{2r}$ with an algebraic identification of the explicit coefficient formulas via a rational-function identity argument. After a parity restriction and a reindexing, the resulting sequences are identified with associated ultraspherical polynomials. We show that, for each admissible~$n$ and $m\ge4$, the corresponding fourth-order equations admit a unique polynomial solution up to scalar multiples.