Superelliptic Affine Lie algebras and orthogonal polynomials II

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 $$\widehat{\mathfrak{g}\otimes A}\cong\mathfrak{g}\otimes A \oplus(Ω<sup>1_A/dA)$$, where $$A=\mathbb{C}[t,t^{{-1},u]/\langle} u<sup>m-(1-2ct<sup>r+t<sup>{2r})\rangle$$. We compute the recursion relations governing a natural cocycle basis in $Ω<sup>1_A/dA$ 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 $u<sup>m=1-2ct<sup>r+t<sup>{2r}$ 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.