Superelliptic Affine Lie algebras and orthogonal polynomials II
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 $\widehat{\mathfrak{g}\otimes A}\cong\mathfrak{g}\otimes A \oplus(Ω^1_A/dA)$, 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 $Ω^1_A/dA$ 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.