Superelliptic Affine Lie algebras and orthogonal polynomials II

Let [Formula: see text] be a finite-dimensional complex simple Lie algebra and [Formula: see text]. The universal central extension of the superelliptic current algebra [Formula: see text] is [Formula: see text], where [Formula: see text]. We compute the recursion relations governing a natural cocycle basis in [Formula: see text] and encode them by generating functions admitting closed integral expressions of superelliptic type. The [Formula: see text] 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 [Formula: see text],. After a parity restriction and a reindexing, the resulting sequences are identified with associated ultraspherical polynomials. We show that, for each admissible [Formula: see text] and every [Formula: see text], the corresponding fourth-order equations admit a unique polynomial solution up to scalar multiples.