Geodesic
Superelliptic Affine Lie algebras and orthogonal polynomials
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e120

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We construct two families of orthogonal polynomials associated with the universal central extensions of the superelliptic Lie algebras. These polynomials satisfy certain fourth-order linear differential equations, and one of the families is a particular collection of associated ultraspherical polynomials. We show that the generating functions of the polynomials satisfy fourth-order linear PDEs. Since these generating functions can be represented by superelliptic integrals, we have examples of linear PDEs of fourth order with explicit solutions without complete integrability. 
Santos, Felipe Albino dos; Neklyudov, Mikhail; Futorny, Vyacheslav. Superelliptic Affine Lie algebras and orthogonal polynomials. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e120. doi: 10.1017/fms.2025.10074
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