Legendre and Chebyshev Polynomials on a Hyperelliptic Curve: an Elementary Account
* Corresponding author: Felipe Albino dos Santos.
Open access. © 2026 the author(s). Published under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
This article belongs to an issue that is still in progress. Bibliographic details may change before formal publication in December, 2026.
Abstract
Cox and Zhao showed that two classical families of orthogonal polynomials are naturally encoded in the coordinate ring A=C[x^1,uu^2=p(x)] of a hyperelliptic curve. The Legendre polynomials arise from the quotient A/A, where =u,ddx, while the Chebyshev polynomials describe the group of units of A. Their results are obtained in the broader context of universal central extensions for arbitrary branch data. We provide a self-contained treatment of these two phenomena in the three smallest cases, using only elementary linear algebra. We also compare the resulting dimensions with the topological quantity 2g+r-1 via the map dx/u. We record two observations that, to our knowledge, have not appeared previously: the dimension of A/A remains unchanged at parameter values where the curve develops a node, and the classical quadratic relation between the generating functions of the two polynomial families arises from a single polynomial relation---the defining equation of the curve in one realization and a Pell-type norm form in the other.
How to cite
Felipe Albino dos Santos, Guillaume Byamwezi Munigwa. “Legendre and Chebyshev Polynomials on a Hyperelliptic Curve: an Elementary Account.” College Mathematics 1, no. 1 (2026): 55-63.