2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/127301In this report, we present a systematic account of mathematical structures of certain special polynomials arisen from the energy study of the superintegrable $N$-state chiral Potts model with a finite number of sizes. The polynomials of low-lying sectors are represented in two different forms, one of which is directly related to the energy description of superintegrable chiral Potts $\ZZ_N$-spin chain via the representation theory of Onsager's algebra. Both two types of polynomials satisfy some $(N+1)$-term recurrence relations, and $N$th order differential equations; polynomials of one kind reveal certain Chebyshev-like properties. Here we provide a rigorous mathematical argument for cases $N=2, 3$, and further raise some mathematical conjectures on those special polynomials for a general $N$.18 pages, Latex ; Typos corrected, Small changes for clearer presentationMathematical PhysicsHigh Energy Physics - TheoryExactly Solvable and Integrable Systems39A10; 33E30; 82B20text