  In 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 $N$.
