Structure of certain Chebyshev-type polynomials in Onsager's algebra representation

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 ZN-spin chain via the representation theory of Onsager's algebra. Both two types of polynomials satisfy some (N+1)-term recurrence relations, and Nth-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.