Structure of certain Chebyshev-type polynomials in Onsagerʹs 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 Z N -spin chain via the representation theory of Onsagerʹs 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.