The superintegrable chiral Potts quantum chain and generalized Chebyshev polynomials

Finite-dimensional representations of Onsager’s algebra are characterized by the zeros of truncation polynomials. The ZN -chiral Potts quantum chain hamiltonians (of which the Ising chain hamiltonian is the N = 2 case) are the main known interesting representations of Onsager’s algebra and the corresponding polynomials have been found by Baxter and Albertini, McCoy and Perk in 1987-89 considering the Yang-Baxter-integrable 2-dimensional chiral Potts model. We study the mathematical nature of these polynomials. We find that for N ≥ 3 and fixed charge Q these don’t form classical orthogonal sets because their pure recursion relations have at least N + 1-terms. However, several basic properties are very similar to those required for orthogonal polynomials. The N + 1-term recursions are of the simplest type: like for the Chebyshev polynomials the coefficients are independent of the degree. We find a remarkable partial orthogonality, for N = 3, 5 with respect to Jacobi-, and for N = 4, 6 with respect to Chebyshev weight functions. The separation properties of the zeros known from orthogonal polynomials are violated only by the extreme zero at one end of the interval.