The superintegrable chiral Potts quantum chain and generalized Chebyshev polynomials

TL;DR

This paper investigates the mathematical properties of polynomials associated with the N-state chiral Potts quantum chain, revealing partial orthogonality and recursion relations similar to classical orthogonal polynomials.

Contribution

It characterizes the nature of these polynomials, showing they are not classical orthogonal sets but exhibit partial orthogonality and simple N+1-term recursion relations.

Findings

Polynomials do not form classical orthogonal sets for N>2.

Partial orthogonality observed for specific N values with respect to Jacobi and Chebyshev weights.

Zeros mostly retain separation properties of orthogonal polynomials, except at an endpoint.

Abstract

Finite-dimensional representations of Onsager's algebra are characterized by the zeros of truncation polynomials. The Z_N-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>2 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.