Onsager's algebra and partially orthogonal polynomials

TL;DR

This paper explores the connection between special polynomials related to Onsager's algebra and Jacobi polynomials, revealing a form of partial orthogonality relevant to the superintegrable chiral Potts model.

Contribution

It demonstrates that polynomials from the Z_3-case of the model are closely related to Jacobi polynomials and exhibit partial orthogonality, extending understanding of their mathematical structure.

Findings

Polynomials satisfy 4-term recursion relations

They are related to Jacobi polynomials

They exhibit partial orthogonality

Abstract

The energy eigenvalues of the superintegrable chiral Potts model are determined by the zeros of special polynomials which define finite representations of Onsager's algebra. The polynomials determining the low-sector eigenvalues have been given by Baxter in 1988. In the Z_3-case they satisfy 4-term recursion relations and so cannot form orthogonal sequences. However, we show that they are closely related to Jacobi polynomials and satisfy a special "partial orthogonality" with respect to a Jacobi weight function.