Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials

TL;DR

This paper introduces cyclic tridiagonal pairs and their associated higher-order Onsager algebras, providing new algebraic structures and connections to orthogonal polynomials beyond Leonard duality.

Contribution

It defines cyclic tridiagonal pairs, constructs divided polynomials, and identifies a higher-order Onsager algebra as their generated algebra, extending the framework of orthogonal polynomials.

Findings

Explicit examples of cyclic tridiagonal pairs are provided.

The algebra generated by divided polynomials generalizes the Onsager algebra.

Connections to orthogonal polynomials and Dunkl shift operators are established.

Abstract

The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N, we associate a pair of `divided polynomials'. The properties of this pair generalize the ones of tridiagonal pairs of Racah type. The algebra generated by the pair of divided polynomials is identified as a higher-order generalization of the Onsager algebra. It can be viewed as a subalgebra of the q-Onsager algebra for a proper specialization at q the primitive 2Nth root of unity. Orthogonal polynomials beyond the Leonard duality are revisited in light of this framework. In particular, certain second-order Dunkl shift operators provide a realization of the divided polynomials at N=2 or q=i.