A family of tridiagonal pairs and related symmetric functions

TL;DR

This paper explores a family of tridiagonal pairs linked to quantum integrable systems, detailing their eigenstructures, recurrence relations, and orthogonal rational functions, generalizing Askey-Wilson polynomials.

Contribution

It introduces a new family of tridiagonal pairs with explicit eigenstructure descriptions and derives associated orthogonal rational functions and their weight functions.

Findings

Eigenvalue sequences and eigenspaces characterized.

Recurrence relations and q-difference equations derived.

Explicit rational solutions and weight functions provided.

Abstract

A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in details. The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described. The overlap functions between the two dual basis are shown to satisfy a coupled system of recurrence relations and a set of discrete second-order $q-$difference equations which generalize the ones associated with the Askey-Wilson orthogonal polynomials with a discrete argument. Normalizing the fundamental solution to unity, the hierarchy of solutions are rational functions of one discrete argument, explicitly derived in some simplest examples. The weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given.