Tridiagonalization of the hypergeometric operator and the Racah-Wilson algebra

TL;DR

This paper explores how the hypergeometric operator's algebraic structure transforms into the Racah-Wilson algebra through tridiagonalization, revealing connections to orthogonal polynomials and their algebraic frameworks.

Contribution

It demonstrates that tridiagonalization of the hypergeometric operator induces a transition from the quadratic Jacobi algebra to the Racah-Wilson algebra, including a degenerate case related to the Hahn algebra.

Findings

Quadratic Jacobi algebra becomes the Racah-Wilson algebra after tridiagonalization.

Degenerate case leads to the Hahn algebra.

Provides algebraic insight into the structure of orthogonal polynomials.

Abstract

The algebraic underpinning of the tridiagonalization procedure is investigated. The focus is put on the tridiagonalization of the hypergeometric operator and its associated quadratic Jacobi algebra. It is shown that under tridiagonalization, the quadratic Jacobi algebra becomes the quadratic Racah-Wilson algebra associated to the generic Racah/Wilson polynomials. A degenerate case leading to the Hahn algebra is also discussed.