Tridiagonalization and the Heun equation

TL;DR

This paper demonstrates how tridiagonalization of the hypergeometric operator produces the Heun operator, introduces Racah-Heun polynomials, and explores their algebraic and physical significance in superintegrable systems.

Contribution

It introduces the Racah-Heun algebra and orthogonal polynomials, connecting hypergeometric and Heun operators through algebraic and physical frameworks.

Findings

Tridiagonalization of hypergeometric operator yields Heun operator.

Racah-Heun algebra is a quadratic extension of Racah algebra.

New Racah-Heun orthogonal polynomials are defined as overlap coefficients.

Abstract

It is shown that the tridiagonalization of the hypergeometric operator $L$ yields the generic Heun operator $M$. The algebra generated by the operators $L,M$ and $Z=[L,M]$ is quadratic and a one-parameter generalization of the Racah algebra. The new Racah-Heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators $L$ and $M$. An interpretation in terms of the Racah problem for $su(1,1)$ algebras and separation of variables in a superintegrable system are discussed.