Tridiagonalization and the Heun equation

Understand

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.