Tridiagonalization and the Heun equation

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 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) and separation of variables in a superintegrable system are discussed.