Algebraic Heun Operator and Band-Time Limiting

We introduce the algebraic Heun operator associated to any bispectral pair of operators. This operator can be presented as a generic bilinear combination of these bispectral operators. We show that the resulting operators are natural generalizations of the ordinary Heun operator. This leads to a simple construction of the operators commuting with the projection operators in problems of band-time limiting and it gives a way to adapt a construction first used by Perline in a purely finite setup to general situations. We also extend this algebraic construction to cover some purely finite cases which lie beyond this scheme, in particular for the anti-Krawtchouk polynomials. In the latter case, the commuting operator is represented by a pentadiagonal matrix which can be constructed explicitly as a polynomial of degree 4 of the pair of bispectral operators.