The Heun operator of Hahn type

TL;DR

This paper introduces the Heun-Hahn operator on uniform grids, demonstrating its polynomial degree-raising property, tridiagonality, and algebraic bilinearity, and explores its algebraic extension and relation to biorthogonal rational functions.

Contribution

It defines the Heun-Hahn operator, analyzes its algebraic properties, and extends the Hahn algebra to include this operator, linking it to biorthogonal rational functions.

Findings

The Heun-Hahn operator maps degree n polynomials to degree n+1.

It is tridiagonal in bases of Pochhammer or Hahn polynomials.

The extended algebra includes the Heun-Hahn operator as a generator.

Abstract

The Heun-Hahn operator on the uniform grid is defined. This operator is shown to map polynomials of degree $n$ to polynomials of degree $n+1$, to be tridiagonal in bases made out of either Pochhammer or Hahn polynomials and to be bilinear in the operators of the Hahn algebra. The extension of this algebra that includes the Heun-Hahn operator as generator is described. Biorthogonal rational functions on uniform grids are shown to be related to this framework.