Algebraic Leonard trio approach to rational functions: the Hahn case

TL;DR

This paper interprets Hahn polynomials and related rational functions algebraically using Leonard trios, introduces the trio Hahn algebra, and establishes its isomorphism with the meta Hahn algebra, elucidating their structural relationship.

Contribution

It introduces the trio Hahn algebra, proves its isomorphism with the meta Hahn algebra, and constructs finite dimensional realizations to explain bispectral and biorthogonality properties.

Findings

Hahn algebra is isomorphic to the meta Hahn algebra

Finite dimensional realizations are constructed using difference operators

Bispectral and biorthogonality properties follow from the algebraic framework

Abstract

The finite families of Hahn polynomials and associated biorthogonal rational functions are interpreted algebraically in the framework of Leonard trios. We introduce the trio Hahn algebra and prove that it is isomorphic to the meta Hahn algebra, thereby clarifying the structural connection between Leonard trios and meta algebras. Finite dimensional realizations in terms of difference operators are constructed, and the functions of interest arise as overlaps between eigensolutions of ordinary eigenvalue problems. Their bispectral and biorthogonality properties follow naturally from the algebraic framework.