Meta Algebras and Biorthogonal Rational Functions: The Hahn Case

TL;DR

This paper introduces a unified algebraic framework for biorthogonal rational functions and Hahn-type orthogonal polynomials using the meta Hahn algebra, revealing their orthogonality and bispectral properties.

Contribution

It develops the three-generated meta Hahn algebra and its finite-dimensional representations to interpret biorthogonal functions and polynomials in a unified algebraic setting.

Findings

Unified algebraic interpretation of Hahn-type functions and polynomials

Derivation of orthogonality relations from the algebraic framework

Establishment of bispectral properties through eigenvalue problems

Abstract

The finite families of biorthogonal rational functions and orthogonal polynomials of Hahn type are interpreted algebraically in a unified way by considering the three-generated meta Hahn algebra and its finite-dimensional representations. The functions of interest arise as overlaps between eigensolutions of generalized and ordinary eigenvalue problems on the representation space. The orthogonality relations and bispectral properties naturally follow from the framework.