Meta algebras and biorthogonal rational functions: the $q$-Hahn case

TL;DR

This paper provides an algebraic framework using the meta $q$-Hahn algebra to unify the understanding of orthogonal polynomials and biorthogonal rational functions of $q$-Hahn type, revealing their properties through algebraic actions.

Contribution

It introduces a unified algebraic interpretation of $q$-Hahn functions and polynomials via the meta $q$-Hahn algebra, connecting their properties to algebraic representations.

Findings

Identifies $q$-Hahn functions as overlaps of eigenbasis solutions.

Derives orthogonality, recurrence, and difference relations algebraically.

Uses finite-dimensional bidiagonal representations of the meta $q$-Hahn algebra.

Abstract

A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of $q$-Hahn type is provided. The approach relies on the meta $q$-Hahn algebra and its finite-dimensional bidiagonal representations. The functions of $q$-Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta $q$-Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta $q$-Hahn algebra on various bases.