Meta Algebras and Special Functions: the Racah Case

TL;DR

This paper develops an algebraic framework using the meta Racah algebra to study Racah-type biorthogonal rational functions and polynomials, revealing their orthogonality and bispectral properties.

Contribution

It introduces a unified algebraic approach based on the meta Racah algebra to analyze Racah-type functions and polynomials, connecting their properties to algebraic representations.

Findings

Identifies these functions as overlap coefficients of eigenvalue problems.

Derives their orthogonality relations naturally.

Establishes bispectral properties of the functions.

Abstract

Finite families of biorthogonal rational functions and orthogonal polynomials of Racah-type are studied within a unified algebraic framework based on the meta Racah algebra and its finite-dimensional representations. These functions are identified as overlap coefficients between eigensolutions of generalized and standard eigenvalue problems posited on the representation space. The approach naturally yields their orthogonality relations and bispectral properties.