The rank two Jacobi algebra

TL;DR

This paper identifies the quadratic rank two Jacobi algebra from bispectral operators of two-variable Jacobi polynomials, revealing its subalgebra structure and dual realizations, and deriving structure relations for these polynomials.

Contribution

It introduces the quadratic rank two Jacobi algebra and explores its subalgebras, dual realizations, and structure relations for two-variable Jacobi polynomials.

Findings

Identification of the quadratic rank two Jacobi algebra.

Existence of Racah and Jacobi subalgebras of rank one.

Derivation of structure relations for two-variable Jacobi polynomials.

Abstract

The quadratic rank two Jacobi algebra is identified from the relations obeyed by the bispectral operators of the two variable Jacobi polynomials orthogonal on the triangle. It is seen to admit as subalgebras Racah and Jacobi algebras of rank one. The dual realizations in terms of differential operators in the variable representation and in terms of difference operators in the degree representation are provided. Structure relations for the two variable Jacobi polynomials are obtained as a by product.