Jacobi polynomials and SU(2,2)

TL;DR

This paper explores the connection between Jacobi polynomials and the SU(2,2) group, revealing a ladder operator structure and a representation of the algebra in terms of linear operators on square-integrable functions.

Contribution

It introduces a ladder structure for Jacobi polynomials linked to SU(2,2) representations and shows the algebraic homomorphism to operators on L^2 functions.

Findings

Jacobi polynomials form a basis for SU(2,2) representations.

The algebra of operators is homomorphic to the universal enveloping algebra of su(2,2).

A ladder operator structure for Jacobi polynomials is established.

Abstract

A ladder structure of operators is presented for the Jacobi polynomials, J_n^(a,b)(x), with parameters n, a and b integers, showing that they are related to the unitary irreducible representation of SU(2,2) with quadratic Casimir C_SU(2,2)=-3/2. As they determine also a base of square-integrable functions, the universal enveloping algebra of su(2,2) is homomorphic to the space of linear operators acting on the L^2 functions defined on (-1,+1) x Z x Z/2.