Algebraic special functions and so(3,2)

TL;DR

This paper explores the algebraic structure of special functions like Legendre polynomials and spherical harmonics, showing they form representations of the Lie algebra so(3,2) and linking Lie algebras to function spaces.

Contribution

It introduces the concept of algebraic special functions as those with ladder structures forming Lie algebra representations, unifying special functions and operator algebras.

Findings

Legendre polynomials and spherical harmonics form irreducible representations of so(3,2)

Universal enveloping algebra of so(3,2) is isomorphic to operator spaces on L^2 functions

Ladder structures characterize algebraic special functions as Lie algebra representations

Abstract

A ladder structure of operators is presented for the associated Legendre polynomials and the spherical harmonics showing that both belong to the same irreducible representation of so(3,2). As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be isomorphic to the space of linear operators acting on the L^2 functions defined on (-1,1) x Z and on the sphere S^2, respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining in this way the "algebraic special functions" that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L^2 functions is isomorphic to the universal enveloping algebra.