Algebraic special functions and so(3,2)

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.