Algebraic special functions and SO(3,2)

• The algebraic ladder structure is constructed for the associated Legendre polynomials (ALP).
• ALP and spherical harmonics support a unitary irreducible SO(3,2)-representation.
• A ladder structure is the condition to get a Lie group representation defining “algebraic special functions”.
• The “algebraic special functions” connect Lie algebras and L2L2 functions.

A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so(3,2) with quadratic Casimir equals to −5/4−5/4. As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be homomorphic to the space of linear operators acting on the L2L2 functions defined on (−1,1)×Z(−1,1)×Z and on the sphere S2S2, 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 L2L2 functions is homomorphic to the universal enveloping algebra.The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO(3,2).