Algebraic special functions and image Original Research Article

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 image with quadratic Casimir equals to image. As both are also bases of square-integrable functions, the universal enveloping algebra of image is thus shown to be homomorphic to the space of linear operators acting on the image functions defined on image and on the sphere image, 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 image 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 image.