  This paper is a contribution to the study of the relations between special
functions, Lie algebras and rigged Hilbert spaces. The discrete indices and
continuous variables of special functions are in correspondence with the
representations of their algebra of symmetry, that induce discrete and
continuous bases coexisting on a rigged Hilbert space supporting the
representation. Meaningful operators are shown to be continuous on the spaces
of test vectors and its dual. Here, the chosen special functions, called
"Algebraic Jacobi Functions" are related to the Jacobi polynomials and the Lie
algebra is su(2,2). These functions with m and q fixed, also exhibit a
su(1,1)-symmetry. Different discrete and continuous bases are introduced. An
extension in the spirit of the associated Legendre polynomials and the
spherical harmonics is presented introducing the "Jacobi Harmonics" that are a
generalization of the spherical harmonics to the three-dimensional hypersphere
S^3.
