Groups, Jacobi functions and rigged Hilbert spaces

TL;DR

This paper explores the deep connections between special functions, Lie algebra representations, and rigged Hilbert spaces, focusing on algebraic Jacobi functions related to su(2,2) and su(1,1) symmetries, and introduces Jacobi Harmonics as a generalization of spherical harmonics.

Contribution

It establishes a framework linking special functions with Lie algebra representations within rigged Hilbert spaces, introducing Jacobi Harmonics as a new generalization.

Findings

Discrete and continuous bases coexist on rigged Hilbert spaces.

Algebraic Jacobi Functions relate to su(2,2) and su(1,1) symmetries.

Introduction of Jacobi Harmonics as a generalization of spherical harmonics.

Abstract

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.