Coupling coefficients for Lie algebra representations and addition formulas for special functions

Representations of the Lie algebra su(1,1) and of a generalization of the oscillator algebra, b(1), are considered. The paper then introduces polynomials which are related by the coupling (or Clebsch–Gordan) coefficients of the Lie algebra in question; by making a proper choice, these polynomials themselves are related to known special functions. The coupling of two or three representations of the Lie algebra then leads to interesting addition formulas for these special functions. The polynomials appearing here are generalized Laguerre and Jacobi polynomials for the su(1,1) case, and Hermite polynomials for the b(1) algebra.