Addition formula for two families of orthogonal polynomials in several variables

An addition formula is a closed-form formula for reproducing kernels of orthogonal polynomials of several variables; the latter are kernels of orthogonal projection operators. We derive addition formulas for two families of orthogonal polynomials, which consist of wrapped products of classical Gegenbauer polynomials on the unit ball with either classical Jacobi polynomials on the simplex or product Laguerre polynomials, each of which is given explicitly as a multilayer of integrals of a classical orthogonal polynomial of one variable. These formulas provide a powerful tool for analysis and allow us, in particular, to reduce a substantial portion of analysis on the Fourier orthogonal expansions on the domains to that of one variable.