Some properties of hyperspherical harmonics

A general formula is given for the canonical decomposition of a homogeneous polynomial of order λ in m variables into a sum of harmonic polynomials. This formula, which involves successive applications of the generalized Laplace operator, is proved in the Appendix. It is shown that the group-theoretical method for constructing irreducible Cartesian tensors follows from the general formula for canonical decomposition. The relationship between harmonic polynomials and hyperspherical harmonics is discussed, and an addition theorem for hyperspherical harmonics is derived. An expansion of a many-dimensional plane wave in terms of Gegenbauer polynomials and Bessel functions is derived and used to construct bicenter expansions of arbitrary functions in many-dimensional spaces. Finally, a formula is derived for the 3λ coefficients of hyperspherical harmonics. These coefficients give the values of integrals involving the products of three harmonics.