Multipole expansions in four-dimensional hyperspherical harmonics

The technique of vector differentiation is applied to the problem of the derivation of multipole expansions in four-dimensional space. Explicit expressions for the multipole expansion of the function r(n)C(j) ((r) over cap) with r = r(1),+ r(2) are given in terms of tensor products of two hyperspherical harmonics depending on the unit vectors (r) over cap (1) and (r) over cap (2). The multipole decomposition of the function (r(1) center dot r(2))(n) is also derived. The proposed method can be easily generalized to the case of the space with dimensionality larger than four. Several explicit expressions for the four-dimensional Clebsch-Gordan coefficients with particular values of parameters are presented in the closed form.