  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 $r^n C_j (\hr)$ with
$\vvr=\vvr_1+\vvr_2$ are given in terms of tensor products of two
hyperspherical harmonics depending on the unit vectors $\hr_1$ and $\hr_2$. The
multipole decomposition of the function $(\vvr_1 \cdot \vvr_2)^n$ is also
derived. The proposed method can be easily generalised 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.
