The solid harmonics are solutions to Laplace's equation in spherical coordinates. The regular and irregular solid harmonics, obtained by rescaling spherical harmonics, are respectively $$R_l^m(\textbf{r})=\sqrt{\frac{4\pi}{2l+1}} r^lY_l^m(\theta,\phi) $$ $$I_l^m(\textbf{r})=\sqrt{\frac{4\pi}{2l+1}} \frac{Y_l^m(\theta,\phi)}{r^{l+1}} $$
They satisfy addition theorems. For example $$R_l^m(\textbf{r}+\textbf{a}) = \sum_{\lambda=0}^l \binom{2l}{2\lambda}^{1/2}\sum_{\mu=-\lambda}^\lambda R_\lambda^\mu(\textbf{r})R_{l-\lambda}^{m-\mu}(\textbf{a}) \langle \lambda, \mu ; l-\lambda, m-\mu | lm\rangle $$ and a similar but infinite expansion holds for $I_l^m(\textbf{r}+\textbf{a})$. Wiki has it here.
I note that if $\textbf{r}\in \mathbb{R}^d$ for $d>3$ then there is a higher dimensional generalization of spherical harmonics (hyperspherical harmonics) which satisfy their own higher dimensional addition theorem. Since the powers of $r$ in the definitions of the solid harmonics aren't "dimension dependent" so to speak, I would expect there to be a generalization of solid harmonics to higher dimensions with it's own version of the above addition theorem. Alas, I cannot find anything of the sort in text or online.
Does anyone know if such an expansion exists? I appreciate any guidance here!