As I see, e.g., in this question, a closed formula for the following integral \begin{equation} \int_{\mathbb{S}^N} Y_{\ell_1\ldots \ell_N}\,Y_{\ell'_1\ldots\ell'_N}\,Y_{\ell''_1\ldots \ell''_N}\, d^N S \end{equation} is well-known in the $N=2$ case. (Here $\mathbb{S}^N=\{x_1^2+\dots+x_{N+1}^2=1\}$ is the $N$-dimensional sphere sitting in $N+1$ dimensional space. The symbol $d^N S$ denotes the $N$-dimensional surface element and the spherical harmonics quantum numbers reduce to $\ell_1=\ell, \ell_2=m$ in the two-dimensional case).
My question is whether a similar formula is known in arbitrary dimension (I am especially interested in the cases $N=3$, $N=5$)?
Hints on how to compute such a formula, or even special cases of it are also warmly appreciated.
