Expansion into a many-dimensional rational series for scalar power functions of vector arguments

For a function of a type $\left| \mathbf{r}_1{+}\ldots {+}\mathbf{r}_{_N} \right|^{-\nu} \in \mathbb{R}$ from the many-dimensional vectors $\mathbf{r}_s$ in Euclidean space, the successive algebraic approach is the derivation of the expansion in the form ${\sim}\sum\limits_{s}\frac{r_1^s}{r_{_{N}}^s}\ldots \frac{r_{_{N-1}}^s}{r_{_{N}}^s}, \,\, (r_k{<}r_{_{N}})$, and also for certain orthogonal functions $H_{\lambda_s}(\mathbf{r}_s)$ as ${\sim}\sum\limits_{\lambda_k} H_{\lambda_1}(\mathbf{r}_1)\ldots H_{\lambda_{_{N}}}(\mathbf{r}_{_{N}})$. The coefficient angular functions are found and determined in both cases.