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 $\mathbb{R}$ $fromthemany-dimensionalvectors$ $\mathbf{r}$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.