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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 thederivation 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 certainorthogonal functions $ H_{\lambda_s}(\mathbf{r}_s) $ as ${\sim}\sum\limits_{\lambda_k} H_{\lambda_1}(\mathbf{r}_1)\ldotsH_{\lambda_{_{N}}}(\mathbf{r}_{_{N}}) $. The coefficient angular functions arefound and determined in both cases.

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