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

TL;DR

This paper develops a new algebraic expansion method for scalar power functions of multiple vector arguments in Euclidean space, expressing them as series involving ratios of vector magnitudes and orthogonal functions, with explicitly determined angular coefficients.

Contribution

It introduces a systematic algebraic approach to expand multi-vector scalar power functions into series with explicit angular coefficients, extending previous methods.

Findings

Derived series expansions for scalar power functions of multiple vectors.

Explicit formulas for angular coefficient functions.

Applicable to orthogonal functions like $H_{\lambda_s}(\mathbf{r}_s)$.

Abstract

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.