Noncommutative Abel-like identities

TL;DR

This paper extends classical Abel--Hurwitz identities to a noncommutative algebra setting, providing new formulas involving sums over subsets with noncommutative elements.

Contribution

It introduces noncommutative generalizations of Abel--Hurwitz identities, broadening their applicability to noncommutative rings and algebraic structures.

Findings

Derived new noncommutative identities involving sums over subsets.

Established formulas connecting sums with products of noncommutative elements.

Extended classical identities to a noncommutative context.

Abstract

We generalize the Abel--Hurwitz identities to an almost entirely noncommutative setting. Namely, let $V$ be a finite set of size $n$, and let $\mathbb{L}$ be any noncommutative ring. For each $s\in V$, let $x_{s}\in\mathbb{L}$. Set $x\left( S\right) :=\sum_{s\in S}x_{s}$ for any $S\subseteq V$. Let $X$ and $Y$ be two elements of $\mathbb{L}$ such that $X+Y$ lies in the center of $\mathbb{L}$. Then, we show that% \begin{align*} & \sum_{S\subseteq V}\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert }\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert }=\sum_{\substack{i_{1},i_{2},\ldots,i_{k}\in V\text{ distinct}% }}\left( X+Y\right) ^{n-k}x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}};\\ & \sum_{S\subseteq V}X\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert -1}\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert }=\left( X+Y\right) ^{n};\\ & \sum_{S\subseteq V}X\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert -1}\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert -1}\left( Y-x\left( V\right) \right) =\left( X+Y-x\left( V\right) \right) \left( X+Y\right) ^{n-1}. \end{align*} (Negative powers are understood to be cancelled by other factors.)