Commutators, matrices and an identity of Copeland

TL;DR

This paper generalizes a formula involving noncommutative ring elements, expressing powers of a product as a matrix-vector product, extending Copeland's work on Pascal-style matrices.

Contribution

It introduces a new matrix expression for powers of noncommutative products, expanding the algebraic tools available for such rings.

Findings

Derived a matrix expression for (ba)^n in noncommutative rings

Generalized Copeland's formula to broader algebraic contexts

Connected the formula to Pascal-style matrices

Abstract

Given two elements $a$ and $b$ of a noncommutative ring, we express $\left( ba\right)^n$ as a "row vector times matrix times column vector" product, where the matrix is the $n$-th power of a matrix with entries $\dbinom{i}{j}\operatorname{ad}_a^{i-j}\left( b\right)$. This generalizes a formula by Tom Copeland used in the study of Pascal-style matrices.