Some binomial formulas for non-commuting operators

TL;DR

This paper derives binomial-like identities for non-commuting operators under specific commutator conditions, with applications to differential operators and multiplication operators, revealing unexpected relations from prior research.

Contribution

It introduces new binomial formulas for operators with particular commutator properties, expanding the algebraic understanding of such operator identities.

Findings

Derived binomial identities for operators with proportional commutators

Identified applications to differentiation and multiplication operators

Revealed unexpected operator relations from previous medical imaging work

Abstract

Let $D$ and $U$ be linear operators in a vector space (or more generally, elements of an associative algebra with a unit). We establish binomial-type identities for $D$ and $U$ assuming that either their commutator $[D,U]$ or the second commutator $[D,[D,U]]$ is proportional to $U$. Operators $D=d/dx$ (differentiation) and $U$- multiplication by $e^{\lambda x}$ or by $\sin \lambda x$ are basic examples, for which some of these relations appeared unexpectedly as byproducts of an authors' previous medical imaging research.