Some binomial formulas for non-commuting operators

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 $D=d\mathrm{/}dx$ (differentiation) and $U$- multiplication by $e^{\lambda x}$orby$\sin \lambda x$ are basic examples, for which some of these relations appeared unexpectedly as byproducts of an authors’ previous medical imaging research.