  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.
