  Three kinds of identities involving non-commutating operators and Euler and
Bernoulli polynomials are studied. The first identity, as given by Bender and
Bettencourt, expresses the nested commutator of the Hamiltonian and momentum
operators as the commutator of the momentum and the shifted Euler polynomial of
the Hamiltonian. The second one, due to J.-C. Pain, links the commutators and
anti-commutators of the monomials of the position and momentum operators. The
third appears in a work by Figuieira de Morisson and Fring in the context of
non-Hermitian Hamiltonian systems. In each case, we provide several proofs and
extensions of these identities that highlight the role of Euler and Bernoulli
polynomials.
