Euler Polynomials and Identities for Non-Commutative Operators

TL;DR

This paper explores identities involving non-commutative operators and Euler and Bernoulli polynomials, providing proofs and extensions that deepen understanding of operator algebra in quantum mechanics.

Contribution

It introduces new proofs and extensions of existing identities linking non-commutative operators with Euler and Bernoulli polynomials.

Findings

Established new proofs for known operator identities.

Extended identities to broader classes of non-Hermitian systems.

Highlighted the role of Euler and Bernoulli polynomials in operator algebra.

Abstract

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.