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Euler polynomials and identities for non-commutative operators

Abstract : Three kinds of identities involving non-commutating operators and Euler and Bernoulli polyno-mials are studied. The first identity, as given in [2] 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 [3], 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 [9] 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. 2
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Valerio de Angelis, Christophe Vignat. Euler polynomials and identities for non-commutative operators. Journal of Mathematical Physics, American Institute of Physics (AIP), 2015, ⟨10.1063/1.4938077⟩. ⟨hal-01248828⟩

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