Research Article
Generating -Commutator Identities and the -BCH Formula
Andrea Bonfiglioli
1
and Jacob Katriel
2
1
Dipartimento di Matematica, Universit` a degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
2
Department of Chemistry, Technion-Israel Institute of Technology, 32000 Haifa, Israel
Correspondence should be addressed to Andrea Bonfglioli; andrea.bonfglioli6@unibo.it
Received 21 June 2016; Accepted 1 August 2016
Academic Editor: Antonio Scarfone
Copyright © 2016 A. Bonfglioli and J. Katriel. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Motivated by the physical applications of -calculus and of -deformations, the aim of this paper is twofold. Firstly, we prove the
-deformed analogue of the celebrated theorem by Baker, Campbell, and Hausdorf for the product of two exponentials. We deal
with the -exponential function exp
() = ∑
∞
=0
(
/[]
!), where []
=1++⋅⋅⋅+
−1
denotes, as usual, the th -integer. We
prove that if and are any noncommuting indeterminates, then exp
()exp
() = exp
(++∑
∞
=2
(,)), where
(,)
is a sum of iterated -commutators of and (on the right and on the lef, possibly), where the -commutator [,]
fl −
has always the innermost position. When [,]
=0, this expansion is consistent with the known result by Sch¨ utzenberger-Cigler:
exp
()exp
() = exp
(+). Our result improves and clarifes some existing results in the literature. Secondly, we provide an
algorithmic procedure for obtaining identities between iterated -commutators (of any length) of and . Tese results can be used
to obtain simplifed presentation for the summands of the -deformed Baker-Campbell-Hausdorf Formula.
1. Introduction
Te celebrated Baker-Campbell-Hausdorf (BCH, for short in
the sequel) Teorem allows the representation of the product
of two exponentials in terms of a single exponential (see
[1] for a comprehensive investigation of this result). Te
applications of the BCH Teorem range over many areas
of mathematics and physics, including theoretical physics,
quantum statistical mechanics, perturbation and transforma-
tion theory, the representation of time-evolution in quantum
mechanics in terms of the exponential of the Hamiltonian,
the study of nonclassical (i.e., coherent, squeezed) states of
light, group theory, control theory, the exponentiation of
Lie algebras into Lie groups, linear subelliptic PDEs, and
geometric integration in numerical analysis. A quite extensive
review of exponential operators and their many roles in
physics was presented by Wilcox [2]. In order to motivate the
main topics of the present paper (i.e., the -deformed BCH
Formula, and an algorithm for generating -commutator
identities), we frst review what is known so far as the -
analogue (or -deformation) of the BCH Teorem, along
with motivations for the physical interest in this subject; see
also [3] by the authors with Achilles.
Te idea of -deformation goes back to Euler in the
mid eighteenth century and to Gauss in the early nineteenth
century. If one defnes the th -integer as []
=1++
2
+⋅⋅⋅+
−1
and, accordingly, if the -factorial is defned as
[]
! = [−1]
!⋅[]
(where [0]
!=1), then the -exponential
is
exp
() =
∞
∑
=0
[]
!
. (1)
It is well known that Jackson’s -derivative, defned by the
ratio
()=
()− ()
(−1)
(2)
satisfes
exp
() = exp
() (see, e.g., the monograph [4]
for an introduction to these topics). Te reader is referred
to the recent monograph [5] for an in-depth analysis and
Hindawi Publishing Corporation
Advances in Mathematical Physics
Volume 2016, Article ID 9598409, 26 pages
http://dx.doi.org/10.1155/2016/9598409