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Advances in Mathematical Physics
Volume 2016 (2016), Article ID 9598409, 26 pages
http://dx.doi.org/10.1155/2016/9598409
Research Article

Generating -Commutator Identities and the -BCH Formula

1Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
2Department of Chemistry, Technion-Israel Institute of Technology, 32000 Haifa, Israel

Received 21 June 2016; Accepted 1 August 2016

Academic Editor: Antonio Scarfone

Copyright © 2016 Andrea Bonfiglioli and Jacob Katriel. This 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.

Abstract

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 Hausdorff for the product of two exponentials. We deal with the -exponential function , where denotes, as usual, the th -integer. We prove that if and are any noncommuting indeterminates, then , where is a sum of iterated -commutators of and (on the right and on the left, possibly), where the -commutator has always the innermost position. When , this expansion is consistent with the known result by Schützenberger-Cigler: . Our result improves and clarifies some existing results in the literature. Secondly, we provide an algorithmic procedure for obtaining identities between iterated -commutators (of any length) of and . These results can be used to obtain simplified presentation for the summands of the -deformed Baker-Campbell-Hausdorff Formula.