Generating q -commutator identities and the q -BCH formula

Abstract

Motivated by the physical applications of q -calculus and of q -deformations, the aim of this paper is twofold. Firstly, we prove the q -deformed analogue of the celebrated theorem by Baker, Campbell, and Hausdorff for the product of two exponentials. We deal with the q -exponential function expq (x) = n = 0∞ (xn / [ n ]q !), where [ n ] q = 1 + q + + qn 1 denotes, as usual, the n th q -integer. We prove that if x and y are any noncommuting indeterminates, then expq (x) expq (y) = expq (x + y + n = 2∞ Qn (x, y)), where Q n (x, y) is a sum of iterated q -commutators of x and y (on the right and on the left, possibly), where the q -commutator [ y, x ]q y x - q x y has always the innermost position. When [ y, x ]q = 0, this expansion is consistent with the known result by Schützenberger-Cigler: expq (x) expq (y) = exp q (x + y). Our result improves and clarifies some existing results in the literature. Secondly, we provide an algorithmic procedure for obtaining identities between iterated q -commutators (of any length) of x and y. These results can be used to obtain simplified presentation for the summands of the q -deformed Baker-Campbell-Hausdorff Formula.