Generating q -Commutator Identities and the q -BCH Formula

Abstract EN

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 exp q ( x ) = ∑ n = 0 ∞ ( x n / [ n ] q ! ) , where [ n ] q = 1 + q + ⋯ + q n - 1 denotes, as usual, the n th q -integer.

We prove that if x and y are any noncommuting indeterminates, then exp q ( x ) exp q ( y ) = exp q ( x + y + ∑ n = 2 ∞ Q n ( 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: exp q ( x ) exp q ( 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.