Generating q-Commutator Identities and the q-BCH Formula. (16th October 2016) - Bonfiglioli, Andrea; Katriel, Jacob
Abstract : Motivated by the physical applications ofq -calculus and ofq -deformations, the aim of this paper is twofold. Firstly, we prove theq -deformed analogue of the celebrated theorem by Baker, Campbell, and Hausdorff for the product of two exponentials. We deal with theq -exponential functionexp q ⁡ ( x ) = ∑ n = 0 ∞ ( x n / [ n ] q ! ), where[ n ] q = 1 + q + ⋯ + q n - 1 denotes, as usual, then thq -integer. We prove that ifx andy are any noncommuting indeterminates, thenexp q ⁡ ( x ) exp q ⁡ ( y ) = exp q ⁡ ( x + y + ∑ n = 2 ∞ Q n ( x, y ) ), whereQ n ( x, y ) is a sum of iteratedq -commutators ofx andy (on the right and on the left, possibly), where theq -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 iteratedq -commutators (of any length) ofx andy . These results can be used to obtain simplified presentation for the summands of theq -deformed Baker-Campbell-Hausdorff Formula.