Generating q-commutator identities and the q-BCH formula

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 exp() = Σ_(^/[]!), where [] = 1 + + ⋅⋅ ⋅ + ^(−1) denotes, as usual, the th -integer. We prove that if and are any noncommuting indeterminates, then exp()exp() = exp( + + Σ_ _(, )), 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 [, ] = 0, this expansion is consistent with the known result by Sch¨utzenberger-Cigler: exp()exp() = exp( + ). 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.