Generating q‐Commutator Identities and the q‐BCH Formula

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 nth 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 Qn(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≔yx − qxy 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) = expq⁡(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.