Generatingq-Commutator Identities and theq-BCH Formula

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 functionexpq⁡(x)=∑n=0∞(xn/[n]q!), where[n]q=1+q+⋯+qn-1denotes, as usual, thenthq-integer. We prove that ifxandyare any noncommuting indeterminates, thenexpq⁡(x)expq⁡(y)=expq⁡(x+y+∑n=2∞Qn(x,y)), whereQn(x,y)is a sum of iteratedq-commutators ofxandy(on the right and on the left, possibly), where theq-commutator[y,x]q≔yx-qxyhas 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 iteratedq-commutators (of any length) ofxandy. These results can be used to obtain simplified presentation for the summands of theq-deformed Baker-Campbell-Hausdorff Formula.