Generating \(q\)-commutator identities and the \(q\)-BCH formula (Q504733)

Summary: 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 \(\mathrm{exp}_q(x) = \sum_{n = 0}^{\infty}(x^n / [n]_q!)\), where \([n]_q = 1 + q + \cdots + q^{n - 1}\) denotes, as usual, the \(n\)th \(q\)-integer. We prove that if \(x\) and \(y\) are any noncommuting indeterminates, then \(\mathrm{exp}_q(x) \mathrm{exp}_q(y) = \mathrm{exp}_q(x + y + \sum_{n = 2}^{\infty} Q_n(x, y))\), 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 \textit{M. P. Schützenberger} [C. R. Acad. Sci., Paris 236, 352--353 (1953; Zbl 0051.09401)] and \textit{J. Cigler} [Monatsh. Math. 88, 87--105 (1979; Zbl 0424.05007)]: \(\mathrm{exp}_q(x) \mathrm{exp}_q(y) = \mathrm{exp}_q(x + y)\). 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.