On the q-analogues of the Zassenhaus formula for dientangling exponential operators

TL;DR

This paper explores q-analogues of the Zassenhaus formula for exponential operators, providing explicit expressions for new operator coefficients and demonstrating alternative factorizations involving different q-exponentials.

Contribution

It introduces a new form of q-analogue of the Zassenhaus formula with explicit expressions for additional operator coefficients, expanding the theoretical framework.

Findings

Derived explicit formulas for coefficients c_2, c_3, c_4

Presented alternative q-exponential factorizations involving different bases

Extended the understanding of disentangling exponential operators in q-calculus

Abstract

Katriel, Rasetti and Solomon introduced a $q$-analogue of the Zassenhaus formula written as $e_q^{(A+B)}$ $=$ $e_q^Ae_q^Be_q^{c_2}e_q^{c_3}e_q^{c_4}e_q^{c_5}...$, where $A$ and $B$ are two generally noncommuting operators and $e_q^z$ is the Jackson $q$-exponential, and derived the expressions for $c_2$, $c_3$ and $c_4$. It is shown that one can also write $e_q^{(A+B)}$ $=$ $e_q^Ae_q^Be_{q^2}^{\C_2}e_{q^3}^{\C_3}e_{q^4}^{\C_4}e_{q^5}^{\C_5}...$. Explicit expressions for $\C_2$, $\C_3$ and $\C_4$ are given.