The BCH formula states that the product of two exponentials of non commuting operators can be combined into a single exponential involving commutators of these operators. One may write that $\ln(e^A e^B) = \sum_{n \geq 1} c_n(A,B)$ where $$c_{n+1} = \frac{1}{n+1} \left( -\frac{1}{2} [c_n, A-B] + \sum_{m=0}^{\lfloor{n/2}\rfloor} \frac{B_{2m}}{(2m)!} \sum_{k_1, \dots, k_{2m} \geq 1,\,\, k_1 + \dots + k_{2m} = n} [c_{k_1}, [ \dots, [c_{k_{2m}}, A+B] \dots]]\right)$$ and $B_n$ are the Bernoulli numbers.

Using this I want to construct the relation $$\ln(e^A e^B) = A+B + \frac{1}{2}[A,B] + \frac{1}{12} \left( [A,[A,B]] - [B,[A,B]]\right)$$

Attempt: So the terms in the expansion are the $c_n(A,B)$ - since $n \geq 1$ using this formula I am not sure if I can directly compute $c_o$ - is it perhaps done recursively? And for the case $n=1$ the first sum there only includes one term corresponding to the case $m=0$ but then what does the sum delimiters mean? In particular, the term in the second sum with $m=0$ involves $k_0$ but what is $k_0$?

Thanks for any tips!


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