Baker–Campbell–Hausdorff (CBH) Formula question

1. May 14, 2014

BlackHole213

2. May 14, 2014

Bill_K

Like it says,

The sum runs over all possible values of sn and rn, where sn + rn > 0. They apparently stand for the number of X's and Y's in the multiple commutator. (Glad I don't have to prove this! )

3. May 14, 2014

BlackHole213

I agree, proving it would be awful, to say to least.

This may be a dumb question, but how do I know what the possible values of $r_n$ and $s_n$ are? I feel like I'm over-thinking this.

For example, if I consider $[X,Y]$, then $r_n=s_n=r_1=s_1=1$. If I just had the BCH formula as written by Dynkin, how would I know that $r_1=s_1=1$ for $n=1$?