Baker–Campbell–Hausdorff (CBH) Formula question

[BlackHole213]

I've been trying to understand the explicit CBH formula

http://en.wikipedia.org/wiki/Baker–...er.E2.80.93Campbell.E2.80.93Hausdorff_formula

However, I don't really understand how to know what the values of $r_i$ and $s_i$ are, where $1\leq i\leq n$.

Thanks.

 

[Bill_K]

I've been trying to understand the explicit CBH formula

http://en.wikipedia.org/wiki/Baker–...er.E2.80.93Campbell.E2.80.93Hausdorff_formula

However, I don't really understand how to know what the values of $r_i$ and $s_i$ are, where $1\leq i\leq n$.

Like it says,

Where s_n and r_n are non-negative integers.

The sum runs over all possible values of s_n and r_n, where s_n + r_n > 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! )

 

[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 $n=1$?