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What could be a reference about binomial expansions for non-commutative elements?

Specifically, where can I find a closed formula for the expansion of $(A+B)^n$ where $[A,B]=C$ and $[C,A]=[C,B]=0$?

I've found some ideas about that and also a proof using PDE's in the following website: link. But I haven't found a such formula in a published scientific paper or book.

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A simple google search led me to this :… I think it pretty much answers your question. As for an actual reference, I'll have to dig a little more! – Somnath Basu Oct 22 '11 at 2:51
@Somnath Basu: This is the same link that I posted with my question! The issue is that there is no actual reference in there! – Chris Oct 22 '11 at 3:36

1 Answer 1

up vote 14 down vote accepted

I don't know if you prefer a particular presentation of the formula, but this is essentially covered by the Baker-Campbell-Hausdorff formula, or actually it's dual, Zassenhaus formula, which in your case reduces to $$e^{(A+B)t}=e^{At}e^{Bt}e^{-[A,B]t^2/2},$$ where one side is the generating function for $(A+B)^n$ while the other has terms of the form $f(n,m,p)A^nB^mC^p$. The binomial theorem here is given by equating the coefficients of $t^n$ on both sides. $$(A+B)^n=\sum_{n\equiv k\pmod{2}} \left(\sum_{r=0}^k \binom{k}{r}A^rB^{k-r}\right)\left(-\frac{C}{2}\right)^{\frac{n-k}{2}}\frac{n!}{k!(\frac{n-k}{2})!}$$

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@Gjergji Zaimi: Thank you so much! That is all I really needed. – Chris Oct 22 '11 at 3:41
@Gjergji Zaimi I am a bit confused by the outmost sum, is k allowed to be grater than n and less than 0? – Barefeg Apr 23 '14 at 22:38

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