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](http://www.voofie.com/content/110/binomial-expansion-for-non-commutative-elements-abn-where-a-b-lambda/). But I haven't found a such formula in a published scientific paper or book.



Binomial Expansion for non-commutative setting
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](http://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula#The_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.
$$(A+B)^n=\sum\_{n\equiv k\pmod{2}} \left(\sum\_{r=0}^k \binom{k}{r}A^rB^{k-r}\right)\left(-\frac{C}{2}\right)^{\frac{n-k}{2}}\frac{n!}{k!(\frac{n-k}{2})!}$$


