Newton's expansion for non-commutative quantities

[ShayanJ]

You probably know that for two commutative quantities x and y,we have:
$(x+y)^n=\sum_{r=0}^n \left( \begin{array}{c} n \\ r \end{array} \right) x^{n-r} y^r$
Now I want to know is there a similar formula for the case when x and y don't commute and we have $[x,y]=c$ and $[x,c]=[y,c]=0$?
Thanks

 

[maajdl]

In the case of commutative types, the terms can be ordered as x^i y^j, as you showed, because x and y commute.
In general, you will have all sorts of ordering, like xxy, xyx, yxx.
Maybe you can work out the special cases n=2 and n=3, to guess an answer.

 

[maajdl]

This might give you an answer:

http://mathoverflow.net/questions/78813/binomial-expansion-for-non-commutative-setting