# Newton's expansion for non-commutative quantities

1. Feb 3, 2014

### 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 wanna know is there a similiar formula for the case when x and y don't commute and we have $[x,y]=c$ and $[x,c]=[y,c]=0$?
Thanks

2. Feb 4, 2014

### 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.

3. Feb 4, 2014