# Closed-Form for commutation relation between powers of raising and lowering operators

1. Jul 21, 2011

### VGen128

Hello,

I am looking to find a closed-form formula for the following commutator
$[J_{-}^{n},J_{+}^{k}]$
where the operators are raising and lowering operators of the $\mathfrak{su}(2)$ algebra for which $[J_{+},J_{-}]=2J_0$ and $[J_{0},J_{\pm}]=\pm J_{\pm}$

I've already made some progress and I obtained the following relations, which can be proved by induction :

$[J_{-}^{n},J_{+}]=-2nJ_0J_{-}^{n-1}-n(n-1)J_-^{n-1}$
$[J_{-},J_{+}^{n}]=-2nJ_{+}^{n-1}J_0-n(n-1)J_+^{n-1}$

The two last relations can be used to find a recurrence relation for the object I wish to compute, but I am hoping for a closed form formula.

Any ideas ?

Thanks

2. Jul 22, 2011

### A. Neumaier

Re: Closed-Form for commutation relation between powers of raising and lowering opera

Use your recurrence formula to get the next few terms, guess from these the general form of the result,
and insert it into your recurrence formula to get recurrences for the unknown coefficients. If a nice closed formula exists, these recurrences should have a simple solution.

3. Jul 23, 2011

### VGen128

Re: Closed-Form for commutation relation between powers of raising and lowering opera

Ok ! Thanks.

I will pursue this...I'll post the result if I get something.