Finding a Closed-Form Formula for the Commutator [J_-^n, J_+^k]

[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

 

[A. Neumaier]

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.

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.

 

[VGen128]

Ok ! Thanks.

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