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I am looking to find a closed-form formula for the following commutator

[itex][J_{-}^{n},J_{+}^{k}][/itex]

where the operators are raising and lowering operators of the [itex]\mathfrak{su}(2)[/itex] algebra for which [itex][J_{+},J_{-}]=2J_0[/itex] and [itex][J_{0},J_{\pm}]=\pm J_{\pm}[/itex]

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

[itex][J_{-}^{n},J_{+}]=-2nJ_0J_{-}^{n-1}-n(n-1)J_-^{n-1}[/itex]

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

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

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

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