On the Lukacs property for free random variables

The Lukacs property of the free Poisson distribution is studied here. We prove that if free $\X$ and $\Y$ are free Poisson distributed with suitable parameters, then $\X+\Y$ and $\left(\X+\Y\right)^{-\frac{1}{2}}\X\left(\X+\Y\right)^{-\frac{1}{2}}$ are free. As as an auxiliary result we give joint cumulants of $\X$ and $\X^{-1}$ for free Poisson distributed $\X$. We also study the Lukacs property of the free gamma distribution.