  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.

Operator Algebras;Probability