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The Lukacs property of the free Poisson distribution is studied. We prove that if free and are free Poisson distributed with suitable parameters, then + and ${(+)}^{-1/2}{(+)}^{-1/2}$ are free. As an auxiliary result we compute the joint cumulants of and ${}^{-1}$ for free Poisson distributed . We also study the Lukacs property of the free Gamma distribution.