Dual Lukacs regressions for non-commutative variables

Dual Lukacs type characterizations of random variables in free probability are studied here. First, we develop a freeness property satisfied by Lukacs type transformations of free-Poisson and free-Binomial non-commutative variables which are free. Second, we give a characterization of non-commutative free-Poisson and free-Binomial variables by properties of first two conditional moments, which mimic Lukacs type assumptions known from classical probability. More precisely, our result is a non-commutative version of the following result known in classical probability: if $U$, $V$ are independent real random variables, such that $E(V(1-U)|UV)$ and $E(V^2(1-U)^2|UV)$ are non-random then $V$ has a gamma distribution and $U$ has a beta distribution.