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)$\mathbb{E}(V(1-U)|UV)$ and E(V²(1−U)²|UV)$\mathbb{E}(V^2{(1-U)}^2|UV)$ are non-random then V has a gamma distribution and U has a beta distribution.