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.