Dual Lukacs regressions of negative orders for non-commutative variables

In the paper we study characterizations of probability measures in free probability. By constancy of regressions for random variable $\V(\I-\U)\V$ given by $\V\U\V$, where $\U$ and $\V$ are free, we characterize free Poisson and free binomial distributions. Our paper is analog in free probability of results known in classical probability \cite{BobWes2002Dual}, where gamma and beta distributions are characterized by constancy of $\E((V(1-U))^{i}|UV)$, for $i \in \{-2,-1,1,2\}$. This paper together with previous results \cite{SzpojanWesol} exhaust all cases of characterizations from \cite{BobWes2002Dual}.