  If $X$ and $Y$ are independent random variables with distributions $\mu$ and
$\nu$ then $U=\psi(X,Y)$ and $V=\phi(X,Y)$ are also independent for some $\psi$
and $\phi$. Properties of this type are known for many important probability
distributions $\mu$ and $\nu$. Also related characterization questions have
been widely investigated: Let $X$ and $Y$ be independent and let $U$ and $V$ be
independent. Are the distributions of $X$ and $Y$ $\mu$ and $\nu$,
respectively? Recently two new properties and characterizations of this kind
involving the Kummer distribution appeared in the literature. For independent
$X$ and $Y$ with gamma and Kummer distributions Koudou and Vallois observed
that $U=(1+(X+Y)^{-1})/(1+X^{-1})$ and $V=X+Y$ are also independent, and Hamza
and Vallois observed that $U=Y/(1+X)$ and $V=X(1+Y/(1+X))$ are independent. In
2011 and 2012 Koudou, Vallois characterizations related to the first property
were proved, while the characterizations in the second setting have been
recently given in Piliszek, Weso{\l}owski (2016). In both cases technical
assumptions on smoothness properties of densities of $X$ and $Y$ were needed.
In 2015, the assumption of independence of $U$ and $V$ in the first setting was
weakened to constancy of regressions of $U$ and $U^{-1}$ given $V$ with no
density assumptions. However, the additional assumption $\mathbb{E}
X^{-1}<\infty$ was introduced. In the present paper we provide a complete
answer to the characterization question in both settings without any additional
technical assumptions regarding smoothness or existence of moments. The
approach is, first, via characterizations exploiting some conditions imposed on
regressions of $U$ given $V$, which are weaker than independence, but for which
moment assumptions are necessary. Second, using a technique of change of
measure we show that the moment assumptions can be avoided.
