Independences of Kummer laws

We prove that if X, Y are positive, independent, non-Dirac random variables and if $\alpha$, $\beta$ $\ge$ 0, $\alpha$ = $\beta$, then the random variables U and V defined by U = Y 1+$\beta$(X+Y) 1+$\alpha$X+$\beta$Y and V = X 1+$\alpha$(X+Y) 1+$\alpha$X+$\beta$Y are independent if and only if X and Y follow Kummer distributions with suitable parameters. In other words, the Kummer distributions are the only invariant measures for lattice recursion models introduced by Croydon and Sasada in [3]. The result extends earlier characterizations of Kummer and gamma laws by independence of U = Y 1+X and V = X 1 + Y 1+X , which is the case of ($\alpha$, $\beta$) = (1, 0).