Independence properties of the Matsumoto--Yor type

We define Letac-Wesolowski-Matsumoto-Yor (LWMY) functions as decreasing functions from $(0,\infty)$ onto $(0,\infty)$ with the following property: there exist independent, positive random variables $X$ and $Y$ such that the variables $f(X+Y)$ and $f(X)-f(X+Y)$ are independent. We prove that, under additional assumptions, there are essentially four such functions. The first one is $f(x)=1/x$. In this case, referred to in the literature as the Matsumoto-Yor property, the law of $X$ is generalized inverse Gaussian while $Y$ is gamma distributed. In the three other cases, the associated densities are provided. As a consequence, we obtain a new relation of convolution involving gamma distributions and Kummer distributions of type 2.