  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.
