We define Letac-Wesolowski-Matsumoto-Yor (LWMY) functions as decreasing functions from (0, ∞) onto (0, ∞) 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, under additional assumptions, that 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.