The following result of Lukacs is known: let $X_1, X_2$ be independent, positive random variables, having the nondegenerate distributions $P_1$ and $P_2$. Suppose that $X_1/X_2$ and $X_1 + X_2$ are independent. Then $P_1$ and $P_2$ are gamma distributions with the same scale parameter. Lukacs' original deduction requires details from complex analysis. Here a simpler proof is given. Instead of $P_1$ and $P_2$ two other probability measures $\mu_1$ and $\mu_2$ are shown to be determined by the independence properties of $X_1$ and $X_2$. It is possible to express $P_i$ and $\mu_i$ by each other, and $\mu_i$ is chosen such that all moments of $\mu_i$ are finite $(i = 1,2)$. Thus the proof reduces to a straight-forward calculation of moments.
"A Simple Proof of a Classical Theorem which Characterizes the Gamma Distribution." Ann. Statist. 6 (5) 1165 - 1167, September, 1978. https://doi.org/10.1214/aos/1176344319