On a functional equation related to an independence property for beta distributions (Q2386098)

A study of a transformation preserving independence for the beta probability distribution, leads to the functional equation: \[ g\Big(\frac{1-x}{1-xy}\Big)-g\Big(\frac{1-y}{1-xy}\Big)=\alpha(x)-\alpha(y), \quad x,y \in (0,1), \] with unknown functions \(g\) and \(\alpha\). The author proves the following theorem concerning a more general functional equation: Theorem: Let \(g_1\), \(g_2\), \(\alpha_1\) and \(\alpha_2\) be locally integrable real functions defined on \((0,1)\) satisfying the equation \[ g_1\Big(\frac{1-x}{1-xy}\Big)+g_2\Big(\frac{1-y}{1-xy}\Big)=\alpha_1(x)+ \alpha_2(y), \quad x,y \in (0,1). \] Then there exist real numbers \(A, B, C, D, E, F, G\) and \(H\), \(A+B+C+D=0\), \(E+F=G+H\), such that for all \(x \in (0,1)\) \[ \begin{aligned} g_1(x)&=A\log(x)+B\log(1-x)+E\\ g_2(x)&=C\log(x)+D\log(1-x)+F\\ \alpha_1(x)&=B\log(x)+(A+D)\log(1-x)+G\\ \alpha_2(x)&=D\log(x)+(B+C)\log(1-x)+H\\ \end{aligned} \] In the proof it is shown that the four unknown functions are actually \(C^1\). Hence, via differentiation, one arrives to other functional equations which are then solved.