On a functional equation related to the Matsumoto-Yor property
Starting from a map discovered by \textit{H. Matsumoto} and \textit{M. Yor} [Nagoya Math. J. 162, 65-86 (2001; Zbl 0983.60075)] which ``preserves a bivariate probability measure which is product of the generalized inverse Gaussian and the gamma distribution'', the author was faced up with an ``intriguing'' functional equation \[ g\bigl(x(x+y) \bigr)-g\bigl(y(x+y) \bigr)= \alpha(x)- \alpha(y),\;\forall x,y\in (0,\infty) \] with unknown functions \(g\) and \(\alpha\). In the present paper he studies a more general version of this equation and proves the following theorem: Let \(g_1,g_2\) and \(\alpha_1, \alpha_2\) be locally integrable real functions defined on \((0,\infty)\) satisfying the equation \[ g_1\bigl(x(x+y) \bigr)+g_2 \bigl(y(x+y) \bigr)= \alpha_1 (x)+ \alpha_2(y),\;\forall x,y\in (0,\infty). \] Then there exist real numbers \(A,B,C\) and \(D\) such that \[ g_1(x)=Ax+B \log(x) +C=-g_2(x), \] \[ \alpha_1 (x)=Ax^2+B \log(x)+D=-\alpha_2 (x)\;\forall x\in (0,\infty). \]
scientific article; zbMATH DE number 1761057