Functional equations in the theory of conditionally specified distributions (Q2714387)

Let \(G_1,G_2,F_1,F_2 : X \to \mathbb R\) be real-valued functions defined on the set \(X\), where \(X = \mathbb R\) (the set of reals) or \(X = \mathbb R_+\). In this paper, the author studies the functional equation NEWLINE\[NEWLINEG_1 ( xy+x) + F_1 (y) = G_2 (xy +y) + F_2 (x) \tag{FE}NEWLINE\]NEWLINE for all \(x, y \in X\). In the case \(X =\mathbb R\), the author proves that all general solutions of (FE) are of the form NEWLINE\[NEWLINEF_i (x) = A(x) + b_i \qquad \text{and} \qquad G_i (x) = A(x) + c_i \quad (i=1,2) NEWLINE\]NEWLINE where \(A : \mathbb R \to \mathbb R\) is an additive function and \(b_i , c_i \) are real arbitrary constants satisfying \(b_1 + c_1 =b_2 + c_2\). The proof is elementary but nontrivial. When \(X =\mathbb R_+\), the author uses a result of \textit{A. Járai} [Publ. Math. Debrecen 26, 17-35 (1979; Zbl 0433.39012)] to show that all measurable solutions of (FE) are of the form NEWLINE\[NEWLINEF_i (x) = C \ln {x \over {x+1}} + \gamma x + b_i\qquad\text{and} \qquad G_i (x) = C \ln x + \gamma x + c_i \quad (i=1,2) NEWLINE\]NEWLINE where \(C, \gamma , b_i , c_i\) are real arbitrary constants satisfying \(b_1 + c_1 = b_2 + c_2\). The functional equation (FE) arises in the characterization problems of joint distributions from conditional distributions.