Extensions of the sine addition law on groups (Q1736320)

The purpose of the present paper is to solve the functional equation \[ f(xy) = g_1(x)h_1(y) + \mu(x)h_2(y), \qquad x,y \in G, \tag{1} \] where \(G\) is a group (not necessarily abelian), and \(\mu : G \to \mathbb{C}\) is a prescribed character, while \(f, g_1, h_1, h_2 : G \to \mathbb{C}\) are the unknown functions to be determined. The main task is a discussion of the particular case \[ f(xy) = f(x)\chi(y) + \mu(x)f(y), \qquad x,y \in G, \tag{2} \] of (1), in which both \(\chi\) and \(\mu\) are given characters and \(f : G \to \mathbb{C}\) is the unknown function. The results obtained for Equation (2) enable the authors to solve Equation (1).