A variant of Wilson's functional equation (Q2834191)

Let \(G\) be a group and \(\sigma\) an automorphism of \(G\) such that \(\sigma(\sigma(x)) = x\) for all \(x \in G\). Let \(\mathbb{C}^*\) denote the multiplicative group of non-zero complex numbers. The authors' main result is that any pair of functions \(f,g:G \to \mathbb{C}\), \(f \neq 0\), that satisfies the following version of Wilson's functional equation NEWLINE\[NEWLINE f(xy) + f(\sigma(y)x) = 2f(x)g(y) \text{ for all }x,y \in G,\tag{1} NEWLINE\]NEWLINE has the form: There exists a homomorphism \(\chi:G \to \mathbb{C}^*\), such that \(g = (\chi + \chi \circ \sigma)/2\). If \(\chi \neq \chi \circ \sigma\), then \(f = \alpha \chi + \beta\chi \circ \sigma\) for some \(\alpha, \beta \in \mathbb{C}\), and if \(\chi = \chi \circ \sigma\), then \(f = (\alpha +a)\chi\) for some \(\alpha \in \mathbb{C}\) and some additive function \(a:G \to \mathbb{C}\) such that \(a \circ \sigma = - a\). Conversely, these formulas for \(f\) and \(g\) define solutions of (\(1\)). The authors also discuss the more complicated instances in which the right hand side of (\(1\)) is replaced by \(2g(x)h(y)\) and in which \(G\) is replaced by a monoid. The paper generalizes earlier works on (\(1\)): The case of \(G\) abelian from [the reviewer, Aequationes Math. 54, No. 1--2, 144--172 (1997; Zbl 0899.39007)], and the case of \(f= g\) in [the reviewer, ibid. 89, No. 3, 657--662 (2015; Zbl 1317.39036)].