A new generalization of Wilson’s functional equation

Let G be a group, let σ : G → G be an involutive automorphism and let χ1, χ2 : G → C∗ be two characters of G such that χ2(xσ(x)) = 1 for all x ∈ G. The aim of this paper is to describe the solutions f, g : G → C of the functional equation χ1(y)f (xy) + χ2(y)f (σ(y)x) = 2f (x)g(y), x,y ∈ G, in terms of characters and additive functions.