A new generalization of Wilson’s functional equation

Abstract

Throughout the paper we work in the following framework and with the following notation and terminology. Let G be a group and S be a semigroup (a set with an associative composition rule). Let σ : G → G be an involutive automorphism, that is an homomorphism such that σ • σ = id. If (G; +) is an abelian group, then the inversion σ(x) := −x is an example of an involutive automorphism. Another example is the complex conjugation map on the multiplicative group of non-zero complex numbers. A function A : G → C is called additive, if it satisfies A(xy) = A(x) + A(y) for all x, y ∈ G. A multiplicative function on S is a map µ : S → C such that µ(xy) = µ(x)µ(y) for all x, y ∈ S. A character on a group G is a homomorphism from G into the multiplicative group of non-zero complex numbers. If S is a topological space, then we let C(S) denote the algebra of continuous functions from S into C.