A New Variant of Wilson’s Functional Equation on Monoids

We find on a monoid M the complex-valued solutions f, g : M → ℂ such that f is central and g is continuous of the functional equation f ( x σ ( y ) ) + f ( τ ( y ) x ) = 2 f ( x ) g ( y ) , x , y ∈ M , where σ : M → M is an involutive automorphism and τ : M → M is an involutive anti-automorphism. The solutions are described in terms of multiplicative functions, additive functions and characters of 2-dimensional representations of M .