On the cosine-sine functional equation on groups

Summary.

We find the solutions f, g, h ∈ C(G) of each of the functional equations

$$\frac{{f(x + y) \pm f(x + \sigma y)}}{2} = f(x)g(y) + h(x)h(y),\forall x,y \in G,$$

((1))

where G is a topological, abelian group, σ : G → G is a continuous involutive automorphism of G, and where C(G) denotes the algebra of continuous, complex-valued functions on G. Our results generalize and extend the ones by Chung, Kannappan, and Ng in A generalization of the Cosine-Sine Functional Equation on Groups, Linear Algebra Appl. 66 (1985), 259-277.