Functional equations on abelian groups with involution, II

Abstract.

We find all solutions f, g, h ₁ and h ₂ of the quadratic-trigonometric functional equation $ f (x + y) + f (x + \sigma y) = 2 f (x) + h_1 (y) + g (x) h_2 (y), \qquad x, y \in G, $ and of certain cases of it, where $\sigma : G \to G$ is an automorphism of the abelian group G such that $\sigma ^2 = I$, by reducing it to the basic functional equations: d'Alembert's equation, Wilson's equation, the quadratic equation and the equation of symmetric second differences.