On quadratic differences that depend on the product of arguments - revisited

Summary.

Let $\mathbb{K}$ be a field of real or complex numbers and $\mathbb{K}_0$ denote the set of nonzero elements of $\mathbb{K}$. Let $\mathbb{G}$ be an abelian group. In this paper, we solve the functional equation f ₁ (x + y) + f ₂ (x - y) = f ₃ (x) + f ₄ (y) + g(xy) by modifying the domain of the unknown functions f ₃, f ₄, and g from $\mathbb{K}$ to $\mathbb{K}_0$ and using a method different from [3]. Using this result, we determine all functions f defined on $\mathbb{K}_0$ and taking values on $\mathbb{G}$ such that the difference f(x + y) + f (x - y) - 2 f(x) - 2 f(y) depends only on the product xy for all x and y in $\mathbb{K}_0$