On a variant of Wilson’s functional equation on groups

Summary.

We study the solutions $f,g:G \to \mathbb{C}$ of the functional equation

$$ f(xy) + f(y^{ - 1} x) = 2f(x)g(y),\quad x,y \in G, $$

where G is a group. We prove that if G is a connected Lie group (or more generally is generated by its squares), and f ≠ 0, then g satisfies d’Alembert’s equation, and f is either proportional to g, or it satisfies Kannappan’s condition.