Extensions of the Sine Addition Formula on Monoids

Abstract

It is known that a pair (f, g) of functions with $$f\ne 0$$ satisfies the sine addition formula $$f(xy)=f(x)g(y)+g(x)f(y)$$ on a semigroup only if $$g = (\mu _1 + \mu _2)/2$$ where $$\mu _1$$ and $$\mu _2$$ are multiplicative functions. Here we solve the variant $$f(xy)=g_1(x)h_1(y)+g(x)h_2(y)$$ for four unknown functions $$f, g_1, h_1, h_2$$ on a monoid, where g is not simply the average of two multiplicative functions but more generally a linear combination of $$n\ge 2$$ distinct multiplicative functions.