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.