A Pexiderized Cosine Functional Equation

Abstract Let S be a semigroup and K be a field. In a recent article we introduced a new cosine functional equation g ( xyz ) − g ( x ) g ( yz ) − g ( y ) g ( xz ) − g ( z ) g ( xy ) + 2 g ( x ) g ( y ) g ( z ) = 0 for an unknown function g : S → K . It was shown that this equation is closely connected to the sine addition formula, and for K = ℂ its solutions are expressible in terms of multiplicative functions. Here we solve the more general functional equation f ( xyz )+ g ( x ) g ( yz )+ g ( y ) g ( xz )+ g ( z ) g ( xy ) + h ( x ) h ( y ) h ( z ) = 0 for three unknown functions f, g, h : S → ℂ, where S is a monoid. The solutions are linear combinations of two multiplicative functions.