Exponential polynomials and the sine addition law on magmas

Abstract

AbstractFor any set X we let $${\mathcal {F}}(X)$$ F ( X ) denote the complex vector space of functions $$f: X \rightarrow \mathbb {C}$$ f : X → C . Let $$X = S$$ X = S be a magma, and let V be a subspace of $${\mathcal {F}}(S)$$ F ( S ) , which is invariant under left or right translations. It is known for an abelian group S that if $$p_1\chi _1, \dots , p_n\chi _n \in {\mathcal {F}}(S)$$ p 1 χ 1 , ⋯ , p n χ n ∈ F ( S ) are nonzero exponential polynomials with distinct exponentials $$\chi _1, \dots , \chi _n$$ χ 1 , ⋯ , χ n then $$p_1\chi _1+ \dots + p_n\chi _n \in V$$ p 1 χ 1 + ⋯ + p n χ n ∈ V $$\Rightarrow $$ ⇒ $$p_1\chi _1, \dots , p_n\chi _n \in V$$ p 1 χ 1 , ⋯ , p n χ n ∈ V and $$\chi _1, \dots , \chi _n \in V$$ χ 1 , ⋯ , χ n ∈ V . We extend this to magmas. Our results imply that any exponential polynomial solution $$f \in {\mathcal {F}}(S)$$ f ∈ F ( S ) of $$f(xy) = f(x)\chi (y) + \chi (x)f(y)$$ f ( x y ) = f ( x ) χ ( y ) + χ ( x ) f ( y ) where $$\chi \in {\mathcal {F}}(S)$$ χ ∈ F ( S ) is an exponential, has the form $$f = a\chi $$ f = a χ where $$a \in {\mathcal {F}}(S)$$ a ∈ F ( S ) is additive, even when $$\chi $$ χ has zeros.