Exponential polynomials and the sine addition law on magmas

Abstract

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

1 Introduction

The theory of exponential polynomials has recently attracted some interest, part of which focuses on such functions on semigroups rather than on groups. With S denoting a semigroup a question is whether translation invariant, finite dimensional subspaces of functions on S consist of exponential polynomials. Ebanks and Ng [5, 6] outline the state of affairs and provide motivation. Furthermore they extend part of the classic theory of exponential polynomials from abelian groups as found in the monograph Székelyhidi [10], in particular [10, Theorem 10.1], to commutative monoids with no prime ideals [5, Theorem 1] and prove the answer is positive. Furthermore they show by a study of an important particular case of the sine addition law that the results of the classic theory can fail for commutative monoids with prime ideals. The particular functional equation is the simple Levi–Civita functional equation (later in the paper labelled (5.2))

$$\begin{aligned} f(xy) = f(x)\chi (y) + \chi (x)f(y), \ x,y \in S, \end{aligned}$$

(1.1)

where $f:S \rightarrow \mathbb {C}$ is the unknown function and $\chi :S \rightarrow \mathbb {C}$ is multiplicative, and the invariant, finite dimensional subspace is $\text {span}\{ f,\chi \}$.

In the present paper we generalize known results about exponential polynomials and translation invariant subspaces of ${\mathcal {F}}(S)$ from a discrete abelian group (Laczkovich [8, Lemma 6]) to a magma S that need not possess an identity (Theorems 4.3 and 4.4). As an application we show that the exponential polynomial solutions f of the functional equation (1.1) are the functions $f = a\chi$, where $a:S \rightarrow \mathbb {C}$ is additive (Theorem 5.6). So the set of exponential polynomial solutions is quite simple, although the set of all solutions of (1.1) is complicated in general (for the current knowledge see the recent paper Ebanks [4]). We apply our results to examples on semigroups, all involving the functional equation (1.1).

In contrast to Ebanks and Ng [5, 6] our objective is not to prove or disprove that finite dimensional varieties consist of exponential polynomials, but to derive and apply general properties of exponential polynomials on semigroups or magmas.

The organization of the paper is as follows. We begin with terminology and notation (Sect. 2) and preliminary discussions of additive and multiplicative functions (Sect. 3). Section 4 derives relations between exponential polynomials and translation invariant spaces of functions on magmas. Section 5 presents the exponential polynomial solutions of (1.1). Finally section 6 contains some illustrative examples.

2 Terminology and notation

This section introduces basic terminology and notation that we shall use in the rest of the paper.

Magmas (see Definition 2.1) occur in the standard literature for functional equations as for example Aczél and Dhombres [2], so we formulate our results for magmas, although the generality of semigroups suffices for our examples. Earlier literature like [2] used the word groupoid for a magma. We shall use the word magma as done in recent literature like [6].

Definition 2.1

A magma is a set S endowed with a map $S \times S \rightarrow S$ called the composition. We write the composition as $(x, y) \mapsto xy$.

Definition 2.2

Let S be a magma.

An element $e \in S$ is called an identity if $e x = x e = x$ for all $x \in S$. S has at most one identity.

An element $z_0 \in S$ of S is called a zero if $z_0x = xz_0 = z_0$ for all $x \in S$. A magma has at most one zero element. A zero is often denoted 0.

For $x \in S$ and $n \in \{1,2 \dots \}$ we shall encounter the expression $x^n$. We interpret it as $x^n:= x \cdot (x \cdot (\cdots (x \cdot (x \cdot x))))$ (n factors).

We say S is a topological magma, if S is endowed with a topology such that the composition $(x,y) \mapsto xy$ is continuous from $S \times S$ into S, when $S \times S$ is equipped with the product topology.

We recall that a semigroup is a magma such that its composition is associative. A monoid is a semigroup which has an identity.

$1_X$ denotes the indicator function of the set X.

All our vector spaces are over $\mathbb {C}$.

For any set X we let ${\mathcal {F}}(X)$ denote the vector space of all complex valued functions on X, and for any topological space X we let C(X) denote the subspace of ${\mathcal {F}}(X)$ consisting of its continuous functions.

$\mathbb {N}:= \{ 1,2, \dots \}$.

For any multi-index $\alpha = (\alpha _1, \dots , \alpha _n) \in (\mathbb {N} \cup \{ 0\}) \times \cdots \times (\mathbb {N} \cup \{ 0\})$ and any $z = (z_1, \dots ,z_n) \in \mathbb {C}^n$ we use the abbreviations $z^{\alpha }:= z_1^{\alpha _1} \cdots z_n^{\alpha _n}$ and $|\alpha |:= \alpha _1 + \dots + \alpha _n$. As is the convention we let $w^0 = 1$ for all $w \in \mathbb {C}$, even for $w = 0$.

3 On additive and multiplicative functions

Section 3 derives some properties of multiplicative functions on a magma. Its main result, Theorem 3.6, extends Artin’s theorem (= Corollary 3.7 for groups.)

Definition 3.1

An additive function on a magma S is a function $a:S \rightarrow \mathbb {C}$ such that $a(xy) = a(x) + a(y)$ for all $x,y \in S$.

Definition 3.2

A multiplicative function on a magma S is a function $\chi :S \rightarrow \mathbb {C}$ such that $\chi (xy) = \chi (x)\chi (y)$ for all $x,y \in S$. An exponential on S is a multiplicative function $\chi$ such that $\chi (x_0) \ne 0$ for some $x_0 \in S$.

On any magma $a = 0$ is an additive function and $\chi = 1$ an exponential.

The term multiplicative function is also used in number theory. Our multiplicative functions on $(\mathbb {N}, \cdot )$ are there said to be completely multiplicative on $\mathbb {N}$, so the concepts differ.

Proposition 3.3

An exponential on a group G vanishes nowhere, so the exponentials of G are its characters, i.e., the homomorphisms of G into the multiplicative group of non-zero complex numbers.

Proof

The proposition is noted in many places, e.g., [9, Lemma 3.4]. $\square$

Examples 3.4

In contrast to the situation on a group exponentials may vanish at some points of a magma, even on a monoid. Two examples are the function $\chi := 1_{\{ 0\}}$ on the monoid $([0,\infty [, +)$ and the function $\chi (x):= x$ on the monoid $(\mathbb {R}, \cdot )$.

Example 3.5

There are other semigroups than groups with the property that their exponentials vanish nowhere. The semigroup $(\,]0,1[, \cdot )$ is an example. Here is a proof. Let $\chi$ be an exponential on $(\,]0,1[, \cdot )$, and suppose for contradiction that $\chi (x_0) = 0$ for some $x_0 \in \, ]0,1[$. Let $x \in \, ]0,1[$ be arbitrary. Choosing $k \in \mathbb {N}$ so large that $x^k < x_0$ we find that

$$\begin{aligned} \chi (x)^k = \chi (x^k) = \chi \left( \frac{x^k}{x_0}x_0\right) = \chi \left( \frac{x^k}{x_0}\right) \chi (x_0) = \chi \left( \frac{x^k}{x_0}\right) 0 = 0. \end{aligned}$$

Thus $\chi (x) = 0$ for all $x \in \,]0,1[$, contradicting $\chi$ is an exponential.

We finish sect. 3 with some facts around the linear independence of the set of exponentials. A result due to E. Artin (in his famous Notre Dame lectures [3, Theorem II.12, pp. 34–35]) says that on any group the set of characters is linearly independent. The same holds for semigroups (Stetkær [9, Theorem 3.18(b)]) and even magmas (Ebanks and Ng [6, Lemma 2.2]), the latter being Corollary 3.7 below.

Theorem 3.6 and its Corollary 3.7 are not the final words. Theorem 4.3 and Corollary 4.6 generalize them from multiplicative functions to exponential polynomials.

Theorem 3.6

Let S be a magma, let V be a left or right translation invariant subspace of ${\mathcal {F}}(S)$ and let $n \in \mathbb {N}$. Let $m_1, \dots ,m_n \in {\mathcal {F}}(S)$ be different multiplicative functions on S and let $c_1, \dots , c_n \in \mathbb {C}$.

If $c_1m_1 + \cdots + c_nm_n \in V$, then each term $c_1m_1, \dots , c_nm_n \in V$.

Proof

We skip the proof (induction on n), since it is the same as the proofs of Artin’s theorem in [9, Theorem 3.18(a)] and [6, Lemma 2.2]. $\square$

Corollary 3.7 arises when you take $V = \{ 0 \}$ in Theorem 3.6. Corollary 4.6(b) is a natural extension of it to exponential polynomials.

Corollary 3.7

The set of exponentials of a magma S is a linearly independent subset of ${\mathcal {F}}(S)$.

4 Some properties of exponential polynomials

In this section we introduce exponential polynomials on topological magmas and derive some of the properties of these functions. The main results are Theorems 4.3 and 4.4. They extend Laczkovich [8, Lemma 6] who deals with discrete abelian groups.

Definition 4.1

Let S be a topological magma. A polynomial on S is a complex valued function on S of the form $P(a_1, \dots , a_k)$ where $P \in \mathbb {C}[x_1, \dots , x_k]$ and where $a_1, \dots a_k \in C(S)$ are additive functions on S. Note that polynomials are continuous by their very definition.

An exponential polynomial is a function of the form $\sum _{i=1}^n p_im_i$ where $n \in \mathbb {N}$, $p_1, \dots , p_n$ are polynomials on S and $m_1, \dots , m_n$ are continuous multiplicative functions on S. Exponential polynomials are continuous.

Székelyhidi [10, p. 28] uses, for commutative topological semigroups, the term normal polynomial for what we here call polynomial.

As mentioned above, in contrast to the situation for groups exponentials on semigroups can vanish at some points (Lemma 3.3 versus Examples 3.4). Thus we can not divide by exponentials on general semigroups as we can and do on groups. This means complications will occur when we pass from groups to semigroups. Example 4.2 singles out one of these complications.

Example 4.2

On the monoid $S:= ([0, \infty [, +)$ with the discrete topology $m:= 1_{\{ 0\}}$ is an exponential, while $p(x):= x$, $x \in [0, \infty [$, is a nonzero additive function. Obviously $pm = 0$. So the exponential polynomial pm vanishes everywhere on S, even though each of its factors is $\ne 0$. This explains why we in the formulation of some results must impose the condition $pm \ne 0$ instead of $p \ne 0$ and $m \ne 0$ (for instance Theorem 4.4 and Corollary 4.6(b)). On a group $p \ne 0$ would suffice by Proposition 3.3.

In Theorems 4.3 and 4.4 we derive connections between exponential polynomials and translation invariant subspaces $V \subseteq {\mathcal {F}}(S)$ on a magma S. Particularly interesting subspaces are $V = \{ 0\}$ which renders results on linear independence (Corollary 4.6), and $V = C(S)$ that implies the continuity of discrete exponential polynomials on topological magmas (Corollary 4.7).

Theorem 4.3 is an extension of Proposition 3.6 from multiplicative functions to exponential polynomials on magmas. It and Theorem 4.4 occur in Laczkovich [8, Lemma 6] for discrete abelian groups. Actually [8, Lemma 6] deals with generalized polynomials, not just polynomials. Our proof of Theorem 4.3 follows Laczkovich’s approach closely, but has more details, since division by exponentials is prohibited.

Theorem 4.3

Let S be a magma with the discrete topology, and let V be a left or right translation invariant subspace of ${\mathcal {F}}(S)$. Let finally $n \in \mathbb {N}$.

If $p_1, \dots , p_n: S \rightarrow \mathbb {C}$ are polynomials and $m_1, \dots m_n: S \rightarrow \mathbb {C}$ are different multiplicative functions, then

$$\begin{aligned} \sum _{i=1}^n p_im_i \in V \ \Rightarrow \ p_1m_1, \dots , p_{n}m_n \in V. \end{aligned}$$

Proof

We assume that V is right translation invariant. The case of a left invariant subspace can be treated analogously, so we leave it out.

In the proof we use the notion of the degree of a polynomial $p:S \rightarrow \mathbb {C}$ on S that we define as follows. Being a polynomial we can write $p = P(a_1, \dots ,a_k)$ for some $P \in \mathbb {C}[x_1, \dots ,x_k]$, some $a_1, \dots , a_k \in C(S)$ that are additive and some $k \in \mathbb {N}$. We let $\deg p$ be the minimal degree of the polynomials P occurring in such expressions for p.

We proceed by induction on n. The case of $n=1$ is a truism. Let $n \ge 2$ be fixed and assume that the theorem holds for all sums of $< n$ terms. We shall prove that the theorem holds for all sums of n terms, so let $f:= \sum _{i=1}^n p_i m_i \in V$, where $p_1, \dots , p_n$ are polynomials and $m_1, \dots m_n$ are different multiplicative functions. In particular $m_1 \ne m_n$. We shall derive that $p_i m_i \in V$ for all $i = 1, \dots , n$. If $p_im_i = 0$ for some $i \in \{ 1, \dots ,n\}$, then the induction hypothesis ensures the conclusion. So we may suppose that $p_im_i \ne 0$ for all $i \in \{ 1, \dots ,n\}$, which in particular implies that the polynomials $p_1, \dots , p_n$ are nonzero, and so that $\deg {p_i} \ge 0$ (we avoid the zero polynomial, which is assigned the degree $-\infty$). By the induction hypothesis it suffices to prove that $p_nm_n \in V$.

We show that $p_nm_n \in V$ by induction on $k:=\sum _{i=1}^n \deg {p_i}$. It is true at the induction start $k = 0$ by Theorem 3.6, because in this case each $p_i$ is a nonzero constant. Assuming that $p_nm_n \in V$ if $\sum _{i=1}^n \deg {p_i} = 0, 1, \dots , k-1$ where $k \ge 1$ is an integer, it suffices to deduce that $p_nm_n \in V$ when $\sum _{i=1}^n \deg {p_i} = k$. So let $\sum _{i=1}^n \deg {p_i} = k$.

For use at the end of the proof we fix $y_0\in S$ such that $m_n(y_0) \ne m_1(y_0)$. We can do so, because $n \ge 2$ and $m_n \ne m_1$.

We find for any $x,y \in S$ that

$$\begin{aligned} f(xy) - m_1(y)f(x)&= m_1(y)[p_1(xy) - p_1(x)]m_1(x) \nonumber \\&\quad + \sum _{i=2}^n[m_i(y)p_i(xy) - m_1(y)p_i(x)]m_i(x). \end{aligned}$$

(4.1)

By assumption $f \in V$. Now V is right translation invariant, so as a function of $x \in S$ the left hand side of (4.1) belongs to V. Taking $y = y_0$ in (4.1) its right hand side is of the form $\sum _{i=1}^n q_im_i$, where

$$\begin{aligned} q_i(x):= m_i(y_0)p_i(xy_0) - m_1(y_0)p_i(x) \end{aligned}$$

(4.2)

for $i = 1, \dots ,n$. The special case $q_1$ simplifies to

$$\begin{aligned} q_1(x):= m_1(y_0)[p_1(xy_0) - p_1(x)]. \end{aligned}$$

(4.3)

Note that $q_1, \dots ,q_n$ are polynomials as functions of $x \in S$.

We see that $\deg {q_i} \le \deg {p_i}$ from the definition of $q_i$, which implies that $\sum _{i=1}^n \deg {q_i} \le \sum _{i=1}^n \deg {p_i} = k$.

Claim:

$$\begin{aligned} q_im_i \in V \text { for } i = 1, \dots , n. \end{aligned}$$

(4.4)

Proof

(of (4.4))

• (A) If at least one $q_im_i = 0$ we may write $\sum _{i=1}^nq_im_i = \sum _{j=1}^N q_{i_j}m_{i_j}$, where $i_{j_1}, \dots , i_{j_N}$ are the indices for which $q_im_i \ne 0$, and $N < n$. From the induction hypothesis we read that $q_{i_1}m_{i_1}, \dots , q_{i_N}m_{i_N} \in V$. The remaining terms $q_im_i$ also belong to V (they are 0), so all the terms $q_im_i$ belong to V. Thus the claim is true in case (A). Due to (A) we may and shall during the rest of the proof of the claim assume that $q_i \ne 0$ for all $i = 1, \dots , n$. We see that there are three possibilities left.

• (B) $m_1(y_0) = 0$. Here we see from the definition of $q_1$ in (4.3) that $q_1 = 0$ which is excluded in this part of the proof, being contained in (A).

• (C) $m_1(y_0) \ne 0$ and $\deg {p_1} = 0$. Here $p_1$ is a constant. By the definition of $q_1$ in (4.3) this implies that $q_1 = 0$ which is excluded in this part of the proof, being contained in (A).

• (D) $m_1(y_0) \ne 0$ and $\deg {p_1}\ge 1$. Since $\deg {p_1} \ge 1$ we have $\deg {q_1} = \deg (x \mapsto p_1(xy_0) - p_1(x)) < \deg {p_1}$, so $\sum _{i=1}^n \deg {q_i} < \sum _{i=1}^n \deg {p_i} = k$. By the induction assumption for k we obtain that $q_im_i \in V$ for $i = 1, \dots , n$, finishing the proof of the claim (4.4).$\square$

We get from (4.4) that $x \mapsto (q_1m_1)(x) = m_1(y_0)[p_1(xy_0) - p_1(x)]m_1(x)$ belongs to V. Similar considerations (again starting with (4.1) but with another index i than 1) reveal that the index 1 is not special and that it can be replaced by i in the formula just derived so that we in particular (for $i = n$) have that

$$\begin{aligned} (x \mapsto m_n(y_0)[p_n(xy_0) - p_n(x)]\,m_n(x)) \in V. \end{aligned}$$

(4.5)

When we rewrite the definition (4.2) of $q_n$ to

$$\begin{aligned} q_n(x)&= m_n(y_0)p_n(xy_0) - m_1(y_0)p_n(x)\\&= m_n(y_0)[p_n(xy_0) - p_n(x)] + [ m_n(y_0) - m_1(y_0)] p_n(x) \end{aligned}$$

and multiply the resulting formula by $m_n(x)$ we get that

$$\begin{aligned}&q_n(x)m_n(x)\\&\quad = m_n(y_0) [p_n(xy_0)- p_n(x)]m_n(x) + [m_n(y_0) - m_1(y_0)] p_n(x)m_n(x). \end{aligned}$$

The left hand side belongs to V as a function of $x \in S$ by (4.4). So does the first term on the right hand side according to (4.5). We conclude that $[m_n(y_0) - m_1(y_0)] p_n m_n \in V$. But $m_n(y_0) \ne m_1(y_0)$, so $p_nm_n \in V$. $\square$

Laczkovich [8, Lemma 6] contains the following Theorem 4.4 for discrete abelian groups, but his proof does not to extend to magmas.

Theorem 4.4

Let S be a magma with the discrete topology and V a left or right translation invariant subspace of ${\mathcal {F}}(S)$. Let $p: S \rightarrow \mathbb {C}$ be a polynomial and $m: S \rightarrow \mathbb {C}$ a multiplicative function such that $pm \ne 0$.

If $pm \in V$, then $m \in V$.

Proof

Let V be right translation invariant. The case of a left invariant subspace can be treated analogously, so we leave it out.

We write the polynomial p in the form

$$\begin{aligned} p(x):= \sum _{\vert k|\le N}c_k \alpha _1(x)^{k_1} \cdots \alpha _n(x)^{k_n} = \sum _{\vert k|\le N}c_k \alpha (x)^{k}, \ x \in S, \end{aligned}$$

with $\alpha = (\alpha _1, \dots , \alpha _n)$ where $\alpha _1, \dots , \alpha _n:S \rightarrow \mathbb {C}$ are additive functions, and $N \in \{ 0,1, \dots \}$. Perhaps $\sum _{\vert k|= N}c_k \alpha ^{k}m = 0$. However, since $pm \ne 0$, at least one term of the decomposition

$$\begin{aligned} pm = \sum _{\vert k|= N}c_k \alpha ^{k} m + \sum _{\vert k|= N-1}c_k \alpha ^{k} m + \cdots + \sum _{\vert k|= 0}c_k \alpha ^{k} m \end{aligned}$$

must be non-zero. Let $N_0$ be the largest index such that $\sum _{\vert k|= N_0}c_k \alpha ^{k}m \ne 0$. Replace N by $N_0$. Thus, possibly replacing p by a polynomial of lower degree than N, we may assume that there exists an $x_0 \in S$ such that $\sum _{\vert k|= N}c_k \alpha (x_0)^{k} m(x_0) \ne 0$. Thus $x_0 \in S$ satisfies that

$$\begin{aligned} m(x_0)\ne 0 \ \text { and } \ \sum _{\vert k|= N}c_k \alpha (x_0)^{k} \ne 0. \end{aligned}$$

(4.6)

Let $l \in \mathbb {N}$ be arbitrary. Since pm belongs to the right translation invariant subspace V the translated function

$$\begin{aligned} x \mapsto&(pm)(x x_0^l) = \sum _{\vert k|\le N}c_k \alpha _1(xx_0^l)^{k_1} \cdots \alpha _n(xx_0^l)^{k_n}m(x)m(x_0)^l \end{aligned}$$

also belongs to V. Since $m(x_0) \ne 0$, we may divide the function by the constant factor $m(x_0)^l$ and still have a function in V. The new function is

$$\begin{aligned} x&\mapsto \sum _{\vert k|\le N}c_k \alpha (xx_0^l)^{k} m(x) = \sum _{\vert k|\le N}c_k [\alpha (x) + l\alpha (x_0)]^{k} m(x)\\&\quad \quad = l^N \sum _{\vert k|= N}c_k \alpha (x_0)^{k} m(x) + \sum _{j=1}^{N-1}l^jw_j(x) + w_0(x), \end{aligned}$$

for some functions $w_0, \dots ,w_{N-1} \in {\mathcal {F}}(S)$. We infer from the following auxiliary Lemma 4.5 that the coefficient of $l^N$ belongs to V, i.e., that

$$\begin{aligned} \sum _{\vert k|= N}c_k \alpha (x_0)^{k} m \in V. \end{aligned}$$

The coefficient $\sum _{\vert k|= N}c_k \alpha (x_0)^{k}$ of m in this expression is $\ne 0$ by (4.6), so $m \in V$ as desired. $\square$

Lemma 4.5

Let V be a subspace of a vector space W. Let $w_0, w_1, \dots , w_N \in W$ for some $N \in \{ 0,1, \dots \}$. If $w_0 + lw_1 + l^2w_2 \cdots + l^N w_N \in V$ for all $l \in \mathbb {N}$, then $w_0, w_1, \dots , w_N \in V$.

Proof

From the assumption we get for any $l_1, \dots ,l_{N+1} \in \mathbb {N}$ that

$$\begin{aligned} \begin{pmatrix} 1 &{} l_1 &{} l_1^2 &{} \dots &{} l_1^N \\ 1 &{} l_2 &{} l_2^2 &{} \dots &{} l_2^N \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 1 &{} l_{N+1} &{} l_{N+1}^2 &{} \dots &{} l_{N+1}^N \end{pmatrix} \begin{pmatrix} w_0\\ w_1\\ \vdots \\ w_N \end{pmatrix} \in V^{N+1}. \end{aligned}$$

On the left we recognize Vandermonde’s matrix, which is known to be invertible if $l_j \ne l_k$ for all $j \ne k$. Taking $l_j = j$ for $j = 1, \dots , l_{N+1}$ we obtain the lemma. $\square$

Corollary 4.6(b) is a natural extension of Artin’s result (Corollary 3.7).

Corollary 4.6

Let S be a magma with the discrete topology. Let $n \in \mathbb {N}$. Let $m_1, \dots , m_n: S \rightarrow \mathbb {C}$ be distinct multiplicative functions and $p_1, \dots , p_n: S \rightarrow \mathbb {C}$ polynomials on S.

• (a) The vectors $p_1m_1, \dots , p_nm_n$ form a direct sum in ${\mathcal {F}}(S)$ in the sense that

  $$\begin{aligned} p_1m_1 + \dots + p_nm_n = 0 \Rightarrow p_1m_1 = \dots = p_nm_n = 0. \end{aligned}$$

• (b) $p_1m_1, \dots , p_nm_n$ are linearly independent, if $p_im_i \ne 0$ for $i = 1, \dots , n$.

Proof

(a) Take $V = \{ 0\}$ in Theorem 4.3. (b) follows from (a). $\square$

Corollary 4.7

Let S be a magma. Let $n \in \mathbb {N}$. Let $m_1, \dots , m_n: S \rightarrow \mathbb {C}$ be distinct multiplicative functions and $p_1, \dots , p_n: S \rightarrow \mathbb {C}$ polynomials on S with respect to the discrete topology on S. If S is a topological magma, then

$$\begin{aligned} p_1m_1 + \dots + p_nm_n \in C(S) \Rightarrow p_1m_1, \dots , p_nm_n \in C(S). \end{aligned}$$

Proof

Take $V = C(S)$ in Theorem 4.3. $\square$

Corollary 4.8

Let S be a magma. Let p be a polynomial (with respect to the discrete topology on S) on S and let m be a multiplicative function, such that $pm \ne 0$. If S is a topological magma, then $pm \in C(S) \Rightarrow m \in C(S)$.

Proof

Take $V = C(S)$ in Theorem 4.4. $\square$

5 On the functional equation (5.2)

Section 5 relates exponential polynomials to solutions of the particular case (5.2) below of the sine addition law (5.1). The derivations in the literature of almost all important properties of the solutions of (5.1) require that S is a semigroup. However, for (5.2) we can work in the generality of magmas.

Definition 5.1

Let S be a magma. We say that the ordered pair (f, g), where $f, g \in {\mathcal {F}}(S)$, satisfies the sine addition law (or formula) if

$$\begin{aligned} f(xy) = f(x)g(y) + f(y)g(x) \text { for all } x,y \in S. \end{aligned}$$

(5.1)

We shall not venture into the theory of the general equation (5.1), but almost exclusively consider the particular case

$$\begin{aligned} f(xy) = f(x)\chi (y) + \chi (x)f(y), \ x,y \in S, \end{aligned}$$

(5.2)

where $f:S \rightarrow \mathbb {C}$ is the unknown and $\chi :S \rightarrow \mathbb {C}$ is multiplicative. (5.2) is the same as the functional equation (1.1) mentioned in the Introduction.

5.1 Solutions of (5.1) and (5.2)

In our opinion the uniqueness result of Lemma 5.2 is a nice new observation about (5.1). We apply it in the proof of Corollary 5.3(a).

Lemma 5.2

Let S be a magma. For any given $f: S \rightarrow \mathbb {C}$, $f \ne 0$, there exists at most one $g: S \rightarrow \mathbb {C}$ such that (f, g) satisfies (5.1).

Proof

If $g_1$ and $g_2$ were two such functions, then $f(x)g(y) = - f(y)g(x)$, where $g:= g_1 -g_2$. Since $f \ne 0$ it follows that $g = 0$. $\square$

Corollary 5.3(b) crops up a couple of times below, so it is worth recording.

Corollary 5.3

Let S be a magma. Let $c \in \mathbb {C}$, and let $\mu : S \rightarrow \mathbb {C}$ be a multiplicative function.

• (a) If $f:= c\mu$ is a nonzero solution of (5.1) for some g, then $g = \mu /2$.

• (b) If $f:= c\mu$ satisfies (5.2) in which $\chi$ is multiplicative, then $f= 0$.

Proof

(a) As is easy to check, the ordered pair $(f,\mu /2)$ is a solution of (5.1), so the result is immediate from the uniqueness of g (Lemma 5.2.)

(b) Suppose for contradiction that $f = c\mu \ne 0$. Then $c \ne 0$ and $\mu \ne 0$. Substituting $f = c\mu$ into (5.2) we get after division by c that $\mu (x)[\mu (y) - \chi (y)] = \chi (x)\mu (y)$. For $\chi = \mu$ or $\chi = 0$ this leads to the contradiction $\mu = 0$, so $\chi$ and $\mu$ are different exponentials. Corollary 3.7 and $\mu (x)[\mu (y) - \chi (y)] = \chi (x)\mu (y)$ imply that $\mu (y) = 0$ for all $y \in S$, contradicting $\mu \ne 0$. $\square$

Examples 5.4

Let S be a magma, and let $\chi \in {\mathcal {F}}(S)$ be multiplicative.

• (a) If $a \in {\mathcal {F}}(S)$ is additive, then $(f,g):= (a\chi , \chi )$ is a solution of (5.2).

• (b) If S is a topological group and $f \in C(S)$ is a solution of (5.2), then $\chi \in C(S)$ and $f = a\chi$ for an additive function $a \in C(S)$, so f is an exponential polynomial. This is well known and follows from Proposition 3.3.

• (c) If S is a topological magma, $f \in C(S)$ is a solution of (5.2) and $\chi$ vanishes nowhere, then $\chi \in C(S)$ and $f = a\chi$ for an additive function $a \in C(S)$. In particular f is an exponential polynomial. (c) is a generalization of (b).

For semigroups Lemma 5.5 is in Ebanks and Ng [5, Lemma 4].

Lemma 5.5

Let S be a magma. Let $f \in {\mathcal {F}}(S)$ be a solution of (5.2) in which $\chi \in {\mathcal {F}}(S)$ is multiplicative. Let V be the subspace of ${\mathcal {F}}(S)$ defined by $V:= \text {span}\{ f,\chi \}$. Then

• (a) V is invariant under both left and right translations.

• (b) $\chi$ and 0 are the only multiplicative functions in V.

Proof

(a) is trivial. (b) Let for contradiction $\mu$ be a multiplicative function in V such that $\mu \ne 0$ and $\mu \ne \chi$. Since $\mu \ne 0$ it is an exponential, and being in V it can be written as $\mu = \alpha f + \beta \chi$. Since f satisfies (5.2) so does $\alpha f$. Substituting $\alpha f = \mu - \beta \chi$ into (5.2) we get $\mu (x)\{ \mu (y) - \chi (y)\} = \chi (x)\{ \mu (y) - \beta \chi (y)\}$. Since $\mu \ne \chi$, this implies $\mu = c\chi$ for some constant $c \in \mathbb {C}$. Since $\mu \ne 0$ we see that $\chi \ne 0$, so that $\chi$ is an exponential as well. Thus $\mu = c\chi$ contradicts the linear independence stated in Corollary 3.7. $\square$

5.2 Exponential polynomial solutions of (5.2)

The main result of subsect. 5.2 is Theorem 5.6 which shows that exponential polynomial solutions of (5.2) are quite simple.

Theorem 5.6

Let S be a topological magma. Let $f: S \rightarrow \mathbb {C}$ be a nonzero, exponential polynomial solution of (5.2) with $\chi :S \rightarrow \mathbb {C}$ multiplicative.

Then $\chi$ is a continuous exponential, and $f = a\chi$, where a is a continuous additive function.

Our proof of Theorem 5.6 is based on the auxiliary Lemmas 5.7, 5.8 and 5.9 below, and we shall derive these lemmas first.

Lemma 5.7

Let S be a topological magma. Let $f: S \rightarrow \mathbb {C}$ be a solution of (5.2) in which $\chi :S \rightarrow \mathbb {C}$ is multiplicative. Let $f = p_1m_1 + \cdots + p_nm_n$, where $n \ge 1$, $p_1, \dots , p_n \in C(S)$ are polynomials and $m_1, \dots , m_n \in C(S)$ are different multiplicative functions such that $p_im_i \ne 0$ for $i = 1, \dots , n$.

Then $n = 1$, $m_1 = \chi$ and $f = p_1\chi$. Furthermore $\chi$ is a continuous exponential, and $f\ne 0$.

Proof

Note that the subspace $V:= \text {span}\{ f, \chi \}$ of ${\mathcal {F}}(S)$ is invariant under right translations (by Corollary 5.5(a).) Note also that $f \ne 0$ since the $p_im_i$’s form a direct sum (Corollary 4.6(1)).

Since $p_1m_1 + \cdots + p_nm_n = f \in V$ we read from Theorem 4.3 that $p_1m_1, \dots , p_nm_n \in V$, and further from Theorem 4.4 that $m_1, \dots , m_n \in V$. Lemma 5.5(b) gives $m_1= \dots = m_n = \chi$. The $m_i$’s are different, so $n = 1$ and $m_1 = \chi$. $\square$

We use Lemma 5.8 for our technical computations in the proof of Lemma 5.9. The expression $y^n$ is defined in sect. 2; what we need here is that $a(y^n) = na(y)$ for any additive function a on S and $\mu (y^n) = \mu (y)^n$ for any multiplicative function $\mu$ on S.

Lemma 5.8

Let S be a magma. Let $N \in \{ 2,3, \dots \}$. Let $a_1, a_2, \dots , a_N:S \rightarrow \mathbb {C}$ be additive functions, let $\chi :S \rightarrow \mathbb {C}$ be a multiplicative function, and put $p:= a_1a_2 \cdots a_N$. Then we get for any $n \in \mathbb {N}$ and all $x,y \in S$ that

$$\begin{aligned}&(p\chi )(xy^n) - p(x)\chi (y^n) - p(y^n)\chi (x)\\&\ = \left\{ n^{N-1}\sum _{j=1}^{N}a_1(y) a_2(y) \cdots a_j(x)\cdots a_N(y) + P(n,x,y)\right\} \chi (x)\chi (y)^n \end{aligned}$$

where P(n, x, y) is a polynomial in n of degree $< N-1$ with coefficients that are functions of x and y.

Proof

Noting that

$$\begin{aligned}&p(xy) - p(x)\chi (y) - p(y)\chi (x)\\&\quad = (a_1a_2 \cdots a_N)(xy)\chi (xy) - (a_1a_2 \cdots a_N)(x)\chi (x)\chi (y)\\&\quad \quad - (a_1a_2 \cdots a_N)(y)\chi (y)\chi (x)\\&\quad = \{(a_1(x) + a_1(y))\cdots (a_N(x) + a_N(y)) - a_1(x)\cdots a_N(x)\\&\quad \quad - a_1(y)\cdots a_N(y)\}\chi (x)\chi (y) \end{aligned}$$

we get the desired result by replacing y by $y^n$ because $a_j(y^n) = na_j(y)$ for all $j = 1, \dots , N$ due to the additivity of $a_j$. Indeed, the coefficient of $\chi (x)\chi (y)^n$ is of the form

$$\begin{aligned}&(a_1(x) + na_1(y))\cdots (a_N(x) + na_N(y))\\&\quad \quad - a_1(x)\cdots a_N(x) - n^Na_1(y)\cdots a_N(y)\\&\quad = n^{N-1}\sum _{j=1}^{N}a_1(y) a_2(y) \cdots a_j(x)\cdots a_N(y) + \sum _{j=1}^{N-2}n^j\psi _j(x,y) \end{aligned}$$

for some $\psi _1, \dots , \psi _{N-2} \in {\mathcal {F}}(S \times S, \mathbb {C})$. This implies the lemma. $\square$

Lemma 5.9

Let S be a magma, let $N \in \{ 2,3, \dots \}$ and let $M \in \mathbb {N}$. Let $a_1^{(k)}, \dots , a_N^{(k)}:S \rightarrow \mathbb {C}$ for $k = 1,2, \dots , M$ be additive functions. Let $\chi : S \rightarrow \mathbb {C}$ be a multiplicative function. Let

$$\begin{aligned} p:= \sum _{k=1}^M a_1^{(k)}\cdots a_N^{(k)} + p_{N-1} + c, \end{aligned}$$

where $p_{N-1}$ is a finite sum of terms each of which is a product of strictly less that N additive functions on S and where $c \in \mathbb {C}$ is a constant.

If $p\chi$ satisfies the special sine addition law (5.2), then the highest order term of $p\chi$ vanishes in the sense that

$$\begin{aligned} \left( \sum _{k=1}^M a_1^{(k)}\cdots a_N^{(k)}\right) \chi = 0. \end{aligned}$$

Proof

Let $p_N:= \sum _{k=1}^M a_1^{(k)}\cdots a_N^{(k)}$. By the help of Lemma 5.8 we obtain for $n = 1, 2, \dots$ the formula

$$\begin{aligned}&(p\chi )(xy^n) - (p\chi )(x)\chi (y^n) - (p\chi )(y^n)\chi (x)\\&\quad = \left\{ n^{N-1} \sum _{k=1}^M \sum _{j=1}^N a_1^{(k)}(y) a_2^{(k)}(y) \cdots a_j^{(k)}(x) \cdots a_N^{(k)}(y) + P'(n,x,y) \right\} \\&\quad \quad \times \chi (x)\chi (y)^n, \end{aligned}$$

where $P'(n,x,y)$ is a polynomial in n of degree $<N-1$ with coefficients that are functions of x and y.

By assumption $p\chi$ satisfies (5.2) so the left hand side of the formula vanishes. Noting that $ab = 0$ $\iff$ $a b^n = 0$ for any $a,b \in \mathbb {C}$ and any fixed $n = 1,2, \dots$ we obtain with $b = \chi (y)$ that

$$\begin{aligned}&\left\{ n^{N-1} \sum _{k=1}^M \sum _{j=1}^N a_1^{(k)}(y) a_2^{(k)}(y) \cdots a_j^{(k)}(x) \cdots a_N^{(k)}(y) + P'(n,x,y)\right\} \\&\quad \quad \times \chi (x)\chi (y) = 0. \end{aligned}$$

For fixed $x,y \in S$ this expression is a polynomial in n. Since it vanishes for all integers $n > 0$ its coefficients are 0. In particular

$$\begin{aligned} \left\{ \sum _{k=1}^M \sum _{j=1}^N a_1^{(k)}(y) a_2^{(k)}(y) \cdots a_j^{(k)}(x) \cdots a_N^{(k)}(y) \right\} \chi (x)\chi (y) = 0. \end{aligned}$$

Taking $y = x$ here we get that

$$\begin{aligned}&\left\{ \sum _{k=1}^M \sum _{j=1}^N a_1^{(k)}(x) a_2^{(k)}(x) \cdots a_N^{(k)}(x) \right\} \chi (x)\chi (x)\\&\quad = N \sum _{k=1}^M a_1^{(k)}(x) a_2^{(k)}(x) \cdots a_N^{(k)}(x)\chi (x)^2 = 0, \end{aligned}$$

which implies the desired relation

$$\begin{aligned} \sum _{k=1}^M a_1^{(k)}(x) a_2^{(k)}(x) \cdots a_N^{(k)}(x)\chi (x) = 0. \end{aligned}$$

$\square$

Proof of Theorem 5.6

By Lemma 5.7$\chi$ is a continuous exponential and $f = p\chi$ for some polynomial $p \in C(S)$. By successive applications of Lemma 5.9 we infer that $f = (a+c)\chi$, where $a:S \rightarrow \mathbb {C}$ is an additive function and $c \in \mathbb {C}$ is a constant. By construction a is a linear combination of some of the additive functions in the expression for the exponential polynomial f. They are continuous by assumption, so $a \in C(S)$. Since f and $a\chi$ both satisfy (5.2) with the given $\chi$ ($a\chi$ according to Example 5.4(a)), so does their difference $c\chi$. Lemma 5.3(a) says that $c\chi = 0$, and so $f = (a+c)\chi = a\chi$. $\square$

Corollary 5.10 applies to important examples, but not in general. It applies to Example 6.4, but not to Example 6.3. Corollary 5.10 is also derived in [5, Examples 1, 5 and 6].

Corollary 5.10

Let S be a topological magma with the property that 0 is the only continuous, additive function on S, which is the case if for example S has a zero element or if S is compact.

If $f \in C(S)$ is a nonzero solution of (5.2) in which $\chi$ is a multiplicative function on S, then f is not an exponential polynomial.

Proof

Suppose for contradiction that f is an exponential polynomial. Then it has by Theorem 5.6 the form $f = a\chi$, where a is a continuous, additive function on S. By the assumption $f = a \chi$ reduces to 0, which contradicts the hypothesis of f being nonzero. Left is the claim that $a = 0$ in the cases mentioned: If $0 \in S$ is a zero element then $a(0) = a(x\cdot 0) = a(x) + a(0)$, and if S is compact then a is bounded being continuous and any bounded, additive function is identically zero. $\square$

6 Selected examples

This section contains some illustrative examples. Ebanks and Ng [5, 6] discuss many other examples.

Example 6.1

In this example exponentials vanish nowhere, because the underlying magma is a group (Cfr. Example 5.4(b)).

Let S be the group $(\mathbb {R},+)$ with its usual topology. The function $x \mapsto \exp (\alpha x)$ where $\alpha \in \mathbb {C}$, is a continuous exponential on $\mathbb {R}$. Applying Theorem 5.6 we find that the exponential polynomials f which are solutions of the corresponding equation (5.2) are the functions $f(x) = cx\exp (\alpha x)$ for fixed $c \in \mathbb {C}$ (We recall that the continuous additive functions on $(\mathbb {R},+)$ are the linear functions [9].) Actually all continuous solutions of (5.2) have according to Example 5.4(b) the form just described, so all continuous solutions f of (5.2) are exponential polynomials.

Example 6.2

Consider the semigroup $S:= (\,]0,1[, \cdot )$ with the topology inherited from $\mathbb {R}$. It might be remarked that the general result [5, Theorem 1] does not apply here, because it requires that S is a monoid.

By Example 3.5 exponentials of S vanish nowhere. We shall first compute the exponentials $\chi \in C(S)$. By Example 3.5 the range of $\chi$ is in the multiplicative group $\mathbb {C}^*$ of non-zero complex numbers. According to Aczél, Baker, Djoković, Kannappan and Radó’s extension theorem [1] (reproduced in [7, Theorem 1.67]) there exists a homomorphism $F: (\mathbb {R}^+,\cdot ) \rightarrow \mathbb {C}^*$ such that $F = \chi$ on ]0, 1[. A homomorphism F between topological groups is continuous, if it is continuous at a single point, so F is continuous since $\chi$ is. Continuous homomorphisms are known for instance from [9, Example 3.9(a)]: There exists a $\lambda \in \mathbb {C}$ such that $F(x) = x^{\lambda }$ for $x \in \mathbb {R}^+$. We conclude that $\chi (x) = x^{\lambda }$ for $x \in \,]0,1[$.

We next note that the continuous additive functions on S are $x \mapsto c\log x$, where c ranges over $\mathbb {C}$ (see for instance [9, Exercise 2.9(a)]).

It follows from Example 5.4(c) that the solutions $f \in C(S)$ of (5.2) are the functions $f(x) = c x^{\lambda }\log x$, and that they are exponential polynomials.

Example 6.3

This example presents by the help of Theorem 5.6 a solution f of (5.2) which is not an exponential polynomial. It is a special instance of Ebanks and Ng [5, Proposition 2].

We consider the commutative monoid $S:= (\mathbb {N},\cdot )$ with the discrete topology. Let $q \in \mathbb {N}$ be a prime and define $I:= q\mathbb {N}$. $I^c = \mathbb {N} {\setminus } I$ is a non-empty proper subsemigroup of S. We introduce the exponential $\chi : S \rightarrow \mathbb {C}$ and the function $f:S \rightarrow \mathbb {C}$ by

$$\begin{aligned} \chi := 1_{I^c} \ \text { and } \ f(x):= {\left\{ \begin{array}{ll} 0 &{} \text {for } x \in I^c\\ \chi (x/q) &{} \text {for } x \in I. \end{array}\right. } \end{aligned}$$

A case by case inspection reveals that the pair $(f,\chi )$ satisfies (5.2). Note that $\chi (q) = 0$ and that $f(q) = 1$.

f is not an exponential polynomial: If it were, then it would according to Theorem 5.6 have the form $f = a\chi$ for some additive function $a:S \rightarrow \mathbb {C}$, and then we obtain the contradiction $1 = f(q) = a(q)\chi (q) = a(q)\cdot 0 = 0$.

Example 6.4

This example illustrates the use of Corollary 5.10. It is from Ebanks and Ng [5, Example 1].

We consider the monoid $S:= ([0,1], \cdot )$ with the topology inherited from $\mathbb {R}$. It has $0 \in [0,1]$ as zero element. $\chi (x):= x$, $x \in S$, is a continuous exponential on S.

$f(x):= x\log x$ (for interpreted as 0) is a continuous function on S. It satisfies (5.2), as is elementary to verify, but according to Corollary 5.10 it is not an exponential polynomial.