1 Introduction

Given a commutative semigroup \((S,+)\) and a commutative group \((H,+)\), we say that \(f:S\rightarrow H\) is a polynomial function (also named generalized polynomial) of degree \(\le m\) if f solves Fréchet’s mixed differences functional equation:

$$\begin{aligned} \Delta _{h_1}\Delta _{h_2}\cdots \Delta _{h_{m+1}}f(x)=0\text { for all } h_1,\ldots , h_{m+1},x\in S. \end{aligned}$$

When we evaluate a polynomial function of degree \(\le m\) on a sum \(u_1+u_2+\cdots +u_{m+1}\) of \(m+1\) variables \(u_1,\ldots , u_{m+1}\), we obtain a sum of functions with the property that each one of them depends on at most m variables \(u_{i_1},\ldots ,u_{i_m}\) (\(i_k\in \{1,\ldots ,m+1\}\) for all k). Indeed, this fact completely characterises polynomial functions, as was proved by Aichinger and Moosbauer [1] for functions defined on abelian groups and by Almira [2] for functions defined on commutative semigroups S which satisfy that \(S+S=S\) and \(0\in S\). Concretely [1, Lemma 4.1] states that, when \((S,+)\) is also a group, the function \(f:S\rightarrow H\) is a generalized polynomial of degree \(\le m\) if and only if it solves Aichinger’s equation:

$$\begin{aligned} f(x_1+\cdots +x_{m+1})=\sum _{i=1}^{m+1}g_i(x_1,x_2,\ldots , \widehat{x_i},\ldots , x_{m+1}) \end{aligned}$$
(1)

for certain functions \(g_i:S^{m}\rightarrow H\), \(i=1,2,\ldots , m+1\). Here \(\widehat{x_{i}}\) means that \(g_i\) does not depend on \(x_i\). Moreover, [2, Theorem 2] proves that, if \(S+S=S\) and f solves (1), then f is a polynomial function of degree \(\le m\). Moreover, if S also satisfies that \(0\in S\), then all polynomial functions of degree \(\le m\) are solutions of (1).

The main goal of this note is to use these results to give simple proofs of several well-known (and useful) characterizations of polynomial functions as solutions of certain functional equations. Concretely, we use that Aichinger’s equation characterizes polynomial functions to solve Ghurye–Olkin’s functional equation (Corollary 1), Wilson’s functional equation (Corollary 2), the Kakutani–Nagumo–Walsh functional equation (Corollary 4), and a general version of Fréchet’s unmixed functional equation (Corollary 3). In all cases we give proofs for general commutative groups or semigroups and we do not worry about the regularity properties of polynomial functions. Although the results we prove are well-known, our proofs are surprisingly easier than the original ones.

2 Characterizations of polynomial functions

We use Aichinger’s equation to prove the main result of this note, which is the following theorem:

Theorem 1

Assume that \((S,+)\) is a commutative semigroup such that \(S+S=S\), \((R,+,\cdot )\) is a commutative ring, \(c:S\rightarrow S\) is an automorphism, and \(f:(S,+)\rightarrow (R,+)\) is a map that satisfies

$$\begin{aligned} f(x+c(y))=\sum _{j=1}^Np_j(x)a_j(y)+\sum _{k=1}^Mq_k(y)b_k(x) \end{aligned}$$
(2)

where \(p_j,q_k:(S,+)\rightarrow (R,+)\) are polynomial functions and \(\deg (p_j)\le r\), \(\deg (q_k)\le s\) for all j, k. Then \(f:(S,+)\rightarrow (R,+)\) is a polynomial function and \(\deg (f) \le r+s+1\).

This theorem is a generalized version of [3, Lemma 2.1] and is the key to proving Corollary 1, which is a generalization of a result proved by Ghurye and Olkin in [7, Lemma 3] (see also [3, Theorem 1.3]), where the equation was used for the characterization of Gaussian probability distributions (see also [14, Chapter 7]). Our proof is not based on the very technical tools used in [7]. Moreover, it simplifies the proof given in [3], since it avoids using exponential polynomials. In particular, it does not depend on [15, Lemma 4.3] and/or [16, Theorem 5.10].

Proof of Theorem 1.

Take \(x_1,\ldots ,x_{r+1},y_1,\ldots ,y_{s+1}\), a set of \(r+s+2\) variables, then:

$$\begin{aligned}{} & {} f(x_1+\cdots +x_{r+1}+y_1+\cdots +y_{s+1}) \\{} & {} \quad = f(x_1+\cdots +x_{r+1}+c(c^{-1}(y_1)+\cdots +c^{-1}(y_{s+1})))\\{} & {} \quad = \sum _{j=1}^Np_j(x_1+\cdots +x_{r+1})a_j(c^{-1}(y_1)+\cdots +c^{-1}(y_{s+1}))\\{} & {} \quad \quad +\, \sum _{k=1}^Mq_k(c^{-1}(y_1)+\cdots +c^{-1}(y_{s+1}))b_k(x_1+\cdots +x_{r+1}), \end{aligned}$$

now, each term \(p_j(x_1+\cdots +x_{r+1})\) can be decomposed as a sum of terms, each one depending on at most r of the variables \(x_1,\ldots ,x_{r+1}\); and each term \(q_k(c^{-1}(y_1)+\cdots +c^{-1}(y_{s+1}))\) can be decomposed as a sum of terms, each one depending on at most s of the variables \(y_1,\ldots ,y_{s+1}\). Thus, f satisfies Aichinger’s equation of order \(r+s+1\), which means that f is a polynomial function of degree \(\le r+s+1\). \(\square \)

Aichinger’s equation can also be used to prove the following

Lemma 1

Assume that \((S,+)\) is a commutative semigroup such that \(0\in S= S+S\), \((H,+)\) is a commutative group, \(c:S\rightarrow S\) is an automorphism, and \(f:(S,+)\rightarrow (H,+)\) is a map such that \(g(x)=f(c(x))\) is a polynomial function of degree \(\le m\). Then f is also a polynomial function of degree \(\le m\).

Proof

Given a set of \(m+1\) variables \(x_1,\ldots , x_{m+1}\), we have that

$$\begin{aligned} f(x_1+\cdots +x_{m+1})= & {} f(c(c^{-1}(x_1)+\cdots +c^{-1}(x_{m+1})))\\= & {} g(c^{-1}(x_1)+\cdots +c^{-1}(x_{m+1}))\\= & {} \sum _{i=1}^{m+1}g_i( c^{-1}(x_1),\ldots ,\widehat{c^{-1}(x_i)},\ldots , c^{-1}(x_{m+1}))\\= & {} \sum _{i=1}^{m+1}G_i( x_1,\ldots ,\widehat{x_i},\ldots x_{m+1}), \end{aligned}$$

which means that f satisfies Aichinger’s equation of order m. \(\square \)

We are now in a good position to study Ghurye–Olkin’s functional equation:

Corollary 1

(Ghurye–Olkin’s functional equation) With the same notation as used in Theorem 1, assume that \((S,+)\) is a commutative group, \(c_i:S\rightarrow S\), \(i=1,\ldots ,n\) are automorphisms and that \(c_i-c_j\) is also an automorphism whenever \(i\ne j\). Let \(f_i:S\rightarrow R\), \(i=1,\ldots ,n\) be such that

$$\begin{aligned} \sum _{i=1}^nf_i(x+c_i(y))=\sum _{j=1}^Np_j(x)a_j(y)+\sum _{k=1}^Mq_k(y)b_k(x) \end{aligned}$$
(3)

where \(p_j,q_k:(S,+)\rightarrow (R,+)\) are polynomial functions and \(\deg (p_j)\le r\), \(\deg (q_k)\le s\) for all j, k. Then each function \(f_i:(S,+)\rightarrow (R,+)\) is a polynomial function and \(\deg (f_i) \le r+s+n\), \(i=1,\ldots ,n\). Moreover, in the particular case that \(p_j=q_k=0\) for all j, k (so that the second member of the equation is 0), the equation is well defined for functions \(f_i\) taking values on a commutative group and each \(f_i\) that solves the equation is a polynomial function of degree at most \(n-1\).

Proof

Assume that the second member of (3) is not 0. Theorem 1 is case \(n=1\), so that it is natural to proceed by induction on n. Indeed, we prove by induction on n that \(f_n\) is a polynomial function of degree \(\le r+s+n\), and a simple rearrangement of the functions \(f_1,\ldots ,f_n\) proves the result for all functions \(f_i\), \(i=1,\ldots ,n\).

Let us assume that \(n>1\) and the result holds for \(n-1\). If we consider both members of the equation as functions F(x, y) defined on \(S\times S\), we can apply the difference operator

$$\begin{aligned} \Delta _{(h_1,-c_1^{-1}(h_1))}F(x,y)=F(x+h_1,y-c_1^{-1}(h_1))-F(x,y) \end{aligned}$$

to both sides of the equation to get that

$$\begin{aligned} \sum _{i=2}^ng_i(x+c_i(y))=\sum _{j=1}^Np_j^*(x)a_j^*(y)+\sum _{k=1}^Mq_k^*(y)b_k^*(x), \end{aligned}$$

where

$$\begin{aligned} g_i(x+c_i(y))= & {} \Delta _{(h_1,-c_1^{-1}(h_1))}f_i(x+c_i(y)) \\= & {} f_i(x+h_1+c_i(y-c_1^{-1}(h_1))) - f_i(x+c_i(y)) \\= & {} f_i(x+h_1+c_i(y)-c_i(c_1^{-1}(h_1)))-f_i(x+c_i(y)) \\= & {} \Delta _{h_1-c_i(c_1^{-1}(h_1))}f_i(x+c_i(y))\\= & {} \Delta _{(1_d-c_i\circ c_1^{-1})(h_1)}f_i(x+c_i(y)) \end{aligned}$$

and \(p_j^*,q_k^*:(S,+)\rightarrow (R,+)\) are polynomial functions satisfying \(\deg (p_j^*)\le r\), \(\deg (q_k^*)\le s\) for all j, k. Thus, the induction hypothesis implies that all functions \(g_i\), \(i=2,\ldots , n\) are polynomial functions of degree \(\le r+s+n-1\). In particular, for each \(h_1\in S\), \(\Delta _{(1_d-c_i\circ c_1^{-1})(h_1)}f_n\) is a polynomial function of degree \(\le r+s+n-1\). Now, \((c_1-c_i)\circ c_1^{-1}= 1_d-c_i\circ c_1^{-1}\) is bijective, so that, for each \(h\in S\), \(\Delta _{h}f_n\) is a polynomial function of degree \(\le r+s+n-1\). In particular, \(f_n\) satisfies Fréchet’s mixed functional equation

$$\begin{aligned} \Delta _{h_1}\Delta _{h_2}\cdots \Delta _{h_{r+s+n}} \Delta _{h}f(x)=0\text { for all } h_1,\ldots , h_{r+s+n-1},h,x\in S. \end{aligned}$$

Hence \(f_n\) is a polynomial function of degree \(\le r+s+n\).

Let us now assume that that \(p_j=q_k=0\) for all j, k and that all functions \(f_i\) take values on a commutative group (not necessarily a ring). Then the equation is of the form

$$\begin{aligned} \sum _{i=1}^nf_i(x+c_i(y))=0. \end{aligned}$$
(4)

Again, we prove by induction on n that \(f_n\) is a polynomial function of degree \(\le n-1\), and a simple rearrangement of the functions \(f_1,\ldots ,f_n\) proves the result for all functions \(f_i\), \(i=1,\ldots ,n\). For \(n=1\), the equation becomes \(f_1(x+c_1(y))=0\) and taking \(y=0\) we get that \(f_1(x)=0\) and \(f_1\) is a polynomial of degree 0. We assume that the result holds true for \(n-1\) and consider the case \(n>1\). Applying to both sides of (4) the operator \(\Delta _{(h_1,-c_1^{-1}(h_1))}\) we obtain that

$$\begin{aligned} \sum _{i=2}^n\Delta _{(1_d-c_i\circ c_1^{-1})(h_1)}f_i(x+c_i(y))=0. \end{aligned}$$

Thus, the induction hypothesis and the fact that \(1_d-c_i\circ c_1^{-1}\) is an automorphism for each \(i\ge 2\) imply that \(\Delta _{h}f_i\) is a polynomial function of degree at most \(n-2\) for each \(i=2,\ldots ,n\) and for every \(h\in S\). In particular, \(f_n\) is a polynomial function of degree at most \(n-1\). \(\square \)

Remark 1

The transformation \(\widetilde{f_i}(x)=f_i(\beta _i(x))\) reduces the equation

$$\begin{aligned} \sum _{i=1}^nf_i(\beta _i(x)+\delta _i(y))=\sum _{j=1}^Np_j(x)a_j(y)+\sum _{k=1}^Mq_k(y)b_k(x) \end{aligned}$$
(5)

to the equation

$$\begin{aligned} \sum _{i=1}^n\widetilde{f_i}(x+(\beta _i^{-1}\circ \delta _i)(y))=\sum _{j=1}^Np_j(x)a_j(y)+\sum _{k=1}^Mq_k(y)b_k(x), \end{aligned}$$
(6)

so that, if \(\beta _i,\delta _i:S\rightarrow S\) are automorphisms such that \(\beta _i^{-1}\circ \delta _i-\beta _j^{-1}\circ \delta _j\) is invertible whenever \(i\ne j\) and \(p_j,q_k:(S,+)\rightarrow (R,+)\) are polynomial functions, \(\deg (p_j)\le r\), \(\deg (q_k)\le s\) for all j, k, then we can use Corollary 1 with \(c_i=\beta _i^{-1}\circ \delta _i\) and Lemma 1 to conclude that the functions \(f_i\) that solve equation (5) are polynomial functions of degree at most \(r+s+n\). Moreover, if the second member of (5) is 0, the functions \(f_i\) that solve the equation are polynomial functions of degree at most \(n-1\).

The following equation was studied one hundred years ago by Wilson [18]:

Corollary 2

(Wilson’s functional equation) Assume that \((S,+)\), \((H,+)\) are commutative groups and \(\beta _i,\delta _i:S\rightarrow S\) are automorphisms such that \(\beta _i^{-1}\circ \delta _i-\beta _j^{-1}\circ \delta _j\) is invertible whenever \(i\ne j\). Assume also that the functions \(f_i,a,b:S\rightarrow H\) (\(i=1,\ldots ,n\)) solve the equation

$$\begin{aligned} \sum _{i=1}^nf_i(\beta _i(x)+\delta _i(y))=a(x)+b(y) \end{aligned}$$

Then a, b are polynomial functions of degree at most n.

Proof

A direct application of Remark 1 with \(r=s=0\) shows that all functions \(f_i\), \(i=1,\ldots ,n\) are polynomial functions of degree at most n. Hence a, b are also polynomial functions of degree at most n. \(\square \)

The following equation was studied by Almira in [3, 5].

Corollary 3

(Generalized Fréchet’s unmixed functional equation) Let \((S,+)\) and \((H,+)\) be commutative groups and assume that \(f:S\rightarrow H\) solves the equation

$$\begin{aligned} q(\tau _h)(f)=\sum _{k=0}^na_kf(x+kh)=0 \text { for all } x,h\in S, \end{aligned}$$
(7)

where \(q(z)=a_0+\cdots +a_nz^n\) is a polynomial and \(\tau _h:S\rightarrow S\) is the translation operator, \(\tau _h(x)=x+h\). Then f is a polynomial function of degree at most \(s=\#\{k:a_k\ne 0\}-1\).

Proof. Set \(\{k:a_k\ne 0\}=\{k_1,k_2,\ldots ,k_s\}\) and apply Corollary 1 with \(p_j=q_k=0\) for all j, k, \(f_i=a_{k_i}f(x)\) and \(c_i(x)=k_ix:=x+\cdots ^{k_i\text { times }}+x\), \(i=1,\ldots ,s\). \(\Box \)

The following equation was introduced by Kakutani and Nagumo [12] and Walsh [17] in the 1930’s. The equation was extensively studied by Haruki [8,9,10] in the 1970’s and 1980’s.

Corollary 4

Let \((H,+)\) be a commutative group, let \(f:{\mathbb {C}}\rightarrow H\) be a solution of the Kakutami–Nagumo–Walsh functional equation

$$\begin{aligned} \frac{1}{N}\sum _{k=0}^{N-1}f(z+w^kh)=f(z) \text { for all } z,h\in {\mathbb {C}}, \end{aligned}$$
(8)

where w is any primitive N-th root of 1. Then f is a polynomial function of degree \(\le N\).

Proof

Use Corollary 2 with \(n=N\), \(f_i(z)=\frac{1}{N}f(z)\), \(\beta _i(z)=z\), \(\delta _i(z)=w^{i-1}z\), \(i=1,\ldots ,N\), and \(a(z)=f(z)\), \(b(z)=0\). Obviously the corollary can be used since \(\beta _i^{-1}(z)=z\), \(\delta _i^{-1}(z)=w^{N+1-i}z\), and

$$\begin{aligned} (\beta _i^{-1}\circ \delta _i-\beta _j^{-1}\circ \delta _j)(z)=(\delta _i-\delta _j)(z)=(w^{i-1}-w^{j-1})z \end{aligned}$$

is an automosphim of \({\mathbb {C}}\) for all \(i\ne j\). \(\square \)

Another special case of Wilson’s equation is

$$\begin{aligned} \sum _{i=1}^{m}f_i(b_ix+c_iy)= \sum _{i=1}^m f_i(b_ix) + \sum _{i=1}^m f_i(c_iy), \end{aligned}$$
(9)

which is a linearized form of the Skitovich-Darmois functional equation

$$\begin{aligned} \prod _{i=1}^m \widehat{\mu _i}(b_ix+c_iy) = \prod _{i=1}^m \widehat{\mu _i}(b_ix) \prod _{i=1}^m \widehat{\mu _i}(c_iy), \end{aligned}$$

an equation which is connected to the characterization problem of Gaussian distributions (see, for example, Linnik [13], Ghurye–Olkin [7, 11]):

Corollary 5

(Linearized Skitovich–Darmois functional equation) Assume that \((S,+)\) and \((H,+)\) are commutative groups and let \(\beta _i,\delta _i:S\rightarrow S\) be automorphisms such that \(\beta _i^{-1}\circ \delta _i-\beta _j^{-1}\circ \delta _j\) is invertible whenever \(i\ne j\). If the functions \(f_i:S\rightarrow H\), \(i=1,\ldots , n\) solve the functional equation

$$\begin{aligned} \sum _{i=1}^nf_i(\beta _i(x)+\delta _i(y))=\sum _{i=1}^nf_i(\beta _i(x))+\sum _{i=1}^nf_i(\delta _i(y)), \end{aligned}$$
(10)

then \(P(x)=\sum _{i=1}^nf_i(\beta _i(x))\) and \(Q(y)= \sum _{i=1}^nf_i(\delta _i(y))\) are polynomial functions of degree \(\le n\).

Proof

Use Corollary 2 with \(a(x)=P(x)\) and \(b(y)=Q(y)\). \(\square \)

Equation (10) has been studied in great detail by Feld’man [6] for functions defined on locally compact commutative groups.

Remark 2

As a final remark we would like to mention that although we have formulated all results in this paper for ordinary functions defined on commutative groups or semigroups, the same proofs can be translated to the distributional setting. In particular, ordinary polynomials of degree \(\le m\) are the unique solutions of Aichinger’s equation (1) when \(f\in {\mathcal {D}}({\mathbb {R}}^d)'\) (so that \(f(x_1+\cdots +x_{m+1}) \in {\mathcal {D}}({\mathbb {R}}^d\times \cdots ^{m+1 \text {times}}\times {\mathbb {R}}^d)'\)) and \(g_i(x_1,\ldots ,\widehat{x_i},\ldots ,x_{m+1}) \in {\mathcal {D}}({\mathbb {R}}^d\times \cdots ^{m+1 \text {times}}\times {\mathbb {R}}^d)'\) but does not depend on \(x_i\)). The proof of this result is a direct consequence of the fact that the translation and difference operators are well defined for distributions and inherit in the distributional framework the properties they have when applied to ordinary functions, and Fréchet’s functional equation also characterizes polynomials in a distributional sense [4].