Abstract
In this paper generalized moment functions are considered. They are closely related to the wellknown functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed.
1 Introduction
Let \((G,+)\) be an Abelian group. Recall that a nonzero function \(m:G\rightarrow {\mathbb {C}}\) is called exponential, if
holds for all x, y in G. Let N be a nonnegative integer. A function \(\varphi :G\rightarrow {\mathbb {C}}\) is termed to be a moment function of order N, if there exist functions \(\varphi _k:G \rightarrow {\mathbb {C}}\) such that \(\varphi _0=1\), \(\varphi _N=\varphi \) and
for all x and y in G and \(k=0,1,\dots ,N\). We generalize this concept by relaxing the assumption \(\varphi _0=1\) to \(\varphi _0(0)=1\). In this case \(\varphi _0\) is an arbitrary exponential function and we say that \(\varphi _0\) generates the generalized moment sequence of order N and the function \(\varphi _k\) is a generalized moment function of order k, or, if we want to specify the exponential \(\varphi _0\), then we say that \(\varphi _k\) is a generalized moment function of order k associated with the exponential \(\varphi _0\). Problems about moment functions have been extensively studied on different type of abstract structures, in particular on hypergroups. They are measure algebras where convolution has specific probabilistic interpretation. In the hypergroupsetting the justification of the name ‘moment problems’ is more visible than on abstract groups without additional structure as we can interpret moment functions by means of properly defined moments of random elements. For a more detailed discussion see e.g. [3] and [8]. More on moment functions can be found e.g. in [2, 4,5,6, 10,11,12] and references therein. As already mentioned groups are special hypergroups and in this paper we are going to focus on a generalized moment problem on Abelian groups.
Equation (1) is closely related to the wellknown functions of binomial type. We are particular interested in (1) defined on abstract structures. A detailed discussion about binomial type equations in abstract setting, which has been the motivation for the present research, can be found in [1]. It was shown in [1] there that if \((G,+)\) is a grupoid and R is a commutative ring, then functions \(\varphi _n:G\rightarrow R\) satisfying (1) for each n in \({\mathbb {N}}\) are of the form
for all t in G and k in \({\mathbb {N}}\) and arbitrary homomorphisms \(a_k\) from \((G,+)\) into \((R,+)\).
The present paper is organized as follows: first we give the definition and a multivariable characterization of generalized moment functions. Next we introduce the notion of generalized moment functions of rank r, a generalization of the solutions of (1), which represent generalized moment functions of rank 1. Our main result is the description of generalized moment functions of higher rank by means of Bell polynomials. As a corollary we get a new characterization of generalized moment functions on Abelian groups, as well.
We recall the definition of the multinomial coefficient which will be used below: let n be a nonnegative integer and l a positive integer. Then we write
for all \(k_{1}, k_2,\ldots , k_{l}\) in \({\mathbb {N}}\) satisfying \( k_{1}+k_2+ \cdots + k_{l}=n\). Observe that for \(l=2\) we obtain the binomial coefficient.
For the sake of completeness we clarify that in this paper the set of complex numbers is denoted by \({\mathbb {C}}\), the set of nonzero complex numbers by \({\mathbb {C}}^{\times }\) and the set of nonnegative integers by \({\mathbb {N}}\).
2 Generalized Moment Functions on Groups
We begin with a multivariable characterization of moment functions.
Proposition 1
Let G be an Abelian group, l, N positive integers with \(l\ge 2\), and let \(\varphi _{0}, \varphi _1,\ldots , \varphi _{N}:G\rightarrow {\mathbb {C}}\) be functions for which
holds for each \(x_{1}, x_2,\ldots , x_{l}\) in G and \(n=0, 1,\ldots , N\). If \(\varphi _{0}(0)=0\), then all the functions \(\varphi _{0}, \varphi _1,\ldots , \varphi _{N}\) are identically zero. If \(\varphi _{0}(0)=1\), then the functions \(\varphi _{0}, \varphi _1,\ldots , \varphi _{N}:G\rightarrow {\mathbb {C}}\) form a generalized moment sequence of order N.
Proof
Assume that the conditions of the proposition are satisfied. If \(l=2\), then equation (3) reduces to (1).
Suppose that \(l>2\) and let x, y be in G. With the substitution \(x_{1}= x\), \(x_{2}= y \) and \(x_{i}=0\) for \(i=3,4,\dots ,n\) we get from equation (3) that
for each \(n=0, 1,\ldots , N\). Thus the only thing we should prove is that \(\varphi _{n}(0)=0\) if \(n\ge 1\). Indeed, in this case the right hand side of the above identity is nothing but \(\displaystyle \sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \varphi _{k}(x)\varphi _{nk}(y)\varphi _0(0)^{l2} \).
With \(n=0\) and \(x_{i}=0\) for \(i=1, 2,\ldots , l\) equation (3) leads to
that is,
from which we derive that either \(\varphi _{0}(0)=0\), or \(\varphi _{0}(0)\) is an \((l1)^\mathrm{th}\) root of unity.
Furthermore, Eq. (3) for \(n=0\) and \(x_{1}, x_2,\ldots , x_{l}\) in G yields that
If \(\varphi _{0}(0)=0\), then \(\varphi _0(x)=0\) for all x in G. If \(\varphi _0^{l1}=1\), then taking \(x_3=\cdots =x_l=0\) and multiplying both sides of the above equation by \(\varphi _0^{l2}\) we get
Thus \(\varphi _{0}(0)^{l2}\varphi _{0}\) is an exponential, which we denote by m. Thus \(\varphi _{0}(0)^{l2}\varphi _{0}(x)=m(x)\) for all x in G. Using the fact that \(\varphi _{0}(0)\) is an \((l1)^\mathrm{th}\) root of unity we obtain that
holds for each x in G.
Similarly, Eq. (3) for \(n=1\) with \(x_{1}=x\) and \(x_{i}=0\) whenever x is in G and \(i=2,3,\dots ,n\) yields that
It follows that

(A)
either \(\varphi _{0}(0)=0\), implying that both \(\varphi _{0}\) and \(\varphi _{1}\) are identically zero;

(B)
or \(\varphi _{0}(0)\) is an \((l1)^\mathrm{th}\) root of unity and
$$\begin{aligned} (l1)\varphi _{0}(x)\varphi _{1}(0)\varphi _{0}(0)^{l2} =0, \end{aligned}$$which happens only when \(\varphi _{1}(0)=0\), otherwise \(\varphi _{0}\equiv 0\), contrary to the fact that \(\varphi _{0}(0)\) is an \((l1)^\mathrm{th}\) root of unity.
Assume now that there exists k in \(\left\{ 1, 2,\ldots , N1\right\} \) such that \(\varphi _{0}(0)\ne 0\) and
In this case Eq. (3), with \(k+1\) instead of n and with the substitution \(x_{1}=x\) in G and \(x_{i}=0\) for \(i=2,3, \ldots , l\), yields that
Again, we have the following two alternatives:

(A)
either \(\varphi _{0}(0)=0\), implying that \(\varphi _{k+1}\) is identically zero;

(B)
or \(\varphi _{0}\) is an \((l1)^\mathrm{th}\) root of unity and
$$\begin{aligned} \varphi _{k}(x)\varphi _{1}(0)\varphi _{0}(0)^{l2}+ \cdots + \varphi _{0}(x)\varphi _{0}(0)^{l2}\cdots \varphi _{k+1}(0)=0, \qquad \end{aligned}$$or equivalently
$$\begin{aligned} (l1)\varphi _{0}(x)\varphi _{0}^{l2}(0)\cdots \varphi _{k+1}(0)=0 \qquad \end{aligned}$$for each x in G.
Due to the induction hypothesis \(\varphi _{1}(0)= \varphi _2(0)=\cdots = \varphi _{k}(0)=0\), thus
Since \(\varphi _{0}(0)\ne 0\), \(\varphi _{0}\) is not identically zero, which shows that the only possibility is that \(\varphi _{k+1}(0)=0\).
Finally we conclude that \(\varphi _{1}(0)=\varphi _{2}(0)= \cdots = \varphi _{N}(0)=0\). However, this implies for x, y in G:
Therefore, if \(\varphi _{0}(0)=1\), then the functions \(\varphi _{0}, \varphi _{1}, \ldots , \varphi _{N}\) form a generalized moment sequence of order N. \(\square \)
Proposition 2
Let G be an Abelian group, and let l, N be positive integers with \(l\ge 2\). If the functions \(\varphi _{0}, \varphi _1,\ldots , \varphi _{N}:G\rightarrow {\mathbb {C}}\) form a generalized moment sequence of order N, then the system of equations
is satisfied for each \(x_{1}, x_2,\ldots , x_{l}\) in G and \(n=0, 1,\ldots , N\).
Proof
We prove the statement by induction on l.
Assume that functions \(\varphi _{0}, \varphi _{1}, \ldots , \varphi _{N}\) constitute a moment sequence of order N, i.e.
is satisfied for each x, y in G. First observe that for \(n=0\) the statement holds trivially, because equation (4) applied for \(n=0\) and the \(\varphi _0\) function in the moment sequence functions are the same, namely
for x, y in G. Thus it is enough to consider the case \(n\ge 1\). Secondly, note that from (5) we have \(\varphi _0(0)=0\) or \(\varphi _0(0)=1\) and for all \(n\ge 1\) we have \(\varphi _n(0)=0\).
Let now n in \(\left\{ 1,2,\dots ,N \right\} \) be arbitrarily fixed and take \(x_1, x_2\) in G. Observe that
for \(x_{1}, x_{2}\) in G, which yields that the statement holds for \(l=2\). Assume that there exists an integer \(l\ge 2\) for which the statement holds. Then
for any \(x_{1}, \ldots , x_{l}, x_{l+1}\) in G. If we use that the statement holds for \(l=2\) and the induction hypothesis, then we get that
for \(x_{1}, \ldots , x_{l}, x_{l+1}\) in G. Thus the statement holds for \(l+1\), too. \(\square \)
3 Generalized Moment Functions of Rank r
In this section we extend the notion of generalized moment functions. At the same time, while stating and proving our results about generalized moment functions of higher rank, notions and results from spectral analysis and synthesis are required.
Let G be a discrete Abelian group. Subsequently, if it is not otherwise stated, the group G is always endowed with this topology.
Given a function f on this group and an element y in G, we define
where \(\delta _{y}\) and \(\delta _{0}\) denote the point masses concentrated at \(y\) and 0, respectively. For the product of modified differences we use the notation
for any positive integer n and for each \(y_{1}, y_2,\ldots , y_{n+1}\) in G. Here the product \(\prod \) on the right hand side is meant as a convolution product of the measures \({{\varDelta }}_{f; y_{i}}\) for all \(i=1, 2,\ldots , n+1\).
In the sequel, the symbol \({\mathbb {C}}G\) denotes the group algebra of G and \({\mathscr {C}}(G)\) the space of all (continuous) complexvalued functions on G. For each function \(f:G\rightarrow {\mathbb {C}}\) the ideal in \({\mathbb {C}}G\) generated by all modified differences of the form \({{\varDelta }}_{f;y}\) with y in G, is denoted by \(M_{f}\). Due to Theorem 12.5 of [9], this ideal \(M_{f}\) is proper if and only if f is an exponential and in this case we call \(M_{f}= \mathrm {Ann\,}\tau (f)\) an exponential maximal ideal. Recall that \(\tau (f)\) denotes the variety of the function f and it is nothing but the smallest closed translation invariant subspace of \({\mathscr {C}}(G)\) that contains the function f.
Let m be an exponential on the group G and n a positive integer. The function \(f:G\rightarrow {\mathbb {C}}\) is a generalized exponential monomial of degree at most n corresponding to the exponential m if
holds for each \(y_{1}, y_2,\ldots , y_{n+1}\) in G. Equivalently, f is a generalized exponential monomial if there exists an exponential m and a positive integer n such that its annihilator includes a positive power of an exponential maximal ideal, that is \(M_{m}^{n+1}\subset \mathrm {Ann\,}\tau (f)\) holds.
A special class of exponential monomials is formed by those corresponding to the exponential identically 1: these are called generalized polynomials.
The function \(f:G\rightarrow {\mathbb {C}}\) is called an exponential monomial if it is a generalized exponential monomial and \(\tau (f)\) is finite dimensional. Finite sums of exponential monomials are called exponential polynomials. Exponential monomials corresponding to the exponential identically 1 are called polynomials.
In connection with exponential polynomials here we also recall Theorem 12.31 from [9].
Theorem 1
Let G be an Abelian group, and let \(f:G\rightarrow {\mathbb {C}}\) be an exponential polynomial. Then there exist positive integers n, k and for each \(i=1, 2,\ldots , n\) and \(j=1, 2,\ldots , k\) there exists a polynomial \(P_{i}:{\mathbb {C}}^{k}\rightarrow {\mathbb {C}}\), an exponential \(m_{i}\) and a homomorphism \(a_{j}\) of G into the additive group of complex numbers such that
for each x in G. Conversely, every function of this form is an exponential polynomial.
A composition of a nonnegative integer n is a sequence of nonnegative integers \(\alpha = \left( \alpha _{k}\right) _{k\in {\mathbb {N}}}\) such that
For a positive integer r, an rcomposition of a nonnegative integer n is a composition \(\alpha = \left( \alpha _{k}\right) _{k\in {\mathbb {N}}}\) with \(\alpha _{k}=0\) for \(k>r\).
Given a sequence of variables \(x=(x_{k})_{k\in {\mathbb {N}}}\) and compositions \(\alpha = \left( \alpha _{k}\right) _{k\in {\mathbb {N}}}\) and \(\beta = \left( \beta _{k}\right) _{k\in {\mathbb {N}}}\) we define
Furthermore, \(\beta \le \alpha \) means that \(\beta _{k}\le \alpha _{k}\) for all \(k\in {\mathbb {N}}\) and \(\beta < \alpha \) stands for \(\beta \le \alpha \) and \(\beta \ne \alpha \).
Definition 1
Let G be an Abelian group, r a positive integer, and for each multiindex \(\alpha \) in \({\mathbb {N}}^r\) let \(f_{\alpha }:G\rightarrow {\mathbb {C}}\) be a continuous function. We say that \((f_{\alpha })_{\alpha \in {\mathbb {N}}^{r}}\) is a generalized moment sequence of rank r, if
holds whenever x, y are in G. The function \(f_0\), where 0 is the zero element in \({\mathbb {N}}^r\), is called the generating function of the sequence.
Remark 1

(i)
For \(r=1\), instead of multiindices, we have ‘ordinary’ indices and (6) is nothing but
$$\begin{aligned} f_{\alpha }(x+y)= & {} \sum _{\beta =0}^{\alpha }\left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) f_{\beta }(x)f_{\alpha \beta }(y) \qquad \end{aligned}$$for each x, y in G and nonnegative integer \(\alpha \), yielding that generalized moment functions of rank 1 are moment sequences.

(ii)
For \(\alpha =(0, \ldots ,0)\) we have
$$\begin{aligned} f_{0, \ldots ,0}(x+y)=f_{0, \ldots ,0}(x)\cdot f_{0, \ldots ,0}(y) \qquad \end{aligned}$$for each x, y in G hence \(f_{0, \ldots ,0}=m\) is an exponential, or identicaly zero. In what follows, when considering generalized moment function sequences of any rank, we always assume that the generating function is not identically zero, hence it is always an expoential.

(iii)
Now let \(\alpha \) be in \({\mathbb {N}}^{r}\) with \(\alpha =1\). In this case we have for each positive integer n that
$$\begin{aligned} f_{n\cdot \alpha }(x+y)=\sum _{k=0}^n \left( {\begin{array}{c}n\\ k\end{array}}\right) f_{k\cdot \alpha }(x)f_{(nk)\cdot \alpha }(y) \qquad \end{aligned}$$for each x, y in G. Hence \((f_{n\cdot \alpha })_{n\in {\mathbb {N}}}\) is a generalized moment function sequence associated with the exponential \(m=f_{0\cdot \alpha }\).

(iv)
Using the above definition for \(r=2\), for example for \(\alpha \le 2\) we have
$$\begin{aligned} \begin{array}{rcl} f_{0,0}(x+y)&{}=&{}f_{0,0}(x)f_{0,0}(y)\\ f_{1,0}(x+y)&{}=&{}f_{1,0}(x)f_{0,0}(y)+f_{1,0}(y)f_{0,0}(x)\\ f_{0,1}(x+y)&{}=&{}f_{0,1}(x)f_{0,0}(y)+f_{0,1}(y)f_{0,0}(x)\\ f_{2,0}(x+y)&{}=&{}f_{2,0}(x)f_{0,0}(y)+2f_{1,0}(x)f_{1,0}(y)+f_{2,0}(y)f_{0,0}(x)\\ f_{0,2}(x+y)&{}=&{}f_{0,2}(x)f_{0,0}(y)+2f_{0,1}(x)f_{0,1}(y)+f_{0,2}(y)f_{0,0}(x)\\ f_{1,1}(x+y)&{}=&{}f_{1,1}(x)f_{0,0}(y)+f_{1,0}(x)f_{0,1}(y)\\ &{}&{}+f_{1,0}(y)f_{0,1}(x)+f_{1,1}(y)f_{0,0}(x) \end{array} \qquad \end{aligned}$$for each x, y in G.
In view of the first equation (or using remark (ii)), we immediately get that the function \(f_{0, 0}=m\) is an exponential. Furthermore, due to the second and the third equation, the functions \(f_{0, 1}\) and \(f_{0, 1}\) are msine functions. In other words, \(\left\{ f_{0, 0}, f_{0, 1}\right\} \) and \(\left\{ f_{0, 0}, f_{1, 0}\right\} \) form a moment sequence of order 1. More generally, due to remark (iii), \((f_{0, n})_{n\in {\mathbb {N}}}\) and also \((f_{n, 0})_{n\in {\mathbb {N}}}\) is a moment function sequence associated with the exponential \(f_{0, 0}=m\).

(v)
Assume that \((f_{\alpha })_{\alpha \in {\mathbb {N}}^{2}}\) is a generalized moment sequence of rank two. For all nonnegative integer n, define the function \(\varphi _{n}:G\rightarrow {\mathbb {C}}\) by
$$\begin{aligned} \varphi _{n}(x)= \sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) f_{k, nk}(x) \qquad \end{aligned}$$for x in G. Then \(\left( \varphi _{n}\right) _{n\in {\mathbb {N}}}\) is a moment function sequence associated with the exponential \(\varphi _{0}=f_{0, 0}=m.\)
Indeed, for \(n=0\),
$$\begin{aligned} \varphi _{0}(x)=f_{0, 0}(x) \qquad \end{aligned}$$for x in G, which is an exponential, as we wrote above.
For each positive n, we have
$$\begin{aligned}&\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \varphi _{k}(x)\varphi _{nk}(y)\\&\quad =\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \cdot \left[ \sum _{i=0}^{k}\left( {\begin{array}{c}k\\ i\end{array}}\right) f_{i, ki}(x)\right] \cdot \left[ \sum _{j=0}^{nk}\left( {\begin{array}{c}nk\\ j\end{array}}\right) f_{j, nkj}(y)\right] \\&\quad =\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left[ \sum _{i=0}^{k} \sum _{j=0}^{nk}\left( {\begin{array}{c}k\\ i\end{array}}\right) \left( {\begin{array}{c}nk\\ j\end{array}}\right) f_{i, ki}(x)f_{j, nkj}(y)\right] \\&\quad =\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) f_{k, nk}(x+y)= \varphi _{n}(x+y) \end{aligned}$$for each x, y in G.

(vi)
The sequence of functions \((f_{\alpha })_{\alpha \in {\mathbb {N}}^{r}}\) is a generalized moment sequence of rank r associated with the nonzero exponential \(f_{0, \ldots , 0}=m\) if and only if \(({f_{\alpha }}/{m})_{\alpha \in {\mathbb {N}}^{r}}\) is a generalized moment sequence of rank r associated with the exponential which is identically one.

(vii)
Let \((f_{\alpha })_{\alpha \in {\mathbb {N}}^{r}}\) be a generalized moment sequence of rank r and denote \(m= f_{0, \ldots , 0}\). Then for all multiindex \(\alpha \) in \({\mathbb {N}}^{r}\), the function \(f_{\alpha }\) is a generalized exponential monomial of degree at most \(\alpha \) corresponding to the exponential m. Thus for any multiindex \(\alpha \) in \({\mathbb {N}}^{r}\), there exists an a generalized polynomial \(p_{\alpha }:G\rightarrow {\mathbb {C}}\) of degree at most \(\alpha \) such that \(f_{\alpha }= p_{\alpha } \cdot m\).
If \(\alpha =0\), that is, if \(\alpha = (0, 0,\ldots , 0)\), then \(f_{0, \ldots , 0}=m\) is an exponential. Thus the above statement holds trivially.
In case \(\alpha =1\), then there exists \(i\in \left\{ 1, 2,\ldots , r\right\} \) such that
$$\begin{aligned} \alpha _{i}=1 \quad \text {and} \quad \alpha _{j}=0 \quad \text {for any } j\in \left\{ 1, \ldots , r\right\} , j\ne i. \end{aligned}$$This, in view of remark (iii) implies that \(f_{\alpha }\) is an msine function, i.e.
$$\begin{aligned} f_{\alpha }(x+y)= f_{\alpha }(x)m(y)+m(x)f_{\alpha }(y) \qquad \end{aligned}$$for x, y in G. This implies however that
$$\begin{aligned} {{\varDelta }}_{m; y_{1}, y_{2}}f_{\alpha }(x)=0, \qquad \end{aligned}$$\(x, y_{1}, y_{2}\) in G, in other words \(f_{\alpha }\) is a generalized exponential monomial of degree at most one corresponding to the exponential m.
Assume now that there exists a multiindex \(\alpha \) in \({\mathbb {N}}^{r}\) such that the statement holds for any multiindex \(\beta \) for which \(\beta < \alpha \). Since
$$\begin{aligned} f_{\alpha }(x+y)= \sum _{\beta \le \alpha }\left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) f_{\beta }(x)f_{\alpha \beta }(y) \end{aligned}$$for each \(x, y\in G\), we have
$$\begin{aligned} {{\varDelta }}_{m; y}f_{\alpha }(x) = \sum _{\beta <\alpha }\left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) f_{\beta }(x)f_{\alpha \beta }(y) \qquad \end{aligned}$$for every x, y in G. Observe that the right hand side of this equation (as a functions of the variable x) is, due to the induction hypothesis, a generalized exponential polynomial of degree at most \(\alpha 1\), showing that \(f_{\alpha }\) is a generalized exponential monomial of degree at most \(\alpha \).

(viii)
The previous remark can be strengthened. Namely, generalized moment sequences of rank r not only generalized exponential monomials, but they are exponential monomials. This means that the variety of any such function is finite dimensional. Indeed, let \((f_{\alpha })_{\alpha \in {\mathbb {N}}^{r}}\) be a generalized moment sequence of rank r. Then for any multiindex \(\alpha \) in \({\mathbb {N}}^{r}\) we have
$$\begin{aligned} f_{\alpha }(x+y)= \sum _{\beta \le \alpha }\left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) f_{\beta }(x)f_{\alpha \beta }(y) \qquad \end{aligned}$$for every x, y in G showing that
$$\begin{aligned} \tau (f_\alpha )\subseteq \mathrm {span}\left\{ f_{\beta }\, \vert \, \beta \le \alpha \right\} . \end{aligned}$$The linear space on the right hand side is obviously finite dimensional, implying the same for \(\tau (f)\).
We conclude that the polynomial \(p_{\alpha }\), appearing in remark (viii), is a complex polynomial of some complexvalued additive functions defined on G.
We can summarize that, for any multiindex \(\alpha \in {\mathbb {N}}^{r}\), there exists a positive integer \(n(\alpha )\) and a \(P_{\alpha }:{\mathbb {C}}^{n(\alpha )}\rightarrow {\mathbb {C}}\) and additive functions \(a_{i}:G\rightarrow {\mathbb {C}}\), \(i=1, \ldots , n(\alpha )\) such that
$$\begin{aligned} f_{\alpha }(x)= P_{\alpha }(a(x))m(x) \end{aligned}$$for each \(x\in G\). Here \(a_{i}\) denotes the \(i^\mathrm{th}\) coordinate functions of the additive function a.

(ix)
If k and \(1\le n_{1}< n_{2}< \cdots < n_{k} \le r\) are positive integers, then let \(\pi _{n_{1}, \ldots , n_{k}}\alpha \) denote that element in \({\mathbb {N}}^{r}\) which differs from \(\alpha \) in only those coordinates that do not belong to the set \(\left\{ n_{1}, \ldots , n_{k}\right\} \), and those coordinates are zero.
With this notation, let k and r be positive integers with \(k\le r\) and \(1\le n_{1}< n_{2}< \cdots < n_{k} \le r\) be also positive integers. If \((f_{\alpha })_{\alpha \in {\mathbb {N}}^{r}}\) is a generalized moment sequence of rank r on the group G, then the sequence of functions \((f_{\pi _{n_{1}, \ldots , n_{k}}\alpha })_{\alpha \in {\mathbb {N}}^{r}}\) form a generalized moment sequence of rank k.
As we will see, while describing the polynomial P appearing in remark (ix), a wellknown sequence of polynomials and an appropriate addition formula will play a distinguished role. Let us consider the sequence of complex polynomials \(\left( B_{n}\right) _{n\in {\mathbb {N}}}\) defined through the following recurrence: for each \(t,t_1,t_2,\dots ,t_{n+1}\) in \({\mathbb {C}}\) we let
for each n in \({\mathbb {N}}\). Alternatively, we can also use the double series expansion of the generating function
We call \(B_{n}\) the \(n^\mathrm{th}\) complete (exponential) Bell polynomial.
For these functions the following addition formula holds:
for any positive integer n, and for each complex numbers \(t_{1}, \ldots , t_{n}\) and \(u_{1}, \ldots , u_{n}\).
The first few complete (exponential) Bell polynomials are
It turns out, however, that these functions are useful in the rank one case, only. To cover the general case as well, a multivariate extension of the Bell polynomials is necessary. Here we follow the notation and the terminology of [7].
The multivariate Bell polynomials can be introduced using the double series expansion of the generating function
Let now \(\alpha = (\alpha _{k})_{k\in {\mathbb {N}}}\) be a composition and \(x=(x_{k})_{k\in {\mathbb {N}}}\) and \(y=(y_{k})_{k\in {\mathbb {N}}}\) be sequences of variables. Then
For instance, if \(r=2\), then the first few multivariate Bell polynomials are the following:
It is important to point out that if \(r=1\) then the multivariate Bell polynomials reduce to to the complete (exponential) Bell polynomials. For any multiindex \(\alpha \in {\mathbb {N}}^{r}\), the polynomial \(B_{\alpha }\) has at most \(\prod _{i=1}^{r}(\alpha _{i}+1)\) variables, where \(\alpha = \left( \alpha _{1}, \ldots , \alpha _{r}\right) \) and its degree is exactely \(\alpha \).
Proposition 3
Let G be a commutative group, r a positive integer, \(m:G\rightarrow {\mathbb {C}}\) an exponential and let \(a= \left( a_{\alpha }\right) _{\alpha \in {\mathbb {N}}^{r}}\) be a sequence of complexvalued additive functions defined on G. We define the functions \(f_{\alpha }:G\rightarrow {\mathbb {C}}\) by
for every x in G. Then \((f_{\alpha })_{\alpha \in {\mathbb {N}}^{r}}\) forms a generalized moment sequence of rank r associated with the exponential m.
Proof
If \(\alpha =0\), then the above formula is
for each x in G, that is, \(f_{0, \ldots , 0}\) is an exponential.
Let now \(\alpha \) be in \({\mathbb {N}}^{r}\) with \(\alpha >0\), then
for each x, y in G. \(\square \)
The following result is about the converse of the previous statement.
Theorem 2
Let G be a commutative group, r a positive integer, and for each \(\alpha \) in \({\mathbb {N}}^{r}\), let \(f_{\alpha }:G\rightarrow {\mathbb {C}}\) be a function. If the sequence of functions \((f_{\alpha })_{\alpha \in {\mathbb {N}}^{r}}\) forms a generalized moment sequence of rank r, then there exists an exponential \(m:G\rightarrow {\mathbb {C}}\) and a sequence of complexvalued additive functions \(a= (a_{\alpha })_{\alpha \in {\mathbb {N}}^{r}}\) such that for every multiindex \(\alpha \) in \({\mathbb {N}}^{r}\) and x in G we have
Proof
Let r be a fixed positive integer, G be a commutative group, let furthermore for each \(\alpha \in {\mathbb {N}}^{r}\), \(f_{\alpha }:G\rightarrow {\mathbb {C}}\) be a function. Assume that the sequence of functions \((f_{\alpha })_{\alpha \in {\mathbb {N}}^{r}}\) forms a generalized moment sequence of rank r. We define a sequence \(a=(a_\alpha )\) and prove the statement by induction on the height of the multiindex \(\alpha \).
If \(\alpha =0\), then we necessarily have \(\alpha = (0, 0,\ldots , 0)\) and
for every x, y in G, which gives immediately that there exists an exponential \(m:G\rightarrow {\mathbb {C}}\) such that \( f_{0, 0,\ldots , 0}= m\). Furthermore, if \(\alpha =1\), then there exists \(i\in \left\{ 1, 2,\ldots , r\right\} \) such that \(\alpha _{i}=1\) and \(\alpha _{j}=0\) for each j in \(\{1, 2,\ldots , r\}\), \(j\ne i\). In view of remark (iii) this implies that \(f_{\alpha }\) is an msine function, that is
holds for x in G.
Now we assume that there exists a multiindex \(\alpha \) in \({\mathbb {N}}^{r}\) such that the statement holds true for every multiindex \(\beta \) with \(\beta < \alpha \). This means that for any multiindex \(\beta <\alpha \) there exists an additive function \(a_{\beta }:G\rightarrow {\mathbb {C}}\) such that the representation in the statement of the theorem holds. We have to show how the additive function \(a_{\alpha }:G\rightarrow {\mathbb {C}}\) should be constructed. Since \(f_{\alpha }\) is a generalized moment function of order \(\alpha \) and of rank r, we have
for every x, y in G. Then, due to the addition formula of the function \(B_{\alpha }\), we have that
holds for each x, y in G. Observe, that this means that the function \(S_{\alpha }:G\rightarrow {\mathbb {C}}\) defined by
for x in G is an msine function. Thus, there exists an additive function \(\eta :G\rightarrow {\mathbb {C}}\) such that \(S(x)= \eta (x)m(x)\) for all x in G, yielding that
for each x in G, since \(B_{\alpha }\) is additive in its last variable.
This means that there exists an additive function \(\xi :G\rightarrow {\mathbb {C}}\) such that
for each x in G. This shows that how the \(\alpha ^\mathrm{th}\) element of the sequence a should be constructed. \(\square \)
As a consequence of this result, the characterization of generalized moment functions (of rank one) follows. Namely, functions of the form \((B_{n}\circ a)\cdot m\) are generalized moment functions. More precisely, we have the following.
Corollary 1
Let G be a commutative group, n an arbitrary positive integer, \(m:G\rightarrow {\mathbb {C}}\) an exponential, and let \(a_{1}, a_2,\ldots , a_{n}:G\rightarrow {\mathbb {C}}\) be additive functions. We define the sequence of functions \((f_{n})_{n\in {\mathbb {N}}}\) for each x in G by
Then \((f_{n})_{n\in {\mathbb {N}}}\) forms a generalized moment sequence associated with the exponential m.
On the other hand, we can deduce the following result:
Corollary 2
Let G be a commutative group, and for each natural number n let \(f_{n}:G\rightarrow {\mathbb {C}}\) be a function. If the sequence \((f_{n})_{n\in {\mathbb {N}}}\) forms a generalized moment sequence, then there exists an exponential \(m:G\rightarrow {\mathbb {C}}\) and there are additive functions \(a_{1}, a_2,\ldots , a_{n}:G\rightarrow {\mathbb {C}}\) such that
holds for every x in G and \(n=1,2,\dots \).
It should be noted that the results form this section coincide with the formula (2) on the page 2 proved by J. Acz’el. [1] in a much more general settings.
4 Summary and Further Research
In this paper we have introduced the notion of generalized moment functions of higher rank defined on a commutative group. We characterize them as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Our next aim is to find a characterization of generalized moment functions of higher rank in the hypergroup settings.
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Funding
The research of E. Gselmann has partially been carried out with the help of the Project 20192.1.11TÉT201900049, which has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary The research of E. Gselmann and L. Székelyhidi has been supported by the NKFIH Grant No. K 134191.
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Fechner, Ż., Gselmann, E. & Székelyhidi, L. Moment Functions on Groups. Results Math 76, 171 (2021). https://doi.org/10.1007/s00025021014676
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DOI: https://doi.org/10.1007/s00025021014676