Abstract
In this paper generalized moment functions are considered. They are closely related to the well-known 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.
Highlights
K j φj (x)φk−j(y) j=0 for all x and y in G and k = 0, 1, . . . , N. We generalize this concept by relaxing the assumption φ0 = 1 to φ0(0) = 1. In this case φ0 is an arbitrary exponential function and we say that φ0 generates the generalized moment sequence of order N and the function φk is a generalized moment function of order k, or, if we want to specify the exponential φ0, we say that φk is a generalized moment function of order k associated with the exponential φ0
Problems about moment functions have been extensively studied on different type of abstract structures, in particular on hypergroups
In this paper we have introduced the notion of generalized moment functions of higher rank defined on a commutative group
Summary
Let (G, +) be an Abelian group. Recall that a nonzero function m : G → C is called exponential, if m(x + y) = m(x)m(y) holds for all x, y in G. Let N be a nonnegative integer. A function φ : G → C is termed to be a moment function of order N , if there exist functions φk : G → C such that φ0 = 1, φN = φ and Z . Fechner et al.
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