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.

本文考虑了广义矩函数。它们与众所周知的二项式函数密切相关，这些函数已经在各种抽象结构上进行了研究。这项工作的主要目的是证明可交换群上广义矩函数的刻画定理。首先给出了定义在交换群上的矩函数的多元表征。接下来介绍高阶广义矩函数的概念，并列出群的一些基本性质。给出了通过完全（指数）贝尔多项式表征指数多项式。主要结果是对交换群上定义的更高阶广义矩函数的描述，该函数是多元贝尔多项式和加性函数的指数和复合的乘积。此外，还陈述了秩为一的广义矩函数的推论。论文最后讨论了一些可能的进一步研究方向。