摘要：Let ( G , ⋅ ) $(G, \cdot)$ be a group, ( H , + ) $(H, +)$ be an Abelian group, and f : G → H $f:G\rightarrow H$ be a function. In this paper, for a positive integer n, we first give a representation of nth-order Cauchy difference of f via the function as C ( n ) f ( x 1 , x 2 , … , x n , x n + 1 ) = ∑ 1 ≤ m ≤ n + 1 ( − 1 ) n + 1 − m ∑ 1 ≤ i 1 < i 2 < ⋯ < i m ≤ n + 1 f ( x i 1 x i 2 ⋯ x i m ) , $$ C^{(n)}f(x_),x_,,\ldots,x_{n},x_{n+1})= \sum_{1\leq m \leq n+1}(-1)^{n+1-m}\sum _{1\leq i_)< i_,< \cdots< i_{m}\leq n+1}f(x_{i_)}x_{i_,}\cdots x_{i_{m}}), $$ where x 1 , x 2 , … , x n + 1 ∈ G $x_), x_,,\ldots, x_{n+1}\in G$ . Then, based on the representation, we get some special solutions of C ( n ) f = 0 $C^{(n)}f=0$ on free groups. Moreover, sufficient and necessary conditions on symmetric groups and finite cyclic groups are also obtained.
关键词：Cauchy difference ; free group ; symmetric group ; cyclic group
