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Note on polynomial functions. (English) Zbl 0805.39010
The main result of the paper is the following: assume that $$(X,+)$$ and $$(Y,+)$$ are commutative groups admiting division by $$(n+1)!$$ and let $$C \subset X$$ be such that $$C + C \subset C$$, $$C - C = X$$ and $$(1/(n + 1)!) C \subset C$$. If $$f:X \to Y$$ is a $$C$$-polynomial function of $$n$$-th order, i.e., $$\sum^{n + 1}_{k = 0} (-1)^{n + 1 - k} {n + 1 \choose k} f(x + kh) = 0$$ for every $$x \in X$$ and $$h \in C$$, then $$f$$ must be a polynomial function of $$n$$-th order, i.e., the previous equality holds for all $$x$$, $$h \in X$$. The author gives an alternative proof of this result, first obtained by Roman Ger.

##### MSC:
 39B52 Functional equations for functions with more general domains and/or ranges