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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.

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