×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite