A generalization of the mean mean value theorem (Q804851)

A proof is sketched that, for an \(f\in C^{n+2}(I)\) (I a real interval of positive length), the n-th divided difference at all \((x_ 0,x_ 1,...,x_ n)\in I^{n+1}\) equals \(f^{(n)}((x_ 0+x_ 1+...+x_ n)/(n+1))/n!\) if, and only if, f is a polynomial of at most \((n+1)\)-st degree on I. \{For similar, but not identical, results see \textit{G. E. Cross} and \textit{Pl. Kannappan} [Aequationes Math. 34, 147-152 (1987; Zbl 0633.39005)]; \textit{Pl. Kannappan} and \textit{B. Crstici} [Prepr., ``Babeş-Bolyai'' Univ., Fac. Math. Phys., Res. Semin. 1989, No.6, 175-180 (1989; Zbl 0683.39006)] and in the abstract from \textit{Pl. Kannappan} and \textit{R. Ger} [Report on the twenty-ninth international symposium on functional equations, June 3-10, 1991, Wolfville, NS, to appear in Aequationes Math. 43 (1992)].\}