A functional equation characterizing cubic polynomials and its stability (Q5957222)

\textit{J. Schwaiger} [Aequationes Math. 48, No. 2-3, 317-323 (1994; Zbl 0810.39007), cf. also \textit{K. M. Andersen}, Math. Mag. 69, No. 2, 137-142 (1996; Zbl 0853.39011); neither of the two quoted in the present paper] gave an elementary proof of the following result. For functions \(f,\) mapping a field \(F\) with sufficiently many elements and of characteristic different from 2 into itself, the \(n\)-th divided difference \(f[x_1,\dots ,x_n]\) is a function \(h\) of (only) \(x_1+\dots +x_n\) \((x_j\neq x_k\) for \(j\neq k\); \(x_k\in F\); \(k\in\{1,\dots ,n\})\) for fixed \(n\geq 2\) if, and only if, \(h\) is linear and \(f\) a polynomial of degree at most \(n,\) the first two coefficients in \(f\) being the same as those in \(h.\) NEWLINENEWLINENEWLINEThe first part of the present paper offers a proof for the particular case \(F=\mathbb{R}\), \(n=3\) by reducing it to the \(n=2\) case. In the second part stability results are offered for the corresponding functional equation.