We present a method of solving functional equations of the type$$F(x)-F(y)=(x-y)[b_1f(\alpha_1x+\beta_1y)+\dots+b_nf(\alpha_nx+\beta_ny)],$$where $f,F:P\to P$ are unknown functions acting on an integral domain $P$ andparameteres $b_1,\dots,b_n;\alpha_1,\dots,\alpha_n;\beta_1,\dots,\beta_n\in P$ are given.We prove that under some assumptions on the parameters involved, all solutionsto such kind of equations are polynomials. We use this method to solve someconcrete equations of this type. For example, the equation\begin{equation}8[F(x)-F(y)]=(x-y)\left[f(x)+3f\left(\frac{x+2y}{3}\right)+3f\left(\frac{2x+y}{3}\right)+f(y)\right]\label{simp}\end{equation}for $f,F:\Rz\to\Rz$ is solved without any regularity assumptions.It is worth noting that (\ref{simp}) stems from a well-known quadrature ruleused in numerical analysis.
@article{35,
title = {Some functional equations characterizing polynomials},
journal = {Tatra Mountains Mathematical Publications},
volume = {43},
year = {2009},
doi = {10.2478/tatra.v44i0.35},
language = {EN},
url = {http://dml.mathdoc.fr/item/35}
}
Koclęga-Kulpa, Barbara; Szostok, Tomasz; Wąsowicz, Szymon. Some functional equations characterizing polynomials. Tatra Mountains Mathematical Publications, Tome 43 (2009) . doi : 10.2478/tatra.v44i0.35. http://gdmltest.u-ga.fr/item/35/