Some functional equations characterizing polynomials

Abstract

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 $P$ and
parameteres
$b_1,\dots,b_n;\alpha_1,\dots,\alpha_n;\beta_1,\dots,\beta_n\in P$
are given.
We prove that under some assumptions on the parameters involved, all solutions
to such kind of equations are polynomials.  We use this method to solve some
concrete 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 rule
used in numerical analysis.