On a functional equation connected to Hermite quadrature rule

In this paper we deal with the functional equationF(y)−F(x)=(y−x)[αf(x)+βf(x+y2)+αf(y)]+(y−x)2[g(y)−g(x)], which is connected to Hermite quadrature rule. It is easy to note that particular cases of this equation generalize many well known functional equations connected to quadrature rules and mean value theorems. Thus the set of solutions is too complicated to be described completely and therefore we prove that (under some assumptions) all solutions of the above equation must be polynomials. We obtain the aforementioned result using a lemma proved by M. Sablik, however this lemma works only in case β≠0. Taking β=0, we obtain the following equationF(y)−F(x)=(y−x)[f(x)+f(y)]+(y−x)2[g(y)−g(x)], which is also solved in the paper.