On a new class of functional equations satisfied by polynomial functions

The classical result of L. Szekelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Szekelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F(x + y) - F(x) - F(y) = yf(x) + xf(y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1-2):143-154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.