On a new class of functional equations satisfied by polynomial functions

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