On solutions of functional equations with linear translations

Abstract

In this paper we study the polynomial functional equations of the form $a f(a_1 x +a_0) +b f(b_1 x +b_0)=g(x)$,where  g(x) is a polynomial of the degree $n \geq  0$. Theorem  \ref{T2.3} affirms that the given equation has a unique polynomial solution provided if $a a_1^i + b b_1^i \not= 0$ for each integer $i \geq  0$. Other non-polynomial solution depends on solutions of the homogeneous equation   $a f(a_1 x +a_0) +b f(b_1 x +b_0) = 0$.