On solutions of functional equations with polynomial translations

In this paper, we study polynomial functional equations of the form af(p(x)) + bf(q(x)) = g(x), where p(x), q(x) are given polynomials and g(x) is a given function. Theorems 21 and 22 contain sufficient conditions under which the functional equation has a solution of the special form. In Section 3 we present an algorithm of constructing polynomial solutions of the functional equations. Other non-polynomial solutions depend on solutions of the homogeneous equation af(p(x)) + bf(q(x)) = 0. That case is analyzed in Section 4. Finally, we present a simple method of constructing examples with desirable properties.