On the construction of functional equations with prescribed general solution

Abstract

Given rational vector spaces V, W a mapping ${f \colon V \to W}$ is called a generalized polynomial of degree at most n, if there are homogeneous generalized polynomials f _i of degree i such that ${f = \sum_{i = 0}^n f_i}$ . Homogeneous generalized polynomials f _i of degree i are mappings of the form ${f_i (x) = f_i^*(x, x, \ldots , x)}$ with ${f_i^* \colon V^i \to W i}$ -linear. In the literature one may find quite a lot of functional equations such that their general solution is of the form f _n or ${f_n + f_{n - 1}}$ where n is a small positive integer ( ≤ 6 or ≤ 4 respectively). In this paper, given an arbitrary positive integer n and an arbitrary subset ${L \subseteq \{0, 1, \ldots, n\}}$ such that ${n \in L}$ , a method is described to find (many) functional equations, such that their general solution is given by ${\sum_{i \in L} f_i}$ . For the cases ${L = \{n\}}$ and ${L = \{n - 1, n\}}$ additional equations are given.