Functional equation characterizing polynomial functions and an algorithm (Q2134905)

The main aim of this paper is to solve the functional equation \[ F(x+y)-F(x)-F(y)= \sum_{i=1}^{n}(a_{i}x+b_{i}y)f(\alpha_{i}x+\beta_{i}y) \qquad \left(x, y\in \mathbb{R}\right),\tag{\(\ast\)} \] for the unknown functions \(f, F\colon \mathbb{R}\to \mathbb{R}\), where for all \(i=1, \ldots, n\), the parameters \(a_{i}, b_{i}\) are fixed and real, while \(\alpha_{i}, \beta_{i}\) are fixed and rational. \textit{T. Nadhomi} et al. [Aequationes Math. 95, No. 6, 1095--1117 (2021; Zbl 1482.39032)] considered the following special case \[ \sum_{i=1}^{n}\gamma_{i}F(\alpha_{i}x+\beta_{i}y)= xf(y)+yf(x). \] The main tool in [loc. cit.] and also in this paper is a lemma about (generalized) polynomials (see Lemma 1.1 in this paper). Based on this, the authors prove for equation \((\ast)\) that if there exists \(i\in \left\{1, \ldots, n \right\}\) such that \[ \det \begin{pmatrix} \alpha_{i} & \beta_{i}\\ a_{i} & b_{i} \end{pmatrix} \neq 0,\tag{D} \] \(f\) is a polynomial function of degree not greater than \(2n\) and \(F\) is a polynomial function of degree not greater than \(2n + 1\). The authors show that condition (D) is essential. More concretely, they provide a counterexample for the case \(n=1\). Finally, the last section of the paper contains a procedure, called ``psfe'' (polynomial solution of functional equation) written in Maple which can be used to obtain the polynomial solutions of functional equations of type \((\ast)\).