On a new class of functional equations satisfied by polynomial functions
The aim of the paper is to present a new class of functional equations whose solutions are polynomial functions. After recalling the notion of polynomial function and the well-known result of \textit{L. Székelyhidi} [Convolution type functional equations on topological abelian groups. Singapore etc.: World Scientific (1991; Zbl 0748.39003)] stating that every solution of a very general linear functional equation is a polynomial function, the authors, as a first step, investigate the nonlinear equation \[ F(x+y)-F(x)-F(y)=yf(x)+xf(y). \tag{1} \] The main tool is a very general lemma extending \textit{L. Székelyhidi}'s result, too long to be reported here. This lemma provides an interesting systematic approach valid for several classes of functional equations. The first consequence of the lemma is the following theorem. Theorem. The pair \((f,F)\) is a solution of the functional equation (1) if and only if \[ f(x)=A_1(x)+a_2x^2, \quad F(x)=B_1(x)+xA_1(x)+\frac{1}{3}a_2x^3, \] for all \(x \in \mathbb{R}\), where \(A_1\) and \(B_1\) are arbitrary additive functions, and \(a_2 \in \mathbb{R}\) is an arbitrary constant. Then the more general functional equation \[ \sum_{i=1}^n \gamma_i F(\alpha_i x+\beta_i y)=yf(x)+xf(y) \tag{2} \] is investigated and the following result is proved. Theorem. Let \(\alpha_i, \beta_i, \gamma_i\), \(i\in \{1,\cdots,n\}\), be real numbers such that \(\sum_{i=1}^n \gamma_i \neq 0\) and \(\alpha_i+\beta_i=1\), \(i\in \{1,\cdots,n\}\). If the pair \((f,F)\) of mapping from \(\mathbb{R}\) into \(\mathbb{R}\) is a solution of the functional equation (2) then the functions \(f\) and \(F\) are polynomial. It is then proved that the solutions of the following functional equations \[ \Delta_y^n F(x)=yf(x)+xf(y), \] \[ \Delta_y^n F(x)-n!F(y)=yf(x)+xf(y), \] \[ \sum_{i=1}^n \gamma_i F(x+\beta_i y)+\gamma_{n+1}F(y)=yf(x)+xf(y), \] are polynomial.
scientific article; zbMATH DE number 7439027
scientific article