On a new class of functional equations satisfied by polynomial functions
type: Item
instance of: scholarly article
title: On a new class of functional equations satisfied by polynomial functions
zbMATH Open document ID: 1482.39032
author: Timothy Nadhomi
author: Chisom Prince Okeke
author: Tomasz Szostok
author: Maciej Sablik
published in: Aequationes Mathematicae
publication date: 2021-12-06T00:00:00Z
review text: 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.
reviewed by: Gian Luigi Forti
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 7439027
zbMATH Keywords: functional equations
zbMATH Keywords: polynomial functions
zbMATH Keywords: monomial functions
zbMATH Keywords: Fréchet operator
zbMATH Keywords: continuity of monomial functions
MaRDI profile type: Publication
full work available at URL: s00010-021-00781-2
OpenAlex ID: W3130005040
cites work: A Mean Value Property of the Derivative of Quadratic Polynomials-without Mean Values and Derivatives
cites work: On two mean value properties and functional equations associated with them
cites work: On polynomial functions on non-commutative groups
cites work: Functional equations and the Cauchy mean value theorem
cites work: Computer assisted solution of systems of two variable linear functional equations
cites work: On Functions Whose Mean Value Abscissas Are Midpoints, with Connections to Harmonic Functions
cites work: A representation theorem for $(X_1-1)(X_2-1)...(X_n-1)$ and its applications
cites work: General and alien solutions of a functional equation and of a functional inequality
cites work: On Sahoo-Riedel equations on a real interval
cites work: Alien Functional Equations: A Selective Survey of Results
cites work: Q4269211
cites work: A Property of Quadratic Polynomials
cites work: On some functional equations connected to Hadamard inequalities
cites work: Q3645793
cites work: On a class of equations stemming from various quadrature rules
cites work: On a functional equation connected to Hermite quadrature rule
cites work: Q3405404
cites work: Q3565768
cites work: Q4914684
cites work: Q4807453
cites work: Q4243455
cites work: Q4277665
cites work: Taylor's theorem and functional equations
cites work: Q4684638
cites work: Q4520097
cites work: A functional equation related to symmetry of operators
cites work: Each semipolynomial on a group is a polynomial
cites work: Q4002300
cites work: Functional equations stemming from numerical analysis
cites work: Alienation of two general linear functional equations
DOI: 10.1007/S00010-021-00781-2
description: scientific article; zbMATH DE number 7439027