A functional identity characterizing polynomials
type: Item
instance of: scholarly article
title: A functional identity characterizing polynomials
zbMATH Open document ID: 0633.39005
DOI: 10.1007/BF01830666
author: George Cross
author: Pl. Kannappan
published in: Aequationes Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137234
review text: If \(f: {\mathbb{R}}\to {\mathbb{R}}\) is differentiable everywhere, then \((1)\quad (f(x)-f(y))/(x-y)=(f'(x)+f'(y))/2,\) x,y\(\in {\mathbb{R}}\), \(x\neq y\), if and only if f is a polynomial of degree less or equal to 2. The authors generalize the result above for higher order derivative. The main result is the following: The functions \(f,g_ k,h: {\mathbb{R}}\to {\mathbb{R}}\), \(k=0,1,...,n\), satisfy \[ f(x)=\sum^{n}_{k=0}(x-y)^ kg_ k(y)+(x- y)^ nh(x),\quad x,y\in {\mathbb{R}},\quad x\neq y,\quad n\geq 1, \] if and only if f is a polynomial of degree less or equal to \(n+1\), \(g_ k(y)=f^{(k)}(y)/k!\) \((k=0,1,...,n-1)\), \(g_ n(y)=f^{(n)}(y)/n!- yf^{(n+1)}(y)/(n+1)!-C,\quad h(x)=xf^{(n+1)}(x)/(n+1)!+C\) and C is an arbitrary constant. To obtain the identity generalizing (1), Peano derivatives are used.
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26C05
zbMATH DE Number: 4030120
zbMATH Keywords: characterization of polynomials by functional equations
zbMATH Keywords: Peano derivative
reviewed by: Gian Luigi Forti
MaRDI profile type: Publication
cites work: A Property of Quadratic Polynomials
cites work: A Riemann-Type Integral of Lebesgue Power
cites work: Q3946362
cites work: The Exact Peano Derivative
OpenAlex ID: W1982696997
description: scientific article; zbMATH DE number 4030120