A functional identity characterizing polynomials
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.
characterization of polynomials by functional equations
scientific article; zbMATH DE number 4030120