Aequationes Mathematicae 48 (1994) 317-323 0001-9054/94/030317-07 $1.50 + 0.20/0
University of Waterloo © 1994 Birkh/iuser Verlag, Basel
On a characterization of polynomials by divided differences
JENS SCHWAIGER
Summary. We consider the functional equation
f[xl,x2 ..... x,l=h(xl+'"+x,) (Xl..... x, eK, xj~xkforjv~k), (O)
where f[xl, x z ..... x,] denotes the (n - l)-st divided difference off and prove
THEOREM. Let n be an integer, n ~ 2, let K be a field, char(K) # 2, with # K >- 8(n - 2) + 2. Let,
furthermore, f, h: K ~ K be functions. Then we have that f, h fulfil (D) if, and only if, there are constants
aj E K, 0 < j -< n (a:= a,, b== a, _ 1) such that
f=ax"+bx'-l+'''+ao and h=ax+b.
D. F. Bailey [B] shows that polynomials of degree at most 3 with real coefficients
can be characterized as the differentiable solutions f of the functional equation
f[x,y,z]=h(x÷y+z) (x,y,z~R,x~yvLz~x). (1)
(The divided difference f[x, y, z] is defined below.)
To get his result Bailey uses a theorem of J. Acz61 [A], stating that all solutions
f, g, h: K ~ K, K any field of characteristic different from 2, of
f(x) - g(y)
= h(x + y) (x, y ~ K, x ~ y) (2)
x-y
are given by f(x) = g(x) = ax 2 + bx + c, h(x) = ax + b. Furthermore Bailey men-
tions that (1) could also be solved--resulting in the same solutions--by assuming
AMS (1991) subject classification: 39B52.
Manuscript received November 4, 1992 and, in final form, April 1, 1993.
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