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On a characterization of polynomials by divided differences
Authors:Jens Schwaiger
Institution:(1) Institut für Mathematik, Karl-Franzens-Universität, A-8010 Graz, Austria
Abstract:Summary We consider the functional equationfx 1,x 2,ctdot, x n ] =h(x 1 + ctdot +x n ) (x 1,ctdot,x n isinK, x j nex k forj ne k), (D) wherefx 1,x 2,ctdot,x n ] denotes the (n – 1)-st divided difference off and prove Theorem. Let n be an integer, n ge 2, let K be a field, char(K) ne 2, with # K ge 8(n – 2) + 2. Let, furthermore, f, h: K rarr K be functions. Then we have that f, h fulfil (D) if, and only if, there are constants aj isin K, 0 le j le n (a := an, b := an – 1) such thatf = ax n +bx n – 1 + ctdot +a 0 and h = ax + b.
Keywords:39B52
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