On a characterization of polynomials by divided differences

Summary

We consider the functional equationf[x ₁,x ₂,⋯, x _n ] =h(x ₁ + ⋯ +x _n ) (x ₁,⋯,x _n ∈K, x _j ≠x _k forj ≠ k), (D) wheref[x ₁,x ₂,⋯,x _n ] denotes the (n − 1)-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 a_j ∈ K, 0 ≤ j ≤ n (a := a_n, b := a_{n − 1}) such thatf = ax ⁿ +bx ^{n − 1} + ⋯ +a ₀ and h = ax + b.