ABSTRACT
Let n be a fixed positive integer and let f be a real valued function defined on the set {x0, . . . , xn} of n + 1 distinct points. The nth order difference is defined by
Qn(f ;x0, . . . , xn) = Qn−1(f ;x0, . . . , xn−1)−Qn−1(f ;x1, . . . , xn)
x0 − xn , n ≥ 2,
(1.1.1) with
Q1(f ;xi, xj) = f(xi)− f(xj)
xi − xj , i 6= j, i, j = 0, . . . , n. (1.1.2)
A simple inductive argument, given below, shows that
Qn(f ;x0, . . . , xn) =
f(xi)∏n j=0 i6=j
(xi − xj) =
f(xi)
ω′(xi) , (1.1.3)
where
ω(x) =
(x − xj).
