Page 28 - Foundations Of Differential Calculus

1. On Finite Differences 11
I II
This result follows immediately from all of the differences of y, y ,y ,
III
y ,... .
16. From what we have seen it is clear that if the exponent n is a positive
integer, sooner or later we obtain a constant difference, and thereafter all
differences vanish. Thus we have

∆.x = ω,
2 2 2
∆ .x =2ω ,
3 3 3
∆ .x =6ω ,
4 4 4
∆ .x =24ω ,

and finally,

n
n
∆ .x =1 · 2 · 3 ··· nω n

(see paragraph 146 for an explanation of this notation). It follows that every
polynomial finally arrives at a constant difference. For instance, the linear
function of x, ax+b, has for a first difference the constant aω. The quadratic
2
2
function ax + bx + c has for second difference the constant 2aω . A third-
degree polynomial has its third difference constant; the fourth degree has
its fourth difference constant, and so forth.
17. The method whereby we find the differences of powers x n can be
further extended to exponents that are negative, a fraction, or even an
irrational number. For the sake of clarity we will discuss only the first
differences of powers with these kinds of exponents, since the law for second
and higher differences is not so easily seen. Let

∆.x = ω,
2 2
∆.x =2ωx + ω ,
3 2 2 3
∆.x =3ωx +3ω x + ω ,
4 3 2 2 3 4
∆.x =4ωx +6ω x +4ω x + ω ,
....