Page 27 - Foundations Of Differential Calculus
P. 27
10 1. On Finite Differences
15. With the aid of this table we can write each of the differences of the
n
powers y = x as follows:
n−1 2 n−2 3 n−3 4 n−4
∆y = Aωx + Bω x + Cω x + Dω x + ··· ,
2 2 n−2 3 n−3 4 n−4
∆ y =2Bω x +6Cω x +14Dω x + ··· ,
3 3 n−3 4 n−4 5 n−5
∆ y =6Cω x +36Dω x + 150Eω x + ··· ,
5 n−5
4 n−4
4
6 n−6
∆ y =24Dω x + 240Eω x + 1560Fω x + ··· .
n
m
In general, the difference of order m of the power x , that is ∆ y,is
expressed in the following way.
Let
n (n − 1) (n − 2) ··· (n − m +1)
I = ,
1 · 2 · 3 ··· m
n − m
K = I,
m +1
n − m − 1
L = K,
m +2
n − m − 2
M = L,
m +3
··· .
Then we let
m m (m − 1)
m m m
α =(m +1) − m + (m − 1)
1 1 · 2
m (m − 1) (m − 2) m
− (m − 2) + ··· ,
1 · 2 · 3
m+1 m m+1 m (m − 2) m+1
β =(m +1) − m + (m − 1)
1 1 · 2
m (m − 1) (m − 2) m+1
− (m − 2) + ··· ,
1 · 2 · 3
m+2 m m+2 m (m − 1) m+2
γ =(m +1) − m + (m − 1)
1 1 · 2
m (m − 1) (m − 2) m+2
− (m − 2) + ··· .
1 · 2 · 3
With these definitions we can write
m n−m
m
x
x
∆ y = αIω x + βKω m+1 n−m−1 + γLω m+2 n−m−2 + ··· .
