Page 55 - Foundations Of Differential Calculus

38 2. On the Use of Differences in the Theory of Series
62. Hence, for various values of n we have the following:
0
S.x = x,
1 1 2 1
S.x = x + x,
2 2
2 1 3 1 2 1
S.x = x + x + x,
3 2 6
3 1 4 1 3 1 2
S.x = x + x + x ,
4 2 4
4 1 5 1 4 1 3 1
Sx = x + x + x − x,
5 2 3 30
5 1 6 1 5 5 4 1 2
S.x = x + x + x − x ,
6 2 12 12
6 1 7 1 6 1 5 1 3 1
S.x = x + x + x − x + x,
7 2 2 6 42
7 1 8 1 7 7 6 7 4 1 2
S.x = x + x + x − x + x ,
8 2 12 24 12
8 1 1 8 2 7 7 5 2 3 1
S.x = + x + x − s + x − x,
9 2 3 15 9 30
9 1 10 1 9 3 8 7 6 1 4 3 2
S.x = x + x + x − x + x − x ,
10 2 4 10 2 20
10 1 11 1 10 5 9 7 5 1 3 5
S.x = x + x + x − x + x − x + x,
11 2 6 2 66
11 1 12 1 11 11 10 11 8 11 6 11 4 5 2
S.x = x + x + x − x + x − x + x ,
12 2 12 8 6 8 12
12 1 13 1 12 11 11 9 22 7 33 5 5 3 691
S.x = x + x + x − x + x − x + x − x,
13 2 6 7 10 3 2730
13 1 14 1 13 13 12 143 10 143 8 143 6 691 2
S.x = x + x + x − x + x − x − x ,
14 2 12 60 28 20 420
14 1 15 1 14 7 13 91 11 143 9 143 7
S.x = x + x + x − x + x − x
15 2 6 30 18 10
91 5 691 3 7
+ x − x + x,
6 90 6
15 1 16 1 15 5 14 91 12 143 10 429 8
S.x = x + x + x − x + x − x
16 2 4 24 12 16
455 6 691 4 35 2
+ x − x + x ,
12 24 4
16 1 17 1 16 4 15 14 13 52 11 143 9
S.x = x + x + x − x + x − x
17 2 3 3 3 3
260 7 1382 5 140 3 3617
+ x − x + x − x,
3 15 3 510
and so forth. These formulas can be continued to the twenty-ninth power.
Indeed, they can be carried to even higher powers, provided that the nu-
merical coefficients have been worked out.