132 
[189 
189. 
NOTE ON A FORMULA IN FINITE DIFFERENCES. 
[From the Quarterly Mathematical Journal, vol. n. (1858), pp. 198—201.] 
In Jacobi’s Memoir “De usu legitimo formula} summatoria} Maclaurinianse,” Grelle, 
t. xil. [1834], pp. 263—273 (1834), expressions are given for the sums of the odd 
powers of the natural numbers 1, 2, 3...« in terms of the quantity. 
viz. putting for shortness 
u = x(x + 1), 
Sx r = F + 2 r + ... + x r , 
the expressions in question are 
Sx 3 = \u 2 , 
Sx 5 = £u 2 (u — 
Sx 7 = ^u 2 (u 2 — fw + §), 
Sx? = -feu 2 (u 3 — |u 2 + 3u — f), 
Sx 11 = feu 2 (u* — 4 v? + fe-u 2 —10 u + 5), 
Sx 13 = feu 2 {u 5 - 'feu 4 + ?fe-u 3 - fe&ii 2 + sfe-u - %%!-), 
&c., 
which, especially as regards the lower powers, are more simple than the ordinary 
expressions in terms of x. 
The expressions are continued by means of a recurring formula, viz. if 
1 
Sx 2p ~ 
Sx 2 ?- 1 = 
2p — 2 
1 
2V 
\uP 1 — aiii p ~ 2 ...+ (—) p_1 a p - 3 u 2 }, 
{u p 
- h,u p ~ 2 ...+ (~) p b p _ 3 u 2 },