189] NOTE ON A FORMULA IN FINITE DIFFERENCES. 133 
then 
2p(2p-l)a 1 = (2p-2)(2p-3)b 1 - pip-1), 
2p (2p -1) a. 2 = (2p - 4) (2p - o)b 2 - (p - 1) (p -2)b lf 
2p (2p -1) a 3 •= (2p- 6) (2p - 7) b 3 - (p -2){p- 3) b,, 
2p (2p — 1) a p _ 3 = 5 . 6 6p_3- 3 . 4 6 P _ 4 , 
0 = 3.4 b p _ 2 - 2 . 3 bp-,, 
by means of which the coefficients b can be determined when the coefficients a are 
known. 
Jacobi remarks also that the expressions for the sums of the even powers may 
be obtained from those for the odd powers by means of the formula 
Sx 2p — - a ^ d x Sx 2p+1 , 
2p + 1 
which shows that any such sum will be of the form (2x + l)u into a rational and 
integral function of u : thus in particular 
Sa f = ^ (2x + 1) u. 
To show a priori that SaJ 2p+1 can be expressed as a rational and integral function 
of u, it may be remarked that Sa? p+1 = fax where fax denotes the summatory integral 
2 (x + 1) 2 -P +1 , taken so as to vanish for x = 0: (j) x x is a rational and integral function 
of x of the degree 2p + 2, and which, as is well known, contains x 2 as a factor. 
Suppose that y is any positive or negative integer less than x, we have 
fax — fay = (y + + (y + 2) 2p+1 ...+ x 2p+1 , 
and in particular putting y = — 1 — x, 
fax — fa (— 1 — x) = (— x) 2p+1 + (1 — x) 2p+1 ...+ x 2p+1 , = 0, 
since the terms destroy each other in pairs; we have therefore fax = fa(-l- x). 
Now u = x 1 + x, or writing this equation under the form oc 2 = — x + u, we see that any 
rational and integral function of x may be reduced to the form Px + Q, where 
P and Q are rational and integral functions of u. Write therefore fax = Px + Q: the 
substitution of — 1 — x in the place of x leaves u unaltered, and the equation 
fax = cj) 1 (— 1 — x) thus shows that P = 0 ; we have therefore fax = Q, a rational and 
integral function of u. Moreover fax as containing the factor x 2 , must clearly contain 
the factor iP, and the expressions for Sx 2p+1 are thus shown to be of the form given 
by Jacobi. 
We may obtain a finite expression for 8x n in terms of the differences of 0 m as 
follows: we have 
Sx n = l n + 2 n ...+x n = {(1 + A) + (1 + A) 2 ...+ (1 + A)*} 0 M = ^ — {(1 + A)*- 1}0 M ,