252 NOTE ON THE SUMMATION OF A CERTAIN FACTORIAL EXPRESSION. [205 
and hence multiplying by dx and integrating from x = 0, and again multiplying by 
dx and integrating from x = 0 to x = 1, we find 
1 1 de. e a - 3 (i - 0)—r+0+i -1 1 ¿0.0«-s (i - 0)-«-y-i + (/3 + 2) f 1 dd. 0“- 2 (i - 0)-«-v-i 
0 3 0 3 0 
if for shortness 
(/3 + 1) (/5 + 2) II (a — 1) II (— a — 7 — 1) 1 a 
H (— 7 — 1) 3 ’ 
a . /3 a. a+1. /3./3 — 1 
3.7"^ 3.4.7.7 — 1 
-f &c. 
Substituting for the definite integrals their values, 
n (a - 3) II (- a - 7 + ¡3 + 1) II (a - 3) II (- a - 7 - 1) II (a - 2) II (- a - 7 - 1) 
IT (— 7 + /3 — I) n (— 7 — 3) + ' n (— 7 — 2) 
whence 
(/3 + 1) (/3 + 2) II (a - 1) II (- a - 7 - 1) , 0 
11 ( — 7 — 1 ) 3 ’ 
i03 + l)(/3 + 2)S = 
II (a — 3) ri(-a-7 + /3 + l) n(- 7 -l) 
Il (a-1) II (— a — 7 — 1) U(-y + /3-l) 
n(a-3) II (— 7 — 1) ox n (a — 2) n(- 7 -l) 
TT (a - 1) n (- 7 - 3) + + ; II (a - I) * II (- 7 - 2)' 
The second and third terms are 
- (a _ i)^ _ 2) (7 + 1) (7 + 2) - (^ + 2) (7 + 1), 
which are 
(a— T) (a — 2) («£ + 2a “ + 7 - 2). 
For the reduction of the first term we have 
II(— et — 7 + /3+1) = [/3 + 1— a — 7]^ +2 II (— a — 7 — 1), 
H (/3-7-1) =[/3-7_rpn(—7 —1); 
and we thus find 
i (0 +1) (0 + 2) (a - 1) 0 - 2) 8 = - (7 + U («0 + 2« - 20 + 7 - 2), 
where, as before, 
a./3 , a.a + l./3./3 — 1