270 
INFINITE PRODUCTS 
are convergent. Hence 
(7 = 
lie 
from which 1) follows at once, using 223. 
225. Further Properties of Gr. 
1. (7(w +1} = u GrQu). 
Let us use the product 
(n — 1) ! 
p n 00 
u (u + 1) ••• (u + n — 1) 
employed in 222. Then 
P„Q + l )= g*g.W. 
u + n 
As 
nu 
= u as n = oo 
u + n 
we get 1) from 2) at once on passing to the limit. 
2. G~(u -f n)= u(u + 1) ••• (u + n — l)(7(w). 
This follows from 1) by repeated applications. 
3. (7(w) = 1 • 2 • • •. n — 1 = (n — 1)! 
where n is a positive integer. 
4. 
For 
6r(w) (7(1 — u) = — 
7T 
Sill 7TM 
(7(1 — w) = — w(7( — w) 
e Cu 
nil- - V 
by 1, 
by 224, 1). 
Hence 
Gr(u) (7(1 — u) = - 
o Cus>Cu 
n[l + ^)e " 
1 — - ]e r 
Us 
a 
(2 
(3 
(4 
(5 
n 1 -