THE HYPERGEOMETRIC FUNCTION 
237 
and this establishes 3). Thus passing to the limit x = 1 in 1) 
gives 
O 
7 (« + ¡3 - 7) F(a, /3, 7) + (7 - «) (7 - P)F(a, /3, 7 + 1) = 0, 
or 
#(«, /8,7) = *'T‘ 0(7 ~ff *(«, A 7 +1)- 
7(7-a-/3) 
Replacing 7 by 7 + 1, this gives 
/3,7+!) 
(7 + 1 — «) (7 + 1 — /3) 
(7 +1) (7 +1 — « - £) 
F(a, /3,y + 2), 
etc. Thus in general 
F(+ /3, 7) = 
(Y-q)(Y + l—<x>--(Y + n —1 —«>(7 —/3)(y + 1 -~/3y--(7 + ft—1—/3) 
7(7 + 1)-.. (7 + w-l)(7-«-^)(7-«-^ + l)... (7 —a —£ + w-l) 
•F(a, ¡3,7+ n). 
Gauss sets now 
71 ! 7l x 
11 (W ’ X>> = (a:+lXa; + 2)...(a; + n)‘ 
Hence the above relation becomes 
F(nfi 7) 
Now 
II (n, 7 — 1) n (w, 7 — a — /3 — 1) 
n (n, 7 — a — 1) n (n, 7 — ¡3 — 1) 
F¡a, /3, 7 + ri). 
lim F(a, /3, 7 + 71) = 1. 
71=00 
For the series 
#(«,/3,7)= 1 +^vg + “-« + 1 -/3-ff+ 1 + ... 
1*7 1 • 2 • 7 - 7 + I 
converges absolutely when 2) holds. Hence 
• |/3| , |«|.|« + l|.|/3|.|/3 + l 
1 + 
1. a 
+ 
1 • 2. a • a + 1 
+ 
(6 
a 
(8 
(9 
is convergent. Now each term in 8) is numerically < the corre 
sponding term in 9) for any 7 > 6r. Hence 8) converges uni 
formly about the point 7 = + 00. We may therefore apply 146, 4. 
As each term of 8) has the limit 0 as 7 = + 00, the relation 7) 
is established.