THE HYPERGEOMETRIC FUNCTION 
233 
Eliminating F(«, /3, 7 — 1, x) from 1), 3) gives 
(/3 - /3, 7, 2;) + uF (a + 1, /3, 7, cr) 
- A^(«, /3 + 1, 7, x) = 0. (4 
Permuting a, ¡3 in 2) gives 
(7 — a — (3)F(a, /3, 7, 2;) + /3(1 — 2:)jP (a, £ + 1, 7, *) 
+ (« — 7)^ (« — 1, A 7, 2;) = 0. (5 
From 3), 5) let us eliminate F (a, /3 + 1, 7, z), getting 
(« — 1 — (7 — /3 — 1)»)^(a, /3, 7, x) + (7 — a)F(a — 1, /3, 7, 2;) 
+ (1 — 7)(1 — xjF(a, /3,7-1,*) = 0. (6 
In 1) let us replace « by a — 1 and 7 by 7 + 1; we get 
(7 — a + 1 ~)F(a — 1, /3, 7 + 1, 2;) + (a — 1 )F(a, /3, 7 + 1, 2;) 
- 7^ 7 (cc - 1, /3, 7, 2;) =0. (a) 
In 6) let us replace 7 by 7 + 1 ; we get 
(«-1—(7 —/3)2-)F(a, A 7+1, 2^) + (7 +1 — a)F(a — 1, /3, 7 + I, x') 
-y(\ — x)F(a,[3,y,x') = Q. (b) 
Subtracting (b) from (a), eliminates ,F(« — 1, /3, 7+ 1, 2:) and 
gives 
7(1 — x^F(a/3yx} — yF(a — 1, /3, 7, 2;) 
+ (7 — fi)xF(a, A 7 + 1» *) = 0. (7 
From 6), 7) we can eliminate F(u — 1, /3, 7, #), getting 
7|7 — 1 +(« + /3 +1 — 2 7)2v\F(a, A 7, *) 
+ (7 — a) (7 — ¡3)xF(ct, A 7 + 1, 2;) 
+ 7(l-7)(l-i)i T (« 1 A7-l+)=0- (8 
In this manner we may proceed, getting the remaining seven. 
188. Conjugate Functions. From the relations between con 
tiguous functions we see that a linear relation exists between any 
three functions 
F(u, A 7> x) F(a’, /3', 7', x) F(a", A', 7", x) 
whose corresponding parameters differ only by integers. Such 
functions are called conjugate.