232 
POWER SERIES 
Between F and two of its contiguous functions exists a linear 
relation. As the number of such pairs of contiguous functions is 
6 • 5 
1 • 2 
= 15, 
there are 15 such linear relations. Let us find one of them. 
We set 
Qn= 
a + 1 • a + 2 • •••«+№ — 1 • /3 • /3 + 1 • • • • /3 + n — 2 
1-2 - •••w-7*7+l-”'7 + w— 1 
Then the coefficient of x n in FÇa/3 r yx') is 
«(£ + n-V)Q n ; 
in F(a + 1, /3, 7, x) it is 
(a + w)(£ + w — 1)<?„; 
in F (a, /3, 7 — 1, æ) it is 
a(/3 + rc-l)(7+n — 1)^ < 
7-1 
Thus the coefficient of x n in 
(7 — a — l)F(a, /3, 7, x) + a F (a + 1, /3, 7, 2;) 
+ (1 - 7)^ T (a, /3, 7 - 1, 2;) 
is 0. This being true for each n, we have 
(7 — a — 1).F(«, /3, 7, 2;) + (« + 1, /3, 7, 2;) 
+ (1 — 7)^(a, /3, 7 — 1, 2;)= 0. (1 
Again, the coefficient of x n in F («, /3 — 1, 7, æ) is a(/3 — 1)$*; 
in xF (a + 1, /3, 7, x) it is n(7 + n — 1) # n . 
Hence using the above coefficients, we get 
(7 — a — /3)F(a, /3, 7, x) + «(1 — 2i)F (« + 1, /3, 7, z) 
+ (/3 - 7)^ («» Æ — 1, 7, z) = 0. (2 
From these two we get others by elimination or by permuting 
the first two parameters, which last does not alter the value of 
the function F^aftyx). 
Thus permuting a, /3 in 1) gives 
(7 — /3 — V)F(a, /3, 7, x) +/3F(a, /3 + 1,7, 2:) 
+ (1 — 7)J T («,/3, 7 — 1, 2:) = 0. (3