THE HYPERGEOMETRIC FUNCTION 
235 
in — xP n it is 
The coefficient of x n in x 2 F" is 
n(n-Y)P n , 
n(a+ w)(/3 + ri) 
in (a + /3 + 1) xF' it is 
in — 7F' it is 
7 + n 
w (a + /3 + T) P n , 
in a/3F it is 
(a+'n)(£ + №) 
7 7 + n Fn ’ 
a/3P n . 
Adding all these gives the coefficient of x n in the left side of 1). 
We find it is 0. 
191. Expression of F( K afi r yx') as an Integral. 
We show that for | x | < 1, 
B(J3, 7 — /3) • F(a/3'yx) = Ç 1 vP- 1 (t — u)y-P- 1 (l — xu)- a du (1 
where B(p, q) is the Beta function of I, 692, 
B(p, q) = uv-'O- — uf^du. 
For by the Binomial Theorem 
a • a + 1 « 
a • a 1 • a + 2 
i .a . a • « -t~ x 9 o , « • «*.-t* x T ^ s i 
(1 — xu)~ a =1 + - xu-\ ——— x 2 u 2 H —-— xru d + 
1 • 2 
1-2-3 
for \xu\ <1. Hence 
J= f u^~ 1 (l — u)^-^ 1 (l — xu)- a du 
Jo 
= V- 1 (1 - uy-P-'du + £ ufi 0 - 
+ “• “ +1 • rr 2 fwVi(l - up-P-'du + ... 
1-2 «/o 
= B (J3, 7 — /8) + «zi? (/3 + 1, 7 — /3) 
+ ^^±I^5(/3 + 2, 7 -/3)+- (2