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§5. Preliminaries on Hypergeometric Functions 181

(5.5) Proposition. The relation

( π/2 −1/2
2 sin 2m θ dθ = π(−1) m
0 m
holds for every natural number m.

Proof. By (5.4) we have

( π/2
1 2
m + 1 2
2 sin 2m θ dθ =
0
(m + 1)

1

1 m − 1 m − 3 ···
1
2 2 2 2 2
=
m(m − 1) ··· 2 · 1
− 1
= π(−1) m 2 .
m
This proves the proposition.
In order to define the hypergeometric series, we will need the following notation:
(a) n where (a) 0 = 1and (a) n = a(a + 1) ··· (a + n − 1) = (a + n − 1)(a) n−1 .
(5.6) Definition. The hypergeometric series for a, b ∈ C and c ∈ C − N is given by

(a) n (b) n n

F(a, b, c; z) = z .
n!(c) n
0≤n
An easy application of the ratio test shows that F(a, b, c; z) is absolutely conver-
gent for |z| < 1 and uniformly convergent for |z|≤ r < 1, and, hence, it represents
an analytic function on the unit disc called the hypergeometric function.
(5.7) Elementary Properties of the Hypergeometric Series.

(1) F(a, b, c; z) = F(b, a, c; z).

(2) F(a, b, b; z) = (1 − z) −a .
2

−a n
(3) F(a, a, 1; z) = z .
n
0≤n

We use the relations (1) n = n!and (a) n /n! = (−1) n −a to verify (2) and (3).
n
We will make use of (3) for the case a = 1/2.
1 2
1 1 − n
F , , 1; z = 2 z .
2 2 n
0≤n