1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions

18. Show that:
    (a) l(a)nl :::; (lal)n
    (b) (1 - z)-a = ~ (a)n Zn (lzl < 1)
    L.. n!
    n=O

8.22 The Hypergeometric Function

Definition 8.8 The power series

1

a·b a(a+l)b(b+l) 2

• l!c z+ 2!c(c+l) z +···

635

+ a(a+1)···(a+n-1)b(b+1)···(b+n-1)zn+··· ( 8. 22 _ 1 )

n!c( c + 1) · · · ( c + n - 1)

where a, b, and c are complex parameters which are neither zero nor a

negative integer, converge~ absolutely/ (by the ratio test) for lzl < 1. If

either a orb, or both, are zero or a negative integer, the series reduces to a
polynomial and so the convergence is trivial for all finite values of z. Also,
from the Example in Section 4.9(h), we have that the series above converges
absolutely on the boundary lzl = 1 provided that Re(c - a - b) > 0.

Hence at least for lzl < 1, the series (8.22-1) represents an analytic

function F( a, b, c; z) called the hypergeometric function. The series (8.22-1)
is called the hypergeometric series. Note that the special case a = c, b = 1 is
z:: zn, i.e., the well-known geometric series. By introducing the factorial
notation, we have

F(a,b c-z) = f: (a)n(b)n Zn= r(c) f: r(a+n)r(b+n) Zn
' ' n=O n!(c)n r(a)r(b) n=O n!r(c + n)
(8.22-2)
at least for lz I < 1. Many properties of the hypergeometric function were
obtained by L. Euler, but a more systematic study of this function was
made by C. F. Gauss [13].
A number of elementary functions are either particular cases or limiting
cases of (8.22-2).

Examples

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


2. zF(l, 1, 2; -z) = Log(l + z)


3. zF(^1 / 2 ,%,%;z^2 ) = Arcsinz


4. zF(^1 / 2 , 1, %;-z^2 ) = Arctanz


5. limb->oo F(a, b, a; z/b) = ez