Sequences, Series, and Special Functions 539
zeros of g (if I< is not empty), and p = oo if g has no zeros in C. Then
· f(z) f n
F(z) = W = CnZ
g n=O(8.4-3)where
bo 0 0 aobi bo (^0) ai
(^1) bz bi bo
Cn = bn+i az (n=0,1,2, ... ) (8.4-4)
0bn bn-i bn-2 anthe power series on the right of (8.4-3) converging at least for lzl < R =
min(p, Ri, Rz).
Proof We note that since g(O) = b 0 -:/:-0, the origin is neither a zero of g
nor an,accumulation point of I< (supposedly not empty), because of the
continuity of g at the origin. Hence, p > O, R > 0, and F(z) = f(z)/g(z)
is analytic in the disk lzl < R by the rule of the derivative of a quotient. It
follows that F( z) has a series expansion of the form z::::: CnZn. Since the
coefficients in the Taylor expansion of an analytic function are uniquely
determined, we may evaluate the coefficients en from g(z)F(z) = f(z), or
which is valid for lzl < R. By Theorem 8.7 we get
= =
I:: (bocn + bicn-i + · · · + bnco) Zn= I:: anZn.
n=O n=O
and the identity principle for power series gives
bocn + bi Cn-i + · · · + bnco = an (8.4-5)
By writing (8.4-5) for n = O, 1, 2, ... , we obtain the triangular infinite
system
boco
bi co+ boci
bzco + bi c1 + boc2(8.4-6)