Page 222 - Determinants and Their Applications in Mathematical Physics

5.5 Determinants Associated with a Continued Fraction 207

Proof. Let
∞

f 2n−1 P 2n−1 − Q 2n−1 = h nr x , (5.5.23)
r
r=0
where f n is defined by the infinite series (5.5.17). Then, from (5.5.8),
h nr =0, all n and r,
where

c r−t p 2n−1,t − q 2n−1,r , 0 ≤ r ≤ n − 1
 r !


h nr = t=0 (5.5.24)
c r−t p 2n−1,t , r ≥ n.
 r !


t=0
The upper limit n in the second sum arises from (5.5.22).
The n equations
h nr =0, n ≤ r ≤ 2n − 1,
yield
n

c r−t p 2n−1,t + c r =0. (5.5.25)
t=1
Solving these equations by Cramer’s formula yields part (a) of the theorem.
Part (b) is proved in a similar manner. Let
∞

f 2n P 2n − Q 2n = k nr x . (5.5.26)
r
r=0
Then,
k nr =0, all n and r,
where

c r−t p 2n,t − q 2n,r , 0 ≤ r ≤ n
 r !


k rn = t=0 (5.5.27)
c r−t p 2n,t , r ≥ n +1.
 n !


t=0
The n equations
k nr =0, n +1 ≤ r ≤ 2n,
yield
n

c r−t p 2n,t + c r =0. (5.5.28)
t=1
Solving these equations by Cramer’s formula yields part (b) of the theorem.