Determinants and Their Applications in Mathematical Physics

5.5 Determinants Associated with a Continued Fraction 207

Proof. Let

f 2 n− 1 P 2 n− 1 −Q 2 n− 1 =

∞

∑

r=0

hnrx

r

, (5.5.23)

wherefnis defined by the infinite series (5.5.17). Then, from (5.5.8),

hnr=0, allnandr,

where

hnr=















∑r

t=0

cr−tp 2 n− 1 ,t−q 2 n− 1 ,r, 0 ≤r≤n− 1

∑r

t=0

cr−tp 2 n− 1 ,t,r≥n.

(5.5.24)

The upper limitnin the second sum arises from (5.5.22).

Thenequations

hnr=0,n≤r≤ 2 n− 1 ,

yield

n

∑

t=1

cr−tp 2 n− 1 ,t+cr=0. (5.5.25)

Solving these equations by Cramer’s formula yields part (a) of the theorem.

Part (b) is proved in a similar manner. Let

f 2 nP 2 n−Q 2 n=

∑∞

r=0

knrx

r

. (5.5.26)

Then,

knr=0, allnandr,

where

krn=















∑r

t=0

cr−tp 2 n,t−q 2 n,r, 0 ≤r≤n

∑n

t=0

cr−tp 2 n,t,r≥n+1.

(5.5.27)

Thenequations

knr=0,n+1≤r≤ 2 n,

yield

n

∑

t=1

cr−tp 2 n,t+cr=0. (5.5.28)

Solving these equations by Cramer’s formula yields part (b) of the theorem.