Determinants and Their Applications in Mathematical Physics

5.5 Determinants Associated with a Continued Fraction 209

5.5.3 Further Determinantal Formulas

Theorem 5.12.

a.P 2 n− 1 =

1

An

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

c 0 c 1 c 2 ··· cn

c 1 c 2 c 3 ··· cn+1

...............................

cn− 1 cn cn+1 ··· c 2 n− 1

x

n

x

n− 1

x

n− 2

··· 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

,

b.P 2 n=

1

Bn

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

c 1 c 2 c 3 ··· cn+1

c 2 c 3 c 4 ··· cn+2

............................

cn cn+1 cn+2 ··· c 2 n

x

n

x

n− 1

x

n− 2

··· 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

.

Proof. Referring to the first line of (5.5.21) and to Theorem 5.11a,

P 2 n− 1 =

1

An

n

∑

r=0

A

(n+1)

n+1,n+1−r

x

r

=

1

An

n+1

∑

j=1

A

(n+1)

n+1,jx

n+1−j

.

Part (a) follows and part (b) is proved in a similar manner with the aid of

the third line in (5.5.21) and Theorem 5.11b.

Lemmas.

a.

n− 1

∑

r=0

ur

r

∑

t=0

cr−tvn+1−t=

n

∑

j=1

vj+1

j− 1

∑

r=0

crun+r−j,

b.

n

∑

r=0

ur

r

∑

t=0

cr−tvn+1−t=

n

∑

j=0

vj+1

j

∑

r=0

crun+r−j.

These two lemmas differ only in some of their limits and could be re-

garded as two particular cases of one lemma whose proof is elementary and

consists of showing that both double sums represent the sum of the same

triangular array of terms.

Let

ψm=

m

∑

r=0

crx

r

. (5.5.31)

Theorem 5.13.

a.Q 2 n− 1 =

1

An

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

c 0 c 1 c 2 ··· cn

c 1 c 2 c 3 ··· cn+1

....................................

cn− 1 cn cn+1 ··· c 2 n− 1

ψ 0 x

n

ψ 1 x

n− 1

ψ 2 x

n− 2

··· ψn

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

,