Determinants and Their Applications in Mathematical Physics

204 5. Further Determinant Theory

=

Qn

Pn

, (5.5.8)

wherePnandQneach satisfy the recurrence relation

Rn=Rn− 1 +anxRn− 2 (5.5.9)

withP 0 =1,P 1 =1+a 1 x,Q 0 = 1, andQ 1 = 1. It follows that

P 2 =1+(a 1 +a 2 )x,

Q 2 =1+a 2 x,

P 3 =1+(a 1 +a 2 +a 3 )x+a 1 a 3 x

2

,

Q 3 =1+(a 2 +a 3 )x,

P 4 =1+(a 1 +a 2 +a 3 +a 4 )x+(a 1 a 3 +a 1 a 4 +a 2 a 4 )x

2

,

Q 4 =1+(a 2 +a 3 +a 4 )x+a 2 a 4 x

2

. (5.5.10)

It also follows from the previous section thatPn=Hn+1, etc., where

Hn+1=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

1 a 1 x

− 11 a 2 x

− 11 a 3 x

.

.

.

.

.

.

.

.

.

− 11 an− 2 x

− 11 an− 1 x

− 11

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

. (5.5.11)

The alternative formula

Hn+1=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

1 x

−a 1 1 x

−a 2 1 x

.

.

.

.

.

.

.

.

.

−an− 3 1 x

−an− 2 1 x

−an− 1 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

n+1

(5.5.12)

can be proved by showing that the second determinant satisfies the same

recurrence relation as the first determinant and has the same initial values.

Also,

Qn=H

(n+1)

11. (5.5.13)

Using elementary methods, it is found that

f 1 =1−a 1 x+a

2

1 x

2

+···,

f 2 =1−a 1 x+a 1 (a 1 +a 2 )x

2

−a 1 (a

2

1 +2a^1 a^2 +a

2

2 )x

3

+···,

f 3 =1−a 1 x+a 1 (a 1 +a 2 )x

2

−a 1 (a

2

1

+2a 1 a 2 +a

2

2

+a 2 a 3 )x

3

+···

+a 1 (a

3

1

+3a

2

1

a 2 +3a 1 a

2

2

+2a

3

2

+2a

2

2

a 3

+a 2 a

2

3

+2a 1 a 2 a 3 )x

4

+···, (5.5.14)