Determinants and Their Applications in Mathematical Physics

202 5. Further Determinant Theory

=

a 1 a 2 a 3 +a 1 b 3 +a 3 b 2

a 1 a 2 a 3 +a 1 b 3 +a 3 b 2 +a 2 a 3 b 1 +b 1 b 3

.

Each of these fractions can be expressed in the formH 11 /H, whereHis a

tridiagonal determinant:

f 1 =

|a 1 |

∣

∣

∣

∣

1 b 1

− 1 a 1

∣

∣

∣

∣

,

f 2 =

∣

∣

∣

∣

a 1 b 2

− 1 a 2

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

1 b 1

− 1 a 1 b 2

− 1 a 2

∣ ∣ ∣ ∣ ∣ ∣

,

f 3 =

∣ ∣ ∣ ∣ ∣ ∣

a 1 b 2

− 1 a 2 b 3

− 1 a 3

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

1 b 1

− 1 a 1 b 2

− 1 a 2 b 3

− 1 a 3

∣ ∣ ∣ ∣ ∣ ∣ ∣

.

Theorem 5.9.

fn=

H

(n+1)

11

Hn+1

,

where

Hn+1=

∣

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

1 b 1

− 1 a 1 b 2

− 1 a 2 b 3

.

.

.

.

.

.

.

.

.

− 1 an− 2 bn− 1

− 1 an− 1 bn

− 1 an

∣

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

. (5.5.2)

Proof. Use the method of induction. Assume that

fn− 1 =

H

(n)

11

Hn

,

which is known to be true for small values ofn. Hence, addingbn/anto

an− 1 ,

fn=

K

(n)

11

Kn

, (5.5.3)