Determinants and Their Applications in Mathematical Physics

5.5 Determinants Associated with a Continued Fraction 205

etc. These formulas lead to the following theorem.

Theorem 5.10.

fn−fn− 1 =(−1)

n

(a 1 a 2 a 3 ···an)x

n

+O(x

n+1

),

that is, the coefficients ofx

r

, 1 ≤r≤n− 1 , in the series expansion offn

are identical to those in the expansion offn− 1.

Proof. Applying the recurrence relation (5.5.9),

Pn− 1 Qn−PnQn− 1 =Pn− 1 (Qn− 1 +anxQn− 2 )−(Pn− 1 +anxPn− 2 )Qn− 1

=−anx(Pn− 2 Qn− 1 −Pn− 1 Qn− 2 )

=an− 1 anx

2

(Pn− 3 Qn− 2 −Pn− 2 Qn− 3 )

.

.

.

=(−1)

n

(a 3 a 4 ···an)x

n− 2

(P 1 Q 2 −P 2 Q 1 )

=(−1)

n

(a 1 a 2 ···an)x

n

(5.5.15)

fn−fn− 1 =

Qn

Pn

−

Qn− 1

Pn− 1

=

Pn− 1 Qn−PnQn− 1

PnPn− 1

=

(−1)

n

(a 1 a 2 ···an)x

n

PnPn− 1

. (5.5.16)

The theorem follows sincePn(x) is a polynomial withPn(0)=1.

Let

fn(x)=

∞

∑

r=0

crx

r

. (5.5.17)

From the third equation in (5.5.14),

c 0 =1,

c 1 =−a 1 ,

c 2 =a 1 (a 1 +a 2 ),

c 3 =−a 1 (a

2

1

+2a 1 a 2 +a

2

2

+a 2 a 3 ),

c 4 =a 1 (a

2

1

a 2 +2a 1 a

2

2

+a

3

2

+2a

2

2

a 3 +a

2

1

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

2

3

+a

2

1

a 4 +a 1 a 2 a 4 +a 2 a 3 a 4 ), (5.5.18)

etc. Solving these equations for thear,

a 1 =−|c 1 |,

a 2 =

∣

∣

∣

∣

c 0 c 1

c 1 c 2

∣

∣

∣

∣

|c 1 |

,