Determinants and Their Applications in Mathematical Physics

5.5 Determinants Associated with a Continued Fraction 203

where

Kn=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

1 b 1

− 1 a 1 b 2

− 1 a 2 b 3

− 1 an− 3 bn− 2

− 1 an− 2 bn− 1

− 1 an− 1 +(bn/an)

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (5.5.4)

Return toHn+1, remove the factoranfrom the last column, and then

perform the column operation

C

′

n

=Cn+Cn+1.

The result is a determinant of order (n+ 1) in which the only element in

the last row is 1 in the right-hand corner.

It then follows that

Hn+1=anKn.

Similarly,

H

(n+1)

11 =anK

(n−1)

11.

The theorem follows from (5.5.3).

Tridiagonal determinants of the formHnare called continuants. They are

also simple Hessenbergians which satisfy the three-term recurrence relation.

ExpandingHn+1by the two elements in the last row, it is found that

Hn+1=anHn+bnHn− 1.

Similarly,

H

(n+1)

11 =anH

(n)

11 +bnH

(n)

11. (5.5.5)

The theorem can therefore be reformulated as follows:

fn=

Qn

Pn

, (5.5.6)

wherePnandQneach satisfy the recurrence relation

Rn=anRn− 1 +bnRn− 2 (5.5.7)

with the initial valuesP 0 =1,P 1 =a 1 +b 1 ,Q 0 = 1, andQ 1 =a 1.

5.5.2 Polynomials andPowerSeries

In the continued fractionfndefined in (5.5.1) in the previous section,

replacearby 1 and replacebrbyarx. Then,

fn=

1

1+

a 1 x

1+

a 2 x

1+

···

an− 1 x

1+

anx

1