Determinants and Their Applications in Mathematical Physics

5.5 Determinants Associated with a Continued Fraction 201

=4

∑

r,s

crcsE

rs,rs

+2

∑

r,s

c

2

sE

rs,rs

−

2

3

∑

r,s,t,u

E

rstu,rstu

. (5.4.48)

This is the second relation betweenQandR, the first being (5.4.39). Iden-

tities (5.4.37), (5.4.38), and (5.3) follow by solving these two equations for

QandR, wherePis given by (5.1).

Exercise.Prove that

∑

r,s

(cr−cs)φn(cr,cs)E

rs

=

∑

r,s

φn(cr,cs)E

rs,rs

,n=1, 2 ,

where

φ 1 (cr,cs)=cr+cs,

φ 2 (cr,cs)=3c

2

r+4crcs+3c

2

s.

Can this result be generalized?

5.5 Determinants Associated with a Continued Fraction

5.5.1 Continuants and the Recurrence Relation

Define a continued fractionfnas follows:

fn=

1

1+

b 1

a 1 +

b 2

a 2 +

···

bn− 1

an− 1 +

bn

an

,n=1, 2 , 3 ,.... (5.5.1)

fnis obtained fromfn− 1 by addingbn/antoan− 1.

Examples.

f 1 =

1

1+

b 1

a 1

=

a 1

a 1 +b 1

,

f 2 =

1

1+

b 1

a 1 +

b 2

a 2

=

a 1 a 2 +b 2

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

,

f 3 =

1

1+

b 1

a 1 +

b 2

a 2 +

b 3

a 3