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