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 |