Page 219 - Determinants and Their Applications in Mathematical Physics

204 5. Further Determinant Theory

= Q n , (5.5.8)
P n
where P n and Q n each satisfy the recurrence relation
(5.5.9)
R n = R n−1 + a n xR n−2
with P 0 =1, P 1 =1 + a 1 x, Q 0 = 1, and Q 1 = 1. It follows that
P 2 =1 + (a 1 + a 2 )x,
Q 2 =1 + a 2 x,
2
P 3 =1 + (a 1 + a 2 + a 3 )x + a 1 a 3 x ,
Q 3 =1 + (a 2 + a 3 )x,
2
P 4 =1 + (a 1 + a 2 + a 3 + a 4 )x +(a 1 a 3 + a 1 a 4 + a 2 a 4 )x ,
2
Q 4 =1 + (a 2 + a 3 + a 4 )x + a 2 a 4 x . (5.5.10)
It also follows from the previous section that P n = H n+1 , etc., where
1 a 1 x

−1 1 a 2 x

−1 1 a 3 x

. . .
H n+1 = . . . . . . . (5.5.11)

−1 1 a n−2 x

−1 1

a n−1 x
−1 1

n+1
The alternative formula
1 x

−a 1 1 x

−a 2 1 x

. . .
H n+1 = . . . . . . (5.5.12)

1 x
−a n−3

1
−a n−2
x
1
−a n−1
n+1
can be proved by showing that the second determinant satisfies the same
recurrence relation as the first determinant and has the same initial values.
Also,
(n+1)
Q n = H . (5.5.13)
11
Using elementary methods, it is found that
2 2
f 1 =1 − a 1 x + a x + ··· ,
1
2
2
3
2
f 2 =1 − a 1 x + a 1 (a 1 + a 2 )x − a 1 (a +2a 1 a 2 + a )x + ··· ,
1
2
2 2 2 3
f 3 =1 − a 1 x + a 1 (a 1 + a 2 )x − a 1 (a +2a 1 a 2 + a + a 2 a 3 )x + ···
2
1
3 2 2 3 2
+a 1 (a +3a a 2 +3a 1 a +2a +2a a 3
2
1
2
1
2
2
4
+a 2 a +2a 1 a 2 a 3 )x + ··· , (5.5.14)
3