160 4. Particular Determinants
Reverting toxand referring to (4.12.17),
xDx
[
Gn− 1
Fn− 1
]
=
FnGn− 2
F
2
n− 1
, (4.12.28)
where the elements in the determinants are nowψm(x),m=0, 1 , 2 ,....
The difference formula
∆
m
ψ 0 =xψm,m=1, 2 , 3 ,..., (4.12.29)
is proved in Appendix A.8. Hence, applying the theorem in Section 4.8.2
on Hankelians whose elements are differences,
En=|ψm|n, 0 ≤m≤ 2 n− 2
=|∆
m
ψ 0 |n
=
∣ ∣ ∣ ∣ ∣ ∣ ∣
ψ 0 xψ 1 xψ 2 ···
xψ 1 xψ 2 xψ 3 ···
xψ 2 xψ 3 xψ 4 ···
....................
∣ ∣ ∣ ∣ ∣ ∣ ∣ n
. (4.12.30)
Every element except the one in position (1,1) contains the factorx. Hence,
removing these factors and applying the relation
ψ 0 /x=ψ 0 +1,
En=x
n
∣ ∣ ∣ ∣ ∣ ∣ ∣
ψ 0 +1 ψ 1 ψ 2 ···
ψ 1 ψ 2 ψ 3 ···
ψ 2 ψ 3 ψ 4 ···
....................
∣ ∣ ∣ ∣ ∣ ∣ ∣ n
=x
n
(
En+E
(n)
11
)
. (4.12.31)
Hence
E
(n)
11
=Gn− 1 =
(
1 −x
n
x
n
)
En. (4.12.32)
Put
un=
Gn
Fn
,
vn=
En− 1
En
. (4.12.33)
The theorem is proved by deducing and solving a differential–difference
equation satisfied byun:
vn
vn+1
=
En− 1 En+1
E
2
n
.
From (4.12.32),
Gn− 1
Gn
=
x(1−x
n
)vn+1
1 −x
n+1