160 4. Particular Determinants
Reverting toxand referring to (4.12.17),
xDx[
Gn− 1Fn− 1]
=
FnGn− 2F
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=xn∣ ∣ ∣ ∣ ∣ ∣ ∣
ψ 0 +1 ψ 1 ψ 2 ···ψ 1 ψ 2 ψ 3 ···ψ 2 ψ 3 ψ 4 ···....................∣ ∣ ∣ ∣ ∣ ∣ ∣ n=xn(
En+E(n)
11)
. (4.12.31)
Hence
E
(n)
11
=Gn− 1 =(
1 −x
nx
n)
En. (4.12.32)Put
un=GnFn,
vn=En− 1En. (4.12.33)
The theorem is proved by deducing and solving a differential–difference
equation satisfied byun:
vnvn+1=
En− 1 En+1E
2
n.
From (4.12.32),
Gn− 1Gn=
x(1−xn
)vn+11 −x
n+1