Page 175 - Determinants and Their Applications in Mathematical Physics

160 4. Particular Determinants

Reverting to x and referring to (4.12.17),

G n−1 F n G n−2
= 2 , (4.12.28)
xD x
F n−1 F
n−1
where the elements in the determinants are now ψ m (x), m =0, 1, 2,....
The difference formula
∆ ψ 0 = xψ m , m =1, 2, 3,..., (4.12.29)
m
is proved in Appendix A.8. Hence, applying the theorem in Section 4.8.2
on Hankelians whose elements are differences,

E n = |ψ m | n , 0 ≤ m ≤ 2n − 2
m
= |∆ ψ 0 | n

ψ 0 xψ 1 xψ 2 ···

xψ 1 xψ 2 xψ 3 ···
. (4.12.30)
=
xψ 2 xψ 3 xψ 4 ···
....................

n
Every element except the one in position (1, 1) contains the factor x. Hence,
removing these factors and applying the relation
ψ 0 /x = ψ 0 +1,

ψ 0 +1 ψ 1 ψ 2 ···

ψ 1 ψ 2 ψ 3
n ···
E n = x

ψ 2 ψ 3 ψ 4
···
....................

n
(n)
= x n E n + E 11 . (4.12.31)
Hence

(n) 1 − x n
E = G n−1 = E n . (4.12.32)
11
x n
Put
u n = G n ,
F n
E n−1
v n = E n . (4.12.33)
The theorem is proved by deducing and solving a differential–difference
equation satisfied by u n :
E n−1 E n+1
= 2 .
v n
v n+1 E
n
From (4.12.32),
G n−1 x(1 − x )v n+1
n
= . (4.12.34)
1 − x n+1
G n