Page 178 - Determinants and Their Applications in Mathematical Physics

4.12 Hankelians 5 163

where f is an arbitrary function of t. Then, it is proved that Section 6.5.2
on Toda equations that

2
D (log A n )= A n+1 A n−1 . (4.12.47)
A 2
n
Put
2
g n = D (log A n ). (4.12.48)
Theorem 4.58. g n satisfies the differential–difference equation
n−1
2
g n = ng 1 + (n − r)D (log g r ).
r=1
Proof. From (4.12.47),

A r+1 A r−1
= g r ,
A 2
r
s s s
A r+1 A r−1

= g r ,
r=1 A r r=1 A r r=1
which simplifies to
s
A s+1
= A 1 g r . (4.12.49)
r=1
A s
Hence,
n−1 n−1 s
A s+1 n−1

= A 1 g r ,
s=1 A s s=1 r=1
n−1

A n = A n g n−r
1
r
r=1
n−1

= A n g r , (4.12.50)
1 n−r
r=1
n−1

log A n = n log A 1 + (n − r) log g r . (4.12.51)
r=1
The theorem appears after differentiating twice with respect to t and
referring to (4.12.48).
In certain cases, the differential–difference equation can be solved and
A n evaluated from (4.12.50). For example, let
e t

p
f =
1 − e t