4.12 Hankelians 5 163
wherefis an arbitrary function oft. Then, it is proved that Section 6.5.2
on Toda equations that
D
2
(logAn)=
An+1An− 1
A
2
n
. (4.12.47)
Put
gn=D
2
(logAn). (4.12.48)
Theorem 4.58. gnsatisfies the differential–difference equation
gn=ng 1 +
n− 1
∑
r=1
(n−r)D
2
(loggr).
Proof. From (4.12.47),
Ar+1Ar− 1
A
2
r
=gr,
s
∏
r=1
Ar+1
Ar
s
∏
r=1
Ar− 1
Ar
=
s
∏
r=1
gr,
which simplifies to
As+1
As
=A 1
s
∏
r=1
gr. (4.12.49)
Hence,
n− 1
∏
s=1
As+1
As
=A
n− 1
1
n− 1
∏
s=1
s
∏
r=1
gr,
An=A
n
1
n− 1
∏
r=1
g
n−r
r
=A
n
1
n− 1
∏
r=1
g
r
n−r
, (4.12.50)
logAn=nlogA 1 +
n− 1
∑
r=1
(n−r) loggr. (4.12.51)
The theorem appears after differentiating twice with respect to tand
referring to (4.12.48).
In certain cases, the differential–difference equation can be solved and
Anevaluated from (4.12.50). For example, let
f=
(
e
t
1 −e
t
)p