Determinants and Their Applications in Mathematical Physics

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