Determinants and Their Applications in Mathematical Physics

254 6. Applications of Determinants in Mathematical Physics

=0,

which proves the theorem whennis even.

Theorem 6.2. The function

yn=D(logun),D=

d

dx

,

is given separately for odd and even values ofnas follows:

y 2 n− 1 =

An− 1 Bn

AnBn− 1

,

y 2 n=

An+1Bn− 1

AnBn

Proof.

y 2 n− 1 =Dlog

(

An

Bn− 1

)

=

1

AnBn− 1

(

Bn− 1 A

′

n−AnB

′

n− 1

)

=

1

AnBn− 1

[

−Bn− 1 A

(n+1)

n+1,n

+AnB

(n)

n,n− 1

]

The first part of the theorem follows from (6.5.3).

y 2 n=Dlog

(

Bn

An

)

=

1

AnBn

(

AnB

′

n

−BnA

′

n

)

=

1

AnBn

[

−AnB

(n+1)

n+1,n+BnA

(n+1)

n+1,n

]

The second part of the theorem follows from (6.5.2).

6.5.2 The Second-Order Toda Equations

Theorem 6.3. The equation

DxDy(logun)=

un+1un− 1

u

2

n

,Dx=

∂

∂x

,etc.

is satisfied by the two-way Wronskian

un=An=

∣

∣Di−^1

x

D

j− 1

y

(f)

∣

∣

n

,

where the functionf=f(x, y)is arbitrary.

Proof. The equation can be expressed in the form

∣

∣

∣

∣

DxDy(An) Dx(An)

Dy(An) An

∣

∣

∣

∣

=An+1An− 1. (6.5.6)