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)