252 6. Applications of Determinants in Mathematical Physics
=2
∑
r
brcrφr
∑
j
e
cju
A
rj
+2
∑
r
brcr
∑
j
φre
cju
A
rj
cj− 1
=2
∑
r
brcrφ
2
r−F
′
= (logA)
′′
−F
′
, (6.4.18)
R=2
∑
j
e
cju
cj− 1
∑
r
brcrφ
′
rA
rj
=2
∑
j
e
cju
cj− 1
∑
r
brcrφr[cjA
rj
−φrφj]
=Q−P. (6.4.19)
Hence, eliminatingP,Q, andRfrom (6.4.16)–(6.4.19),
d
2
F
du
2
− 2
dF
du
+2F(logA)
′′
=0. (6.4.20)
Put
F=e
u
y. (6.4.21)
Then, (6.4.20) is transformed into
d
2
y
du^2
−y+2y
d
2
du^2
(logA)=0. (6.4.22)
Finally, putu=ωεx,(ω
2
=−1). Then, (6.4.22) is transformed into
d
2
y
dx
2
+ε
2
y+2y
d
2
dx
2
(logA)=0,
which is identical with (6.4.1), the Kay–Moses equation. This completes
the proof of the theorem.
6.5 The Toda Equations......................
6.5.1 The First-Order Toda Equation...........
Define two Hankel determinants (Section 4.8)AnandBnas follows:
An=|φm|n, 0 ≤m≤ 2 n− 2 ,
Bn=|φm|n, 1 ≤m≤ 2 n− 1 ,
A 0 =B 0 =1. (6.5.1)
The algebraic identities
AnB
(n+1)
n+1,n−BnA
(n+1)
n+1,n+An+1Bn−^1 =0, (6.5.2)
Bn− 1 A
(n+1)
n+1,n
−AnB
(n)
n,n− 1
+An− 1 Bn= 0 (6.5.3)