Page 267 - Determinants and Their Applications in Mathematical Physics

252 6. Applications of Determinants in Mathematical Physics

φ r e c j u A rj
=2 e c j u A rj +2
c j − 1
b r c r φ r
b r c r
r j r j
2
=2 b r c r φ − F
r
r

= (log A) − F , (6.4.18)
e c j u

R =2 b r c r φ A rj
c j − 1 r
j r
e c j u
=2 b r c r φ r [c j A rj − φ r φ j ]
c j − 1
j r
= Q − P. (6.4.19)
Hence, eliminating P, Q, and R from (6.4.16)–(6.4.19),
2
d F dF

− 2 +2F(log A) =0. (6.4.20)
du 2 du
Put
F = e y. (6.4.21)
u
Then, (6.4.20) is transformed into
2
d y d 2
− y +2y (log A)=0. (6.4.22)
du 2 du 2
2
Finally, put u = ωεx,(ω = −1). Then, (6.4.22) is transformed into
2
d y 2 d 2
+ ε y +2y (log A)=0,
dx 2 dx 2
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) A n and B n as follows:
A n = |φ m | n , 0 ≤ m ≤ 2n − 2,
B n = |φ m | n , 1 ≤ m ≤ 2n − 1,
A 0 = B 0 =1. (6.5.1)
The algebraic identities
(n+1) (n+1)
A n B − B n A + A n+1 B n−1 =0, (6.5.2)
n+1,n n+1,n
(n+1) (n)
B n−1 A − A n B n,n−1 + A n−1 B n = 0 (6.5.3)
n+1,n