Page 268 - Determinants and Their Applications in Mathematical Physics

6.5 The Toda Equations 253

are proved in Theorem 4.30 in Section 4.8.5 on Turanians.
Let the elements in both A n and B n be defined as
φ m (x)= f (m) (x), f(x) arbitrary,

so that
φ = φ m+1 (6.5.4)
m
and both A n and B n are Wronskians (Section 4.7) whose derivatives are
given by
(n+1)

A = −A ,
n n+1,n
(n+1)

B = −B . (6.5.5)
n n+1,n
Theorem 6.1. The equation
u n u n+1

u =
n
u n−1
is satisfied by the function defined separately for odd and even values of n
as follows:
u 2n−1 = A n ,
B n−1
u 2n = B n .
Proof. A n

B 2 u = B n−1 A − A n B
n−1 2n−1 n n−1
(n+1) (n)
= −B n−1 A + A n B n,n−1
n+1,n

B 2 u 2n−1 u 2n = A n−1 B n .
n−1
u 2n−2
Hence, referring to (6.5.3),

2
B n−1 u 2n−1 u 2n − u 2n−1 = A n−1 B n + B n−1 A (n+1) − A n B (n)
n,n−1
u 2n−2 n+1,n
=0,
which proves the theorem when n is odd.
2

A u = A n B − B n A
n 2n n n
(n+1) (n+1)
= −A n B + B n A n,n+1 ,
n+1,n

A 2 u 2n u 2n+1 = A n+1 B n−1 .
n
u 2n−1
Hence, referring to (6.5.2),

A 2 u 2n u 2n+1 − u = A n+1 B n−1 + A n B (n+1) − B n A (n+1)
n 2n n+1,n n,n+1
u 2n−1