Page 269 - Determinants and Their Applications in Mathematical Physics

254 6. Applications of Determinants in Mathematical Physics

=0,
which proves the theorem when n is even.
Theorem 6.2. The function
d
y n = D(log u n ), D = ,
dx
is given separately for odd and even values of n as follows:

y 2n−1 = A n−1 B n ,
A n B n−1
A n+1 B n−1
y 2n = .
Proof. A n B n

y 2n−1 = D log A n
B n−1
1
= B n−1 A − A n B n−1

n
A n B n−1
1 (n+1) (n)
= −B n−1 A + A n B n,n−1 .
n+1,n
A n B n−1
The first part of the theorem follows from (6.5.3).

y 2n = D log B n
1 A n

= A n B − B n A
n n
A n B n
1 (n+1) (n+1)
= −A n B + B n A .
n+1,n n+1,n
A n B 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

u n+1 u n−1 ∂
D x D y (log u n )= , D x = , etc.
u 2 ∂x
n
is satisfied by the two-way Wronskian
i−1 j−1
u n = A n = D D (f) ,

x y
n
where the function f = f(x, y) is arbitrary.
Proof. The equation can be expressed in the form

D x D y (A n ) D x (A n )
= A n+1 A n−1 . (6.5.6)
D y (A n ) A n