Page 270 - Determinants and Their Applications in Mathematical Physics

6.5 The Toda Equations 255

The derivative of A n with respect to x, as obtained by differentiating the
rows, consists of the sum of n determinants, only one of which is nonzero.
That determinant is a cofactor of A n+1 :
(n+1)
D x (A n )= −A n,n+1 .
Differentiating the columns with respect to y and then the rows with respect
to x,
(n+1)
D y (A n )= −A ,
n+1,n
D x D y (A n )= A (n+1) . (6.5.7)
nn
Denote the determinant in (6.5.6) by E. Then, applying the Jacobi identity
(Section 3.6) to A n+1 ,

(n+1)
−A
(n+1)
A nn n,n+1
(n+1) (n+1)
E =
−A A
n+1,n+1
n+1,n
(n+1)
= A n+1 A
n,n+1;n,n+1
which simplifies to the right side of (6.5.6).
It follows as a corollary that the equation
d
u n+1 u n−1
2
D (log u n )= , D = ,
u 2 dx
n
is satisfied by the Hankel–Wronskian
u n = A n = |D i+j−2 (f)| n ,
where the function f = f(x) is arbitrary.
Theorem 6.4. The equation
1 u n+1 u n−1 d
D ρ ρD ρ (log u n ) = , D ρ = ,
ρ u 2 dρ
n
is satisfied by the function
u n = A n = e −n(n−1)x B n , (6.5.8)
where
i+j−2
B n = (ρD ρ ) f(ρ) , f(ρ) arbitrary.

n
Proof. Put ρ = e . Then, ρD ρ = D x and the equation becomes
x
2
2
D (log A n )= ρ A n+1 A n−1 . (6.5.9)
A
x 2
n
Applying (6.5.8) to transform this equation from A n to B n ,
2
2
D (log B n )= D (log A n )
x x
2
= ρ B n+1 B n−1 −[(n+1)n+(n−1)(n−2)−2n(n−1)]x
e
B 2
n