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 bothAnandBnbe defined as

φm(x)=f

(m)

(x),f(x) arbitrary,

so that

φ

′

m=φm+1 (6.5.4)

and bothAnandBnare Wronskians (Section 4.7) whose derivatives are

given by

A

′

n=−A

(n+1)

n+1,n,

B

′

n=−B

(n+1)

n+1,n

. (6.5.5)

Theorem 6.1. The equation

u

′

n

=

unun+1

un− 1

is satisfied by the function defined separately for odd and even values ofn

as follows:

u 2 n− 1 =

An

Bn− 1

,

u 2 n=

Bn

An

.

Proof.

B

2

n− 1 u

′

2 n− 1 =Bn− 1 A

′

n−AnB

′

n− 1

=−Bn− 1 A

(n+1)

n+1,n+AnB

(n)

n,n− 1

B

2

n− 1

(

u 2 n− 1 u 2 n

u 2 n− 2

)

=An− 1 Bn.

Hence, referring to (6.5.3),

B

2

n− 1

[

u 2 n− 1 u 2 n

u 2 n− 2

−u

′

2 n− 1

]

=An− 1 Bn+Bn− 1 A

(n+1)

n+1,n

−AnB

(n)

n,n− 1

=0,

which proves the theorem whennis odd.

A

2

n

u

′

2 n

=AnB

′

n

−BnA

′

n

=−AnB

(n+1)

n+1,n

+BnA

(n+1)

n,n+1

,

A

2

n

(

u 2 nu 2 n+1

u 2 n− 1

)

=An+1Bn− 1.

Hence, referring to (6.5.2),

A

2

n

[

u 2 nu 2 n+1

u 2 n− 1

−u

′

2 n

]

=An+1Bn− 1 +AnB

(n+1)

n+1,n−BnA

(n+1)

n,n+1