Determinants and Their Applications in Mathematical Physics

6.5 The Toda Equations 257

whereAnandBnare Hankelians defined as

An=|φm|n, 0 ≤m≤ 2 n− 2 ,

Bn=|φm|n, 1 ≤m≤ 2 n− 1 ,

φ

′

m=(m+1)φm+1.

Proof.

B

(n)

1 n

=(−1)

n+1

A

(n)

11

,

A

(n+1)

1 ,n+1

=(−1)

n

Bn. (6.5.10)

It follows from Theorems 4.35 and 4.36 in Section 4.9.2 on derivatives of

Turanians that

D(An)=−(2n−1)A

(n+1)

n+1,n,

D(Bn)=− 2 nB

(n+1)

n,n+1

,

D(A

(n)

11 )=−(2n−1)A

(n+1)

1 ,n+1;1n,

D(B

(n)

11

)=− 2 nB

(n+1)

1 ,n+1;1,n

. (6.5.11)

The algebraic identity in Theorem 4.29 in Section 4.8.5 on Turanians is

satisfied by bothAnandBn.

B

2

n

y

′

2 n− 1

=BnD(B

(n)

11

)−B

(n)

11

D(Bn)

=2n

[

B

(n)

11

B

(n+1)

n,n+1

−BnB

(n+1)

1 ,n+1;1n

]

=2nB

(n)

1 nB

(n+1)

1 ,n+1

=− 2 nA

(n)

11

A

(n+1)

11

.

Applying the Jacobi identity,

A

(n)

11

A

(n+1)

11

(y 2 n−y 2 n− 2 )=An+1A

(n)

11

−AnA

(n+1)

11

=An+1A

(n+1)

1 ,n+1;1,n+1

−A

n+1

n+1,n+1

A

(n+1)

11

=−

[

A

(n+1)

1 ,n+1

] 2

=−B

2

n.

Hence,

y

′

2 n− 1

(y 2 n−y 2 n− 2 )=2n,

which proves the theorem whennis odd.

[

A

(n+1)

1 ,n+1

] 2

y

′

2 n=A

(n+1)

11

D(An+1)−An+1D(A

(n+1)

11

)

=(2n+1)

[

An+1A

(n+2)

1 ,n+2;1,n+1−A

(n+1)

11 A

(n+2)

n+2,n+1

]

=−(2n+1)A

(n+1)

1 ,n+1

A

(n+2)

1 ,n+2

.