Determinants and Their Applications in Mathematical Physics

110 4. Particular Determinants

Proof. The identity is a particular case of Jacobi variant (A) (Sec-

tion 3.6.3),

∣

∣

∣

∣

∣

T

(n)

ip

T

(n+1)

i,n+1

T

(n)

jp

T

(n+1)

j,n+1

∣

∣

∣

∣

∣

−TnT

(n+1)

ij;p,n+1=0, (4.8.20)

where (i, j, p)=(1,n,1).

Let

An=T

(n,r)

,

Bn=T

(n,r+1)

.

Then Theorem 4.29 is satisfied by bothAnandBn.

Theorem 4.30. For all values ofr,

a.AnB

(n+1)

n+1,n

−BnA

(n+1)

n+1,n

+An+1Bn− 1 =0.

b.Bn− 1 A

(n+1)

n+1,n−AnB

(n)

n,n− 1 +An−^1 Bn=0.

Proof.

Bn=(−1)

n

A

(n+1)

1 ,n+1

,

B

(n+1)

n+1,n=(−1)

n

A

(n+1)

n 1 ,

Bn− 1 =(−1)

n− 1

A

(n)

1 n

=(−1)

n

A

(n+1)

n,n+1;1,n+1,

A

(n+1)

1 n;n,n+1

=A

(n)

1 ,n− 1

=(−1)

n− 1

B

(n)

n,n− 1

. (4.8.21)

Denote the left-hand side of (a) byYn. Then, applying the Jacobi identity

toAn+1,

(−1)

n

Yn=

∣

∣

∣

∣

∣

A

(n+1)

n 1

A

(n+1)

n,n+1

A

(n+1)

n+1, 1

A

(n+1)

n+1,n+1

∣

∣

∣

∣

∣

−An+1A

(n+1)

n,n+1;1,n+1

=0,

which proves (a).

The particular case of (4.8.20) in which (i, j, p)=(n, 1 ,n) andT is

replaced byAis

∣

∣

∣

∣

∣

An− 1 A

(n+1)

n,n+1

A

(n)

1 n A

(n+1)

1 ,n+1

∣

∣

∣

∣

∣

−AnA

(n+1)

n1;n,n+1

=0. (4.8.22)

The application of (4.8.21) yields (b).

This theorem is applied in Section 6.5.1 on Toda equations.