Page 125 - 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),

(n)
T T (n+1) (n+1)

ip i,n+1 − T n T =0, (4.8.20)
(n) (n+1) ij;p,n+1
T T
jp j,n+1
where (i, j, p)=(1,n, 1).
Let
A n = T (n,r) ,
B n = T (n,r+1) .

Then Theorem 4.29 is satisfied by both A n and B n .
Theorem 4.30. For all values of r,
(n+1) (n+1)
a. A n B − B n A + A n+1 B n−1 =0.
n+1,n n+1,n
(n+1) (n)
b. B n−1 A − A n B n,n−1 + A n−1 B n =0.
n+1,n
Proof.
(n+1)
B n =(−1) A ,
n
1,n+1
(n+1) (n+1)
B =(−1) A n1 ,
n
n+1,n
B n−1 =(−1) n−1 A (n)
1n
(n+1)
=(−1) A ,
n
n,n+1;1,n+1
(n+1) (n)
A 1n;n,n+1 = A 1,n−1
=(−1) n−1 B (n) . (4.8.21)
n,n−1
Denote the left-hand side of (a) by Y n . Then, applying the Jacobi identity
to A n+1 ,

(n+1)
A A (n+1) (n+1)

n n1 n,n+1 − A n+1 A
(n+1) (n+1) n,n+1;1,n+1
(−1) Y n =
A A

n+1,1 n+1,n+1
=0,
which proves (a).
The particular case of (4.8.20) in which (i, j, p)=(n, 1,n) and T is
replaced by A is

A (n+1)
A n−1 (n+1)
n,n+1 − A n A =0. (4.8.22)
(n) (n+1) n1;n,n+1
A A
1,n+1
1n
The application of (4.8.21) yields (b).
This theorem is applied in Section 6.5.1 on Toda equations.