Page 273 - Determinants and Their Applications in Mathematical Physics

258 6. Applications of Determinants in Mathematical Physics

Hence, referring to the first equation in (4.5.10),
(n+1) 2

B y =(2n +1)B n B n+1 ,
1,n+1 2n
(n+1) (n)
B n B n+1 (y 2n−1 − y 2n+1 )= B n B 11 − B n+1 B 11
(n+1) (n+1) (n+1)
= B n+1,n+1 B 11 − B n+1 B 1,n+1;1,n+1
(n+1) 2

= B .
1,n+1
Hence,
y (y 2n−1 − y 2n+1 )=2n +1,

2n
which proves the theorem when n is even.
6.6 The Matsukidaira–Satsuma Equations

6.6.1 A System With One Continuous and One Discrete
Variable

Let A (n) (r) denote the Turanian–Wronskian of order n defined as follows:

A (n) (r)= f r+i+j−2 , (6.6.1)

n
where f s = f s (x) and f = f s+1 . Then,

s
(n) (n−1)
A (r)= A (r +2),
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(n) (n−1)
A (r)= A (r +1).
1n
Let
τ r = A (n) (r). (6.6.2)
Theorem 6.7.

τ τ τ τ τ
r+1 τ r r r = r+1 τ r+1 r τ r

τ τ τ
τ r−1 τ r τ r r−1 τ r−1

τ r
r r
for all values of n and all differentiable functions f s (x).
Proof. Each of the functions

τ r±1 ,τ r+2 ,τ ,τ ,τ r±1

r r
can be expressed as a cofactor of A (n+1) with various parameters:
(n+1)
τ r = A (r),
n+1,n+1
(n+1)
τ r+1 =(−1) A 1,n+1 (r)
n
(n+1)
=(−1) A (r)
n
n+1,1
(n+1)
τ r+2 = A (r).
11