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),

[

B

(n+1)

1 ,n+1

] 2

y

′

2 n

=(2n+1)BnBn+1,

BnBn+1(y 2 n− 1 −y 2 n+1)=BnB

(n+1)

11

−Bn+1B

(n)

11

=B

(n+1)

n+1,n+1

B

(n+1)

11

−Bn+1B

(n+1)

1 ,n+1;1,n+1

=

[

B

(n+1)

1 ,n+1

] 2

Hence,

y

′

2 n(y^2 n−^1 −y^2 n+1)=2n+1,

which proves the theorem whennis even.

6.6 The Matsukidaira–Satsuma Equations

6.6.1 A System With One Continuous and One Discrete

Variable

LetA
(n)
(r) denote the Turanian–Wronskian of orderndefined as follows:

A

(n)

(r)=

∣

∣f

r+i+j− 2

∣

∣

n

, (6.6.1)

wherefs=fs(x) andf

′

s=fs+1. Then,

A

(n)

11 (r)=A

(n−1)

(r+2),

A

(n)

1 n

(r)=A

(n−1)

(r+1).

Let

τr=A

(n)

(r). (6.6.2)

Theorem 6.7.
∣
∣
∣
∣

τr+1 τr

τr τr− 1

∣

∣

∣

∣

∣

∣

∣

∣

τ

′′

r τ

′

r

τ

′

r τr

∣

∣

∣

∣

=

∣

∣

∣

∣

τ

′

r+1 τr+1

τ

′

r τr

∣

∣

∣

∣

∣

∣

∣

∣

τ

′

r τr

τ

′

r− 1 τr−^1

∣

∣

∣

∣

for all values ofnand all differentiable functionsfs(x).

Proof. Each of the functions

τr± 1 ,τr+2,τ

′

r,τ

′′

r,τ

′

r± 1

can be expressed as a cofactor ofA

(n+1)

with various parameters:

τr=A

(n+1)

n+1,n+1

(r),

τr+1=(−1)

n

A

(n+1)

1 ,n+1

(r)

=(−1)

n

A

(n+1)

n+1, 1 (r)

τr+2=A

(n+1)

11

(r).