Determinants and Their Applications in Mathematical Physics

5.2 The Generalized Cusick Identities 181

s 2 =ai+1,j+φiψj,

s 3 =ai+1,j+1.

The lemma follows.

Let

A

∗

2 n

=|a

∗

ij

| 2 n,

Pf

∗

n=

(

A

∗

2 n

)

1 / 2

. (5.2.15)

Lemma 5.3.

2 n− 1

∑

i=1

(−1)

i+1

Pf

(n)

i

x

2 n−i− 1

=Pf

∗

n− 1

.

Proof. Denote the sum byFn. Then, referring to (5.2.13) and Section 3.7

on bordered determinants,

F

2

n=

2 n− 1

∑

i=1

2 n− 1

∑

j=1

(−1)

i+j

Pf

(n)

i

Pf

(n)

j

x

4 n−i−j− 2

=

2 n− 1

∑

i=1

2 n− 1

∑

j=1

A

(2n−1)

ij

x

4 n−i−j− 2

=−

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a 11 a 12 ··· a 1 , 2 n− 1 x

2 n− 2

a 21 a 22 ··· a 2 , 2 n− 1 x

2 n− 3

.........................................

a 2 n− 1 , 1 a 2 n− 1 , 2 ··· a 2 n− 1 , 2 n− 1 1

x

2 n− 2

x

2 n− 3

··· 1 •

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

2 n

. (5.2.16)

(It is not necessary to putaii= 0, etc., in order to prove the lemma.)

Eliminate thex’s from the last column and row by means of the row and

column operations

R

′

i=Ri−xRi+1,^1 ≤i≤^2 n−^2 ,

C

′

j=Cj−xCj+1,^1 ≤j≤^2 n−^2. (5.2.17)

The result is

F

2

n

=−

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a

∗

11

a

∗

12

··· a

∗

1 , 2 n− 1

•

a

∗

21

a

∗

22

··· a

∗

2 , 2 n− 1

•

.....................................

a

∗

2 n− 1 , 1

a

∗

2 n− 1 , 2

··· a

∗

2 n− 1 , 2 n− 1

1

• • ··· 1 •

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

2 n

=+|a

∗

ij|^2 n−^2

=A

∗

2 n− 2.

The lemma follows by taking the square root of each side.