Determinants and Their Applications in Mathematical Physics

5.2 The Generalized Cusick Identities 185

From (5.2.4),

∂ai, 2 n

∂φ 2 n− 1

=ψi.

Also,

∂Hn

∂φ 2 n− 1

=Hn− 1.

Hence,

2 n− 1

∑

i=1

(−1)

i+1

Pf

(n)

i

ψi=Hn− 1 Kn. (5.2.24)

Similarly,

2 n− 1

∑

i=1

(−1)

i+1

Pf

(n)

i φi=HnKn−^1. (5.2.25)

The following three theorems express modified forms of|aij|nin terms of

the Hankelians.

LetBn(φ) denote the determinant which is obtained from|aij|nby

replacing the last row by the row

[

φ 1 φ 2 φ 3 ...φn

]

.

Theorem 5.6.

a.B 2 n− 1 (φ)=Hn− 1 HnK

2

n− 1 ,

b.B 2 n− 1 (ψ)=H

2

n− 1 Kn−^1 Kn,

c. B 2 n(φ)=−H

2

n

Kn− 1 Kn,

d.B 2 n(ψ)=−Hn− 1 HnK
2
n

.

Proof. ExpandingB 2 n− 1 (φ) by elements from the last row and their

cofactors and referring to (5.2.13), (5.2.14), and (5.2.25),

B 2 n− 1 (φ)=

2 n− 1

∑

j=1

φjA

(2n−1)

2 n− 1 ,j

=Pf

(n)

2 n− 1

2 n− 1

∑

i=1

(−1)

i+1

Pf

(n)

i φi

=Pfn− 1 HnKn− 1. (5.2.26)

Part (a) now follows from Theorem 5.1 and (b) is proved in a similar

manner.

ExpandingB 2 n(φ) with the aid of Theorem 3.9 on bordered determinants

(Section 3.7) and referring to (5.2.11) and (5.2.25),

B 2 n(φ)=−

2 n− 1

∑

i=1

2 n− 1

∑

j=1

ai, 2 nφjA

(2n−1)

ij