Determinants and Their Applications in Mathematical Physics

5.2 The Generalized Cusick Identities 179

In particular,

ai, 2 n=

2 n−i

∑

s=1

φs+i− 1 ψ 2 n−s, 1 ≤i≤ 2 n− 1. (5.2.4)

LetA 2 ndenote the skew-symmetric determinant of order 2ndefined as

A 2 n=|aij| 2 n, (5.2.5)

whereaijis defined by (5.2.3) for 1≤i≤j≤ 2 nandaji=−aij, which

impliesaii=0.

LetHnandKndenote Hankelians of orderndefined as

Hn=

{

|hij|n,hij=φi+j− 1

|φm|n, 1 ≤m≤ 2 n−1;

(5.2.6)

Kn=

{

|kij|n,kij=ψi+j− 1

|ψm|n, 1 ≤m≤ 2 n−1.

(5.2.7)

All the elementsφrandψrwhich appear inHnandKn, respectively, also

appear ina 1 , 2 nand therefore also inA 2 n. The principal identity is given

by the following theorem.

Theorem 5.1.

A 2 n=H

2

n

K

2

n

.

However, since

A 2 n=Pf

2

n

,

wherePfnis a Pfaffian (Section 4.3.3), the theorem can be expressed in the

form

Pfn=HnKn. (5.2.8)

Since Pfaffians are uniquely defined, there is no ambiguity in sign in this

relation.

The proof uses the method of induction. It may be verified from (4.3.25)

and (5.2.3) that

Pf 1 =a 12 =φ 1 ψ 1 =H 1 K 1 ,

Pf 2 =

∣

∣

φ 1 ψ 1 φ 1 ψ 2 +φ 2 ψ 1 φ 1 ψ 3 +φ 2 ψ 2 +φ 3 ψ 1

φ 2 ψ 2 φ 2 ψ 3 +φ 3 ψ 2

φ 3 ψ 3

∣ ∣ ∣ ∣ ∣ ∣ =

∣

∣

∣

∣

φ 1 φ 2

φ 2 φ 3

∣

∣

∣

∣

∣

∣

∣

∣

ψ 1 ψ 2

ψ 2 ψ 3

∣

∣

∣

∣

=H 2 K 2 (5.2.9)

so that the theorem is known to be true whenn= 1 and 2.