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.