Page 194 - Determinants and Their Applications in Mathematical Physics

5.2 The Generalized Cusick Identities 179

In particular,

2n−i

a i,2n = φ s+i−1 ψ 2n−s , 1 ≤ i ≤ 2n − 1. (5.2.4)
s=1
Let A 2n denote the skew-symmetric determinant of order 2n defined as
A 2n = |a ij | 2n , (5.2.5)
where a ij is defined by (5.2.3) for 1 ≤ i ≤ j ≤ 2n and a ji = −a ij , which
implies a ii =0.
Let H n and K n denote Hankelians of order n defined as

|h ij | n , h ij = φ i+j−1
H n = (5.2.6)
|φ m | n , 1 ≤ m ≤ 2n − 1;

|k ij | n , k ij = ψ i+j−1
K n = (5.2.7)
|ψ m | n , 1 ≤ m ≤ 2n − 1.
All the elements φ r and ψ r which appear in H n and K n , respectively, also
appear in a 1,2n and therefore also in A 2n . The principal identity is given
by the following theorem.
Theorem 5.1.
2
2
A 2n = H K .
n n
However, since
2
A 2n =Pf ,
n
where Pf n is a Pfaffian (Section 4.3.3), the theorem can be expressed in the
form
Pf n = H n K n . (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 ψ 1
= φ 2 ψ 2
φ 2 φ 3 ψ 2 ψ 3

(5.2.9)
= H 2 K 2
so that the theorem is known to be true when n = 1 and 2.