Page 195 - Determinants and Their Applications in Mathematical Physics

180 5. Further Determinant Theory

Assume that
Pf m = H m K m , m < n. (5.2.10)
The method by which the theorem is proved for all values of n is outlined
as follows.
Pf n is expressible in terms of Pfaffians of lower order by the formula
2n−1
i+1 (n)
Pf n = (−1) Pf a i,2n , (5.2.11)
i
i=1
where, in this context, a i,2n is defined as a sum in (5.2.4) so that Pf n
is expressible as a double sum. The introduction of a variable x enables
the inductive assumption (5.2.10) to be expressed as the equality of two
polynomials in x. By equating coefficients of one particular power of x,an
(n)
identity is found which expresses Pf as the sum of products of cofactors
i
of H n and K n (Lemma 5.5). Hence, Pf n is expressible as a triple sum
containing the cofactors of H n and K n . Finally, with the aid of an identity
in Appendix A.3, it is shown that the triple sum simplifies to the product
H n K n .
The following Pfaffian identities will also be applied.

(n) (2n−1) 1/2

Pf = A , (5.2.12)
i ii
(n) (n) (2n−1)
(−1) i+j Pf Pf = A , (5.2.13)
i j ij
(n)
Pf =Pf n−1 . (5.2.14)
2n−1
The proof proceeds with a series of lemmas.
5.2.2 Four Lemmas
Let a ∗ be the function obtained from a ij by replacing each φ r by
ij
(φ r − xφ r+1 ) and by replacing each ψ r by (ψ r − xψ r+1 ).
Lemma 5.2.
2
a = a ij − (a i,j+1 + a i+1,j )x + a i+1,j+1 x .
∗
ij
Proof.
j−i

a = (φ s+i−1 − xφ s+i )(ψ j−s − xψ j−s+1 )
∗
ij
s=1
2
= a ij − (s 1 + s 2 )x + s 3 x ,
where
j−i

s 1 = φ s+i−1 ψ j−s+1
s=1
= a i,j+1 − φ i ψ j ,