Determinants and Their Applications in Mathematical Physics

180 5. Further Determinant Theory

Assume that

Pfm=HmKm,m<n. (5.2.10)

The method by which the theorem is proved for all values ofnis outlined

as follows.

Pfnis expressible in terms of Pfaffians of lower order by the formula

Pfn=

2 n− 1

∑

i=1

(−1)

i+1

Pf

(n)

i

ai, 2 n, (5.2.11)

where, in this context,ai, 2 nis defined as a sum in (5.2.4) so that Pfn

is expressible as a double sum. The introduction of a variablexenables

the inductive assumption (5.2.10) to be expressed as the equality of two

polynomials inx. By equating coefficients of one particular power ofx,an

identity is found which expresses Pf

(n)

i

as the sum of products of cofactors

ofHnandKn(Lemma 5.5). Hence, Pfn is expressible as a triple sum

containing the cofactors ofHnandKn. Finally, with the aid of an identity

in Appendix A.3, it is shown that the triple sum simplifies to the product

HnKn.

The following Pfaffian identities will also be applied.

Pf

(n)

i =

(

A

(2n−1)

ii

) 1 / 2

, (5.2.12)

(−1)

i+j

Pf

(n)

i

Pf

(n)

j

=A

(2n−1)

ij

, (5.2.13)

Pf

(n)

2 n− 1

=Pfn− 1. (5.2.14)

The proof proceeds with a series of lemmas.

5.2.2 Four Lemmas......................

Let a

∗

ij be the function obtained from aij by replacing each φr by

(φr−xφr+1) and by replacing eachψrby (ψr−xψr+1).

Lemma 5.2.

a

∗

ij=aij−(ai,j+1+ai+1,j)x+ai+1,j+1x

2

.

Proof.

a

∗

ij=

j−i

∑

s=1

(φs+i− 1 −xφs+i)(ψj−s−xψj−s+1)

=aij−(s 1 +s 2 )x+s 3 x

2

,

where

s 1 =

j−i

∑

s=1

φs+i− 1 ψj−s+1

=ai,j+1−φiψj,