Page 126 - Determinants and Their Applications in Mathematical Physics

4.8 Hankelians 1 111

4.8.6 Partial Derivatives with Respect to φ m
In A n , the elements φ m , φ 2n−2−m ,0 ≤ m ≤ n − 2, each appear in (m +1)
positions. The element φ n−1 appears in n positions, all in the secondary
diagonal. Hence, ∂A n /∂φ m is the sum of a number of cofactors, one for
each appearance of φ m . Discarding the suffix n,
∂A
= A pq . (4.8.23)
p+q=m+2
∂φ m
For example, when n ≥ 4,
∂A
=
∂φ 3 A pq
p+q=5
= A 41 + A 32 + A 23 + A 14 .
By a similar argument,

= A ip,jq , (4.8.24)
∂A ij
p+q=m+2
∂φ m

= A irp,jsq . (4.8.25)
∂A ir,js
p+q=m+2
∂φ m
Partial derivatives of the scaled cofactors A ij and A ir,js can be obtained
from (4.8.23)–(4.8.25) with the aid of the Jacobi identity:
∂A ij
= − A A pj (4.8.26)
iq
p+q=m+2
∂φ m

A ij A
iq
= . (4.8.27)
A pj
•
p+q=m+2
The proof is simple.
Lemma.

A ij A is A iq
∂A ir,js
= A rj A rs A rq , (4.8.28)

p+q=m+2 A pj A ps •
∂φ m

which is a development of (4.8.27).
Proof.
∂A ir,js 1 ∂A
= A ∂A ir,js
A 2 ∂φ m − A ir,js ∂φ m
∂φ m
1
=
A 2 AA irp,jsq − A ir,js A pq
p,q

= A irp,jsq − A ir,js A pq . (4.8.29)
p,q