4.8 Hankelians 1 1114.8.6 Partial Derivatives with Respect toφm.......
InAn, the elementsφm,φ 2 n− 2 −m,0≤m≤n−2, each appear in (m+1)
positions. The elementφn− 1 appears innpositions, all in the secondary
diagonal. Hence,∂An/∂φmis the sum of a number of cofactors, one for
each appearance ofφm. Discarding the suffixn,
∂A
∂φm=
∑
p+q=m+2Apq. (4.8.23)For example, whenn≥4,
∂A
∂φ 3=
∑
p+q=5Apq=A 41 +A 32 +A 23 +A 14.
By a similar argument,
∂Aij∂φm=
∑
p+q=m+2Aip,jq, (4.8.24)∂Air,js∂φm=
∑
p+q=m+2Airp,jsq. (4.8.25)Partial derivatives of the scaled cofactorsA
ij
andAir,js
can be obtainedfrom (4.8.23)–(4.8.25) with the aid of the Jacobi identity:
∂A
ij∂φm=−
∑
p+q=m+2A
iq
Apj
(4.8.26)=
∑
p+q=m+2∣
∣
∣
∣
A
ij
AiqA
pj∣
∣
∣
∣
. (4.8.27)
The proof is simple.
Lemma.
∂A
ir,js∂φm=
∑
p+q=m+2∣ ∣ ∣ ∣ ∣ ∣
A
ij
Ais
AiqA
rj
Ars
ArqA
pj
A
ps∣ ∣ ∣ ∣ ∣ ∣
, (4.8.28)
which is a development of (4.8.27).
Proof.
∂A
ir,js∂φm=
1
A
2[
A
∂Air,js∂φm−Air,js∂A
∂φm]
=
1
A
2∑
p,q[
AAirp,jsq−Air,jsApq]
=
∑
p,q[
A
irp,jsq
−Air,js
Apq