Determinants and Their Applications in Mathematical Physics

112 4. Particular Determinants

The lemma follows from the second-order and third-order Jacobi identi-

ties.

4.8.7 Double-Sum Relations

WhenAnis a Hankelian, the double-sum relations (A)–(D) in Section 3.4

withfr=gr=

1

2

can be expressed as follows. Discarding the suffixn,

A

′

A

=D(logA)=

(^2) ∑n− 2
m=0
φ
′
m

∑

p+q=m+2

A

pq

, (A 1 )

(A

ij

)

′

=−

2 n− 2

∑

m=0

φ

′

m

∑

p+q=m+2

A

ip

A

jq

, (B 1 )

2 n− 2

∑

m=0

φm

∑

p+q=m+2

A

pq

=n, (C 1 )

2 n− 2

∑

m=0

φm

∑

p+q=m+2

A

ip

A

jq

=A

ij

. (D 1 )

Equations (C 1 ) and (D 1 ) can be proved by puttingaij=φi+j− 2 in (C)

and (D), respectively, and rearranging the double sum, but they can also

be proved directly by taking advantage of the first kind of homogeneity of

Hankelians and applying the Euler theorem in Appendix A.9.

AnandA

(n)

ij

are homogeneous polynomial functions of their elements of

degreesnandn−1, respectively, so thatA

ij

nis a homogeneous function of

degree (−1). Hence, denoting the sums in (C 1 ) and (D 1 )byS 1 andS 2 ,

AS 1 =

2 n− 2

∑

m=0

φm

∂A

∂φm

=nA,

S 2 =−

2 n− 2

∑

m=0

φm

∂A

ij

∂φm

=A

ij

.

which prove (C 1 ) and (D 1 ).

Theorem 4.31.

2 n− 2

∑

m=1

mφm

∑

p+q=m+2

A

pq

=n(n−1), (C 2 )