Page 127 - 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

When A n is a Hankelian, the double-sum relations (A)–(D) in Section 3.4
with f r = g r = 1 can be expressed as follows. Discarding the suffix n,
2
2n−2
A
= D(log A)= φ A , (A 1 )
pq
A m
m=0 p+q=m+2
2n−2

ij ip jq
(A ) = − φ A A , (B 1 )
m
m=0 p+q=m+2
2n−2

A pq = n, (C 1 )
φ m
m=0 p+q=m+2
2n−2

A A jq = A . (D 1 )
ip
ij
φ m
m=0 p+q=m+2
Equations (C 1 ) and (D 1 ) can be proved by putting a ij = φ i+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.
(n)
A n and A are homogeneous polynomial functions of their elements of
ij
degrees n and n − 1, respectively, so that A ij is a homogeneous function of
n
degree (−1). Hence, denoting the sums in (C 1 ) and (D 1 )by S 1 and S 2 ,
2n−2
∂A
AS 1 = φ m
m=0 ∂φ m
= nA,
2n−2
∂A ij

S 2 = − φ m
m=0 ∂φ m
= A .
ij
which prove (C 1 ) and (D 1 ).
Theorem 4.31.
2n−2

A pq = n(n − 1), (C 2 )
mφ m
m=1 p+q=m+2