Determinants and Their Applications in Mathematical Physics

4.8 Hankelians 1 113

2 n− 2

∑

m=1

mφm

∑

p+q=m+1

A

ip

A

jq

=(i+j−2)A

ij

. (D 2 )

These can be proved by puttingaij=φi+j− 2 andfr=gr=r−1 in (C)

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

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

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

Proof. AnandA
ij
n
are homogeneous functions of degreen(n−1) and

(2−i−j), respectively, in the suffixes of their elements. Hence, denoting

the sums byS 1 andS 2. respectively,

AS 1 =

2 n− 2

∑

m=1

mφm

∂A

∂φm

=n(n−1)A,

S 2 =−

2 n− 2

∑

m=1

mφm

∂A

ij

∂φm

=−(2−i−j)A

ij

.

The theorem follows.

Theorem 4.32.

n

∑

r=1

n

∑

s=1

(r+s−2)φr+s− 3 A

rs

=0, (E)

which can be rearranged in the form

2 n− 2

∑

m=1

mφm− 1

∑

p+q=m+2

A

pq

=0 (E 1 )

and

n

∑

r=1

n

∑

s=1

(r+s−2)φr+s− 3 A

ir

A

sj

=iA

i+1,j

+jA

i,j+1

=0, (i, j)=(n, n).

(F)

which can be rearranged in the form

2 n− 2

∑

m=1

mφm− 1

∑

p+q=m+2

A

ip

A

jq

=iA

i+1,j

+jA

i,j+1

=0, (i, j)=(n, n).

(F 1 )