114 4. Particular Determinants
Proof of (F).Denote the sum bySand apply the Hankelian relation
φr+s− 3 =ar,s− 1 =ar− 1 ,s.
S=
∑n
s=1
(s−1)A
sj
∑n
r=1
ar,s− 1 A
ir
+
∑n
r=1
(r−1)A
ir
∑n
s=1
ar− 1 ,sA
sj
=
n
∑
s=1
(s−1)A
sj
δs− 1 ,i+
n
∑
r=1
(r−1)A
ir
δr− 1 ,j.
The proof of (F) follows. Equation (E) is proved in a similar manner.
Exercises
Prove the following:
1.
∑
p+q=m+2
A
ij,pq
=0.
2.
2 n− 2
∑
m=0
φm
∑
q+q=m+2
Aip,jq=(n−1)Aij.
3.
2 n− 2
∑
m=1
mφm
∑
p+q=m+2
Aip,jq=(n
2
−n−i−j+2)Aij.
4.
2 n− 2
∑
m=0
φm
∑
p+q=m+2
A
ijp,hkq
=nA
ij,hk
.
5.
2 n− 2
∑
m=1
mφm
∑
p+q=m+2
A
ijp,hkq
=(n
2
−n−i−j−h−k−4)A
ij,hk
.
6.
2 n− 2
∑
m=1
mφm− 1
∑
p+q=m+2
A
ijp,hkq
=iA
i+1,j;hk
+jA
i,j+1;hk
+hA
ij;h+1,k
+kA
ij;h,k+1
.
7.
2 n− 2
∑
m=0
∑
p+q=m+2
φp+r− 1 φq+r− 1 A
pq
=φ 2 r, 0 ≤r≤n−1.
8.
2 n− 2
∑
m=1
m
∑
p+q=m+2
φp+r− 1 φq+r− 1 A
pq
=2rφ 2 r, 0 ≤r≤n−1.
9.Prove that
n− 1
∑
r=1
rA
r+1,j
n
∑
m=1
φm+r− 2 A
im
=iA
i+1,j
by applying the sum formula for Hankelians and, hence, prove (F 1 )
directly. Use a similar method to prove (E 1 ) directly.