4.10 Henkelians 3 131
=
∑
i,j,r,s
AisAjr
(2i−1)(2j−1)
[p
2
x
2(i+j−1)
+q
2
y
2(i+j−1)
−1]
=
∑
i,j,r,s
(i+j−1)aijAisArj
(2i−1)(2j−1)
=
1
2
∑
i,j,r,s
(
1
2 i− 1
+
1
2 j− 1
)
aijAisArj
=
∑
i,j,r,s
aijAisArj
2 i− 1
=
∑
i,s
Ais
2 i− 1
∑
r
∑
j
aijArj
=A
∑
i,s
Ais
2 i− 1
∑
r
δir
=−AW
which proves the theorem.
Theorem 4.40.
p
2
V
2
(x)+q
2
V
2
(y)=W
2
−AW.
This theorem resembles Theorem 4.39 closely, but the following proof
bears little resemblance to the proof of Theorem 4.39. Applying double-sum
identity (D) in Section 3.4 withfr=randgs=s−1,
∑
r
∑
s
[
p
2
x
2(r+s−1)
+q
2
y
2(r+s−1)
− 1
]
A
is
A
rj
=(i+j−1)A
ij
,
p
2
[
∑
s
A
is
x
2 s− 1
][
∑
r
A
rj
x
2 r− 1
]
+q
2
[
∑
s
A
is
y
2 s− 1
][
∑
r
A
rj
y
2 r− 1
]
−
[
∑
s
A
is
][
∑
r
A
rj
]
=(i+j−1)A
ij
.
Put
λi(x)=
∑
j
A
ij
x
2 j− 1
.
Then,
p
2
λi(x)λj(x)+q
2
λi(y)λj(y)−λi(1)λj(1)=(i+j−1)A
ij
.
Divide by (2i−1)(2j−1), sum overiandjand note that
∑
i
λi(x)
2 i− 1
=−
V(x)