Determinants and Their Applications in Mathematical Physics

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)

A

.