Page 146 - Determinants and Their Applications in Mathematical Physics

4.10 Henkelians 3 131

2 2(i+j−1) 2 2(i+j−1)
= A is A jr [p x + q y − 1]
(2i − 1)(2j − 1)
i,j,r,s

= (i + j − 1)a ij A is A rj
(2i − 1)(2j − 1)
i,j,r,s

1 1
1

= +
2 2i − 1 2j − 1 a ij A is A rj
i,j,r,s

= a ij A is A rj
2i − 1
i,j,r,s

= A is
2i − 1 a ij A rj
r
i,s j

= A A is
2i − 1 δ ir
i,s r
= − AW
which proves the theorem.
Theorem 4.40.
2
2
2
2
2
p V (x)+ q V (y)= W − 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 with f r = r and g s = s − 1,
2 2(r+s−1)
2 2(r+s−1)

p x + q y − 1 A A rj =(i + j − 1)A ,
is
ij
r s

is 2s−1
rj 2r−1
rj 2r−1
is 2s−1
p 2 A x A x + q 2 A y A y
s r s r

ij
− A is A rj =(i + j − 1)A .
s r
Put
ij 2j−1
λ i (x)= A x .
j
Then,
2
2
p λ i (x)λ j (x)+ q λ i (y)λ j (y) − λ i (1)λ j (1)=(i + j − 1)A .
ij
Divide by (2i − 1)(2j − 1), sum over i and j and note that
λ i (x) V (x)
= − .
2i − 1 A
i