Page 147 - Determinants and Their Applications in Mathematical Physics

132 4. Particular Determinants

The result is
i + j − 1
1 2 2 2 2 2
p V (x)+ q V (y) − W = A ij
A 2 (2i − 1)(2j − 1)
i j

1 1
= 1 + A ij
2 2i − 1 2j − 1
i j
W
= − .
A
The theorem follows. The determinant W appears in Section 5.8.6.
Theorem 4.41. In the particular case in which (p, q)=(1, 0),
V (x)=(−1) n+1 U(x).
Proof.
x 2(i+j−1) − 1
a ij = = a ji ,
i + j − 1
which is independent of y. Let

1

1

,
[c ij ] n 1
Z =

···

1
xx x ··· x •
3 5 2n−1
n+1
where
c ij =(i − j)a ij
= −c ji .
The proof proceeds by showing that U and V are each simple multiples of
Z. Perform the column operations

C = C j − x 2j−1 C n+1 , 1 ≤ j ≤ n,
j
on U. This leaves the last column and the last row unaltered, but [a ij ] n is
replaced by [a ] n , where

ij
x 2(i+j−1)

a = a ij − .
2i − 1
ij
Now perform the row operations
1
R = R i + R n+1 , 1 ≤ i ≤ n.

2i − 1
i
The last column and the last row remain unaltered, but [a ] n is replaced

ij
by [a ] n , where

ij
1
a = a +

2i − 1
ij ij