Determinants and Their Applications in Mathematical Physics

132 4. Particular Determinants

The result is

1

A^2

[

p

2

V

2

(x)+q

2

V

2

(y)−W

2

]

=

∑

i

∑

j

i+j− 1

(2i−1)(2j−1)

A

ij

=

1

2

∑

i

∑

j

(

1

2 i− 1

+

1

2 j− 1

)

A

ij

=−

W

A

.

The theorem follows. The determinantWappears 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.

aij=

x

2(i+j−1)

− 1

i+j− 1

=aji,

which is independent ofy. Let

Z=

∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

∣

∣

∣

1

1

[cij]n 1

···

1

xx

3

x

5

··· x

2 n− 1

• 
  

∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

∣

∣

∣

n+1

,

where

cij=(i−j)aij

=−cji.

The proof proceeds by showing thatUandVare each simple multiples of

Z. Perform the column operations

C

′

j=Cj−x

2 j− 1

Cn+1, 1 ≤j≤n,

onU. This leaves the last column and the last row unaltered, but [aij]nis

replaced by [a
′
ij
]n, where

a

′

ij=aij−

x

2(i+j−1)

2 i− 1

.

Now perform the row operations

R

′

i=Ri+

1

2 i− 1

Rn+1, 1 ≤i≤n.

The last column and the last row remain unaltered, but [a
′
ij
]nis replaced

by [a
′′
ij
]n, where

a

′′

ij

=a

′

ij

+

1

2 i− 1