Determinants and Their Applications in Mathematical Physics

134 4. Particular Determinants

Let

A=|φm|n, 0 ≤m≤ 2 n− 2 ,

where

φm=

x

2 m+2

− 1

m+1

.

Ais identical to|aij|n, whereaijis defined in Theorem 4.41. LetYdenote

the determinant of order (n+ 1) obtained by borderingAby the row

[

111 ... 1 •

]

n+1

below and the column

[

1

1

3

1

5

...

1

2 n− 1

•

]T

n+1

on the right.

Theorem 4.42.

Y=−nKnφ

n(n−1)

0

n

∑

i=1

2

2 i− 1

(n+i−1)!

(n−i)!(2i)!

φ

n−i

0 ,

whereKnis the simple Hilbert determinant.

Proof. Perform the column operations

C

′

j

=Cj−Cj− 1

in the orderj=n, n− 1 ,n− 2 ,...,2. The result is a determinant in which

the only nonzero element in the last row is a 1 in position (n+1,1). Hence,

Y=(−1)

n

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∆φ 0 ∆φ 1 ∆φ 2 ··· ∆φn− 2 1

∆φ 1 ∆φ 2 ∆φ 3 ··· ∆φn− 1

1

3

∆φ 2 ∆φ 3 ∆φ 4 ··· ∆φn

1

5

............................................

∆φn− 1 ∆φn ∆φn+1 ··· ∆φ 2 n− 3

1

2 n− 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

.

Perform the row operations

R

′

i

=Ri−Ri− 1

in the orderi=n, n− 1 ,n− 2 ,...,2. The result is

Y=(−1)

n

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∆φ 0 ∆φ 1 ∆φ 2 ··· ∆φn− 2 1

∆

2

φ 0 ∆

2

φ 1 ∆

2

φ 2 ··· ∆

2

φn− 1 ∆α 0

∆

2

φ 1 ∆

2

φ 2 ∆

2

φ 3 ··· ∆

2

φn ∆α 1

..................................................

∆

2

φn− 2 ∆

2

φn− 1 ∆

2

φn ··· ∆

2

φ 2 n− 4 ∆αn− 2

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

,

where

αm=

1

2 m+1

.