Page 149 - Determinants and Their Applications in Mathematical Physics

134 4. Particular Determinants

Let
A = |φ m | n , 0 ≤ m ≤ 2n − 2,
where
x 2m+2 − 1
φ m = .
m +1
A is identical to |a ij | n , where a ij is defined in Theorem 4.41. Let Y denote
the determinant of order (n + 1) obtained by bordering A by the row

111 ... 1 •
n+1
below and the column
1 1 1
T
1 ... •
3 5 2n − 1
n+1
on the right.
Theorem 4.42.
n 2i−1
n(n−1) 2 (n + i − 1)!
Y = −nK n φ φ n−i ,
0 (n − i)!(2i)! 0
i=1
where K n is the simple Hilbert determinant.
Proof. Perform the column operations

C = C j − C j−1
j
in the order j = 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,

∆φ 0 ∆φ 1 ∆φ 2 ··· ∆φ n−2 1
1

∆φ 1 ∆φ 2 ∆φ 3 ··· ∆φ n−1
3
n 1
Y =(−1) ∆φ 2 ∆φ 3 ∆φ 4 ··· .
5
∆φ n

............................................
1

∆φ n−1 ∆φ n+1 ··· ∆φ 2n−3
∆φ n
2n−1 n
Perform the row operations

R = R i − R i−1
i
in the order i = n, n − 1,n − 2,..., 2. The result is

∆φ 0 ∆φ 1 ∆φ 2 ··· ∆φ n−2 1
2 2 2 2

∆ φ 0 ∆ φ 1 ∆ φ 2 ··· ∆ φ n−1 ∆α 0
2 2 2 2
n
Y =(−1) ∆ φ 1 ∆ φ 2 ∆ φ 3 ··· ∆ φ n ∆α 1 ,

..................................................
2 2 2 2

∆ φ n−2 ∆ φ n−1 ∆ φ n ··· ∆ φ 2n−4 ∆α n−2 n
where
1
α m = .
2m +1