56 4. Particular Determinants
This completes the proof of Theorem 4.3 which can be expressed in the
form
Hrs
Xn
=
∏n
i=1
(ys−xi)
(ys−xr)
∏n
i=1
i=r
(xr−xi)
.
Let
An=|σ
(m)
i,j− 1
|n
=
∣
∣
∣
∣
∣
∣
∣
∣
σ
(m)
10
σ
(m)
11
... σ
(m)
1 ,n− 1
σ
(m)
20 σ
(m)
21 ... σ
(m)
2 ,n− 1
........................
σ
(m)
n 0 σ
(m)
n 1 ... σ
(m)
n,n− 1
∣
∣
∣
∣
∣
∣
∣
∣
n
,m≥n.
Theorem 4.4.
An=(−1)
n(n−1)/ 2
Xn.
Proof.
An=
∣
∣C
0 C 1 C 2 ...Cn− 1 ,
∣
∣
n
where, from the lemma in Appendix A.7,
Cj=
[
σ
(m)
1 j
σ
(m)
2 j
σ
(m)
3 j
... σ
(m)
nj
]T
=
j
∑
p=0
σ
(m)
p
[
v
j−p
1 v
j−p
2 v
j−p
3 ... v
j−p
n
]T
,vr=−xr,σ
(m)
0 =1.
Applying the column operations
C
′
j
=Cj−
j
∑
k=1
σ
(m)
k
Cj−k
in the orderj=1, 2 , 3 ,...so that each new column created by one operation
is applied in the next operation, it is found that
C
′
j
=
[
v
j
1
v
j
2
v
j
3
... v
j
n
]T
,j=0, 1 , 2 ,....
Hence
An=|v
j− 1
i
|n
=(−1)
n(n−1)/ 2
|x
j− 1
i
|n.
Theorem 4.4 follows.