54 4. Particular Determinants
Postmultiply the VandermondianVn(x)orVn(x 1 ,x 2 ,...,xn)byAn,
prove the reduction formula
Vn(x 1 ,x 2 ,...,xn)=Vn− 1 (x 2 −x 1 ,x 3 −x 1 ,...,xn−x 1 )
n
∏
p=2
(xp−x 1 ),
and hence evaluateVn(x).
2.Prove that
|x
j− 1
i y
n−j
i |n=
∏
1 ≤r<s≤n
∣
∣
∣
∣
yr xr
ys xs
∣
∣
∣
∣
3.If
xi=
z+ci
ρ
,
prove that
|x
j− 1
i
|n=ρ
−n(n−1)/ 2
|c
j− 1
i
|n,
which is independent ofz. This relation is applied in Section 6.10.3 on
the Einstein and Ernst equations.
4.1.3 Cofactors of the Vandermondian...........
Theorem 4.1. The scaled cofactors of the VandermonianXn=|xij|n,
wherexij=x
j− 1
i
are given by the quotient formula
X
ij
n=
(−1)
n−j
σ
(n)
i,n−j
gni(xi)
,
where
gnr(x)=
n− 1
∑
s=0
(−1)
s
σ
(n)
rs
x
n− 1 −s
.
Notes on the symmetric polynomialsσ
(n)
rs and the functiongnr(x)are given
in Appendix A.7.
Proof. Denote the quotient byFij. Then,
n
∑
k=1
xikFjk=
1
gnj(xj)
n
∑
k=1
(−1)
n−k
σ
(n)
j,n−k
x
k− 1
i
(Putk=n−s)
=
1
gnj(xj)
n− 1
∑
s=0
(−1)
s
σ
(n)
js
x
n−s− 1
i
=
gnj(xi)
gnj(xj)
=δij.