Page 312 - Determinants and Their Applications in Mathematical Physics

6.10 The Einstein and Ernst Equations 297

Proof. Proof of (a). Denote the determinant on the left by W m .

2n
i+j−2
w i+j−2 + w i+j = y k x ,
k
k=1
where
y k =(−1) k+1 ε k M k (c). (6.10.33)
Hence, applying the lemma in Section 4.1.7 with N → 2n and n → m,

2n

y k x
i+j−2
W m =
k

k=1

2n m
r−1 j−1
= x x ,
k r k i
Y m
m
k 1 ,k 2 ,...,k m =1 r=2
where
m

Y m = y k r . (6.10.34)
r=1
Hence, applying Identity 4 in Appendix A.3,

k 1 ,k 2 ,...k m
1 r−1
2n
m
W m = x V (x j 1 ,x j 2 ,...,x j m ).
m! Y m j r
k 1 ,k 2 ,...,k m =1 j 1 ,j 2 ,...,j m r=2
(6.10.35)
Applying Theorem (b) in Section 4.1.9 on Vandermondian identities,
1 - . 2
2n
W m = Y m V (x k 1 ,x k 2 ,...,x k m ) . (6.10.36)
m!
k 1 ,k 2 ,...,k m =1
Due to the presence of the squared Vandermondian factor, the conditions of
Identity 3 in Appendix A.3 with N → 2n are satisfied. Also, eliminating the
x’s using (6.10.26) and (6.10.28) and referring to Exercise 3 in Section 4.1.2,
- . 2 −m(m−1) - . 2
) = ρ ) . (6.10.37)
(V (x k 1 ,x k 2 ,...,x k m V (c k 1 ,c k 2 ,...,c k m
Hence,
-
W m = ρ −m(m−1) Y m V (c k 1 ,c k 2 ,...,c k m ) . 2 . (6.10.38)
1≤k 1 <k 2 <...<k m ≤2n
From (6.10.33) and (6.10.34),
m

(c),
K
M k r
Y m =(−1) E m
r=1