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 byWm.

wi+j− 2 +wi+j=

2 n

∑

k=1

ykx

i+j− 2

k

,

where

yk=(−1)

k+1

εkMk(c). (6.10.33)

Hence, applying the lemma in Section 4.1.7 withN→ 2 nandn→m,

Wm=

∣

∣

∣

∣

∣

2 n

∑

k=1

ykx

i+j− 2

k

∣ ∣ ∣ ∣ ∣ =

2 n

∑

k 1 ,k 2 ,...,km=1

Ym

(

m

∏

r=2

x

r− 1

kr

)

∣

∣

x

j− 1

ki

∣

∣

m

,

where

Ym=

m

∏

r=1

yk

r

. (6.10.34)

Hence, applying Identity 4 in Appendix A.3,

Wm=

1

m!

2 n

∑

k 1 ,k 2 ,...,km=1

Ym

k 1 ,k 2 ,...km

∑

j 1 ,j 2 ,...,jm

(

m

∏

r=2

x

r− 1

jr

)

V(xj 1 ,xj 2 ,...,xjm).

(6.10.35)

Applying Theorem (b) in Section 4.1.9 on Vandermondian identities,

Wm=

1

m!

2 n

∑

k 1 ,k 2 ,...,km=1

Ym

{

V(xk

1

,xk

2

,...,xk

m

)

} 2

. (6.10.36)

Due to the presence of the squared Vandermondian factor, the conditions of

Identity 3 in Appendix A.3 withN→ 2 nare satisfied. Also, eliminating the

x’s using (6.10.26) and (6.10.28) and referring to Exercise 3 in Section 4.1.2,

{

(V(xk

1

,xk

2

,...,xk

m

)

} 2

=ρ

−m(m−1)

{

V(ck

1

,ck

2

,...,ck

m

)

} 2

. (6.10.37)

Hence,

Wm=ρ

−m(m−1)

∑

1 ≤k 1 <k 2 <...<km≤ 2 n

Ym

{

V(ck 1 ,ck 2 ,...,ckm)

} 2

. (6.10.38)

From (6.10.33) and (6.10.34),

Ym=(−1)

K

Em

m

∏

r=1

Mkr(c),