Determinants and Their Applications in Mathematical Physics

A.3 Multiple-Sum Identities 313

4.IfFk

1 k 2 ...km

is invariant under any permutation of the parameterskr,

1 ≤r≤m, then

∑

k 1 ,k 2 ,...,km

Fk

1 k 2 ...km

Gk

1 k 2 ...km

=

1

m!

∑

k 1 ,k 2 ,...,km

Fk

1 k 2 ...km

k 1 ,k 2 ,...,km

∑

j 1 ,j 2 ,...,jm

Gj

1 j 2 ...jm

,

where the sum on the left ranges over them! permutations of the param-

eters and, in the inner sum on the right, the parametersjr,1≤r≤m,

range over them! permutations of thekr.

Proof. Denote the sum on the left byS. Them! permutations of the

parameterskrgivem! alternative formulae forS, which differ only in the

order of the parameters inGk
1 k 2 ...km

. The identity appears after summing

thesem! formulas.

Illustration.Putm= 3 and use a simpler notation. Let

S=

∑

i,j,k

FijkGijk.

Then,

S=

∑

i,k,j

FikjGikj=

∑

i,j,k

FijkGikj

..................

S=

∑

k,j,i

FkjiGkji=

∑

i,j,k

FijkGkji.

Summing these 3! formulas forS,

3!S=

∑

i,j,k

Fijk(Gijk+Gikj+···+Gkji),

S=

1

3!

∑

i,j,k

Fijk

i,j,k

∑

p,q,r

Gpqr.

5.

n

∑

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

Fk 1 k 2 ...kmGk 1 k 2 ...km

=

1

m!

n

∑

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

Fk 1 k 2 ...km

k 1 ,k 2 ,...,km

∑

j 1 ,j 2 ,...,jm

Gj 1 j 2 ...jm.

The inner sum on the right is identical with the inner sum on the right

of Identity 4 and the proof is similar to that of Identity 4. In this case,

the number of terms in the sum on the left ism

n

, but the number of

alternative formulas for this sum remains atm!.

The identities given in 3–5 are applied in Section 6.10.3 on the Einstein

and Ernst equations.