Page 328 - Determinants and Their Applications in Mathematical Physics

A.3 Multiple-Sum Identities 313

is invariant under any permutation of the parameters k r ,
4. If F k 1 k 2 ...k m
1 ≤ r ≤ m, then
k 1 ,k 2 ,...,k m
1
= ,
F k 1 k 2 ...k m G k 1 k 2 ...k m F k 1 k 2 ...k m G j 1 j 2 ...j m
m!
k 1 ,k 2 ,...,k m k 1 ,k 2 ,...,k m j 1 ,j 2 ,...,j m
where the sum on the left ranges over the m! permutations of the param-
eters and, in the inner sum on the right, the parameters j r ,1 ≤ r ≤ m,
range over the m! permutations of the k r .
Proof. Denote the sum on the left by S. The m! permutations of the
parameters k r give m! alternative formulae for S, which differ only in the
. The identity appears after summing
order of the parameters in G k 1 k 2 ...k m
these m! formulas.
Illustration. Put m = 3 and use a simpler notation. Let

S = F ijk G ijk .
i,j,k
Then,

S = F ikj G ikj = F ijk G ikj
..................
i,j,k
i,k,j

S = F kji G kji = F ijk G kji .
k,j,i i,j,k
Summing these 3! formulas for S,

3! S = F ijk (G ijk + G ikj + ··· + G kji ),
i,j,k
1 i,j,k

S = G pqr .
3! F ijk
p,q,r
i,j,k
5.
n

F k 1 k 2 ...k m G k 1 k 2 ...k m
k 1 ,k 2 ,...,k m =1
k 1 ,k 2 ,...,k m
1
n
= F k 1 k 2 ...k m G j 1 j 2 ...j m .
m!
k 1 ,k 2 ,...,k m =1 j 1 ,j 2 ,...,j m
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 is m , but the number of
n
alternative formulas for this sum remains at m!.
The identities given in 3–5 are applied in Section 6.10.3 on the Einstein
and Ernst equations.