Page 326 - Determinants and Their Applications in Mathematical Physics

A.3 Multiple-Sum Identities 311

which proves the lemma when i> 1.

Cyclic Permutations

The cyclic permutations of the r-set {i 1 i 2 i 3 ... i r } are alternately odd
and even when r is even, and are all even when r is odd. Hence, the signs as-
sociated with the permutations alternate when r is even but are all positive
when r is odd.
Examples. If

sgn{ij} =1,
then
sgn{ji} = −1.
If
sgn{ij k} =1,
then
sgn{jk i} =1,
sgn{ki j} =1.

If
sgn{ij k m} =1,
then
sgn{jk m i} = −1,
sgn{k mij} =1,
sgn{mij k} = −1.
Cyclic permutations appear in Section 3.2.4 on alien second and higher
cofactors and in Section 4.2 on symmetric determinants.
Exercise. Prove that

r 1 r 2 r 3 ···
|δ r i s j n = sgn r n .
|
s 1 s 2 s 3 ··· s n
1≤i,j≤n
A.3 Multiple-Sum Identities

1. If

n

f i = c i+1−j,j , 1 ≤ i ≤ 2n − 1,
j=1