Determinants and Their Applications in Mathematical Physics

A.3 Multiple-Sum Identities 311

which proves the lemma wheni>1.

Cyclic Permutations

The cyclic permutations of ther-set{i 1 i 2 i 3 ... ir}are alternately odd

and even whenris even, and are all even whenris odd. Hence, the signs as-

sociated with the permutations alternate whenris even but are all positive

whenris odd.

Examples.If

sgn{ij}=1,

then

sgn{ji}=− 1.

If

sgn{ijk}=1,

then

sgn{jki}=1,

sgn{kij}=1.

If

sgn{ijkm}=1,

then

sgn{jkmi}=− 1 ,

sgn{kmij}=1,

sgn{mijk}=− 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

|δr

isj

|n= sgn

{

r 1 r 2 r 3 ··· rn

s 1 s 2 s 3 ··· sn

}

.

1 ≤i,j≤n

A.3 Multiple-Sum Identities

1.If

fi=

n

∑

j=1

ci+1−j,j, 1 ≤i≤ 2 n− 1 ,