Page 342 - Determinants and Their Applications in Mathematical Physics

A.7 Symmetric Polynomials 327

Also,
g nj (x i )=0, j = i. (A.7.5)

Examples

(3)
σ = x 1 x 2 + x 1 x 3 + x 2 x 3 ,
2
(n)
σ =1, 1 ≤ r ≤ n,
r0
(3)
σ = x 1 + x 3 ,
21
(3)
σ = x 1 x 3 ,
22
(4)
σ = x 1 + x 2 + x 4 ,
31
(4)
σ = x 1 x 2 + x 1 x 4 + x 2 x 4 ,
32
(4)
σ = x 1 x 2 x 4 .
33
Lemma.
s

σ (n) = σ (n) (−x r ) s−p .
rs p
p=0
Proof. Since

−1
1 x
g r (x)= − x r 1 − x r f(x)
∞
f(x) x
q
= − ,
q=0 x r
x r
it follows that
n−1 n ∞
p (n) n−p+q
(−1) σ p x
x
(−1) s+1 (n) n−1−s = .
σ
rs q+1
s=0 p=0 q=0 x r
Equating coefficients of x n−1−s ,
n

σ
(−1) s+1 (n) = (−1) σ x .
p (n) s−p
rs p r
p=s+1
Hence
s n

(−1) s+1 (n) + (−1) σ x = (−1) σ x
σ
p (n) s−p
p (n) s−p
rs p r p r
p=0 p=0
= x s−n f(x r )
r
=0.
The lemma follows.
Symmetric polynomials appear in Section 4.1.2 on Vandermondians.