Page 341 - Determinants and Their Applications in Mathematical Physics

326 Appendix
∂V
= V (V +1 − x)
∂u
from which it follows that S m satisfies the nonlinear recurrence relation
m
m
S m+1 =(1 − x)S m + S r S m−r .
r
r=0
It then follows that
m
m
∆ψ m = ψ m+1 − ψ m = ψ r ψ m−r .
r
r=0

A.7 Symmetric Polynomials

(n)
in the n variables x i ,
Let the function f n (x) and the polynomials σ p
1 ≤ i ≤ n, be defined as follows:
n n

f n (x)= (x − x i )= (−1) σ x . (A.7.1)
p (n) n−p
p
i=1 p=0
Examples

(n)
σ 0 =1,
n
(n)
σ 1 = x r ,
r=1
(n)
σ = x r x s ,
2
1≤r<s≤n
(n)
σ 3 = x r x s x t ,
... 1≤r<s<t≤n
............
σ (n) = x 1 x 2 x 3 ...x n .
n
These polynomials are known as symmetric polynomials.
(n)
Let the function g nr (x) and the polynomials σ rs in the (n−1) variables
x i ,1 ≤ i ≤ n, i = r, be defined as follows:
n−1
f n (x)
g nr (x)= = (−1) σ x , (A.7.2)
s (n) n−1−s
rs
s=0
x − x r
g nn (x)= f n−1 (x) (A.7.3)
for all values of x. Hence,
σ (n) = σ (n−1) . (A.7.4)
ns s