Determinants and Their Applications in Mathematical Physics

326 Appendix

∂V

∂u

=V(V+1−x)

from which it follows thatSmsatisfies the nonlinear recurrence relation

Sm+1=(1−x)Sm+

m

∑

r=0

(

m

r

)

SrSm−r.

It then follows that

∆ψm=ψm+1−ψm=

m

∑

r=0

(

m

r

)

ψrψm−r.

A.7 Symmetric Polynomials

Let the functionfn(x) and the polynomialsσ

(n)

p in thenvariablesxi,

1 ≤i≤n, be defined as follows:

fn(x)=

n

∏

i=1

(x−xi)=

n

∑

p=0

(−1)

p

σ

(n)

p

x

n−p

. (A.7.1)

Examples

σ

(n)

0

=1,

σ

(n)

1 =

n

∑

r=1

xr,

σ

(n)

2 =

∑

1 ≤r<s≤n

xrxs,

σ

(n)

3

=

∑

1 ≤r<s<t≤n

xrxsxt,

... ............

σ

(n)

n =x^1 x^2 x^3 ...xn.

These polynomials are known as symmetric polynomials.

Let the functiongnr(x) and the polynomialsσ

(n)

rs in the (n−1) variables

xi,1≤i≤n,i=r, be defined as follows:

gnr(x)=

fn(x)

x−xr

=

n− 1

∑

s=0

(−1)

s

σ

(n)

rs

x

n− 1 −s

, (A.7.2)

gnn(x)=fn− 1 (x) (A.7.3)

for all values ofx. Hence,

σ

(n)

ns

=σ

(n−1)

s

. (A.7.4)