Determinants and Their Applications in Mathematical Physics

4.1 Alternants 63

4.1.9 Further Vandermondian Identities..........

The notation

Nm={ 12 ···m},

Jm={j 1 j 2 ···jm},

Km={k 1 k 2 ···km},

whereJmandKmare permutations ofNm, is used to simplify the following

lemmas.

Lemma 4.5.

V(x 1 ,x 2 ,...,xm)=

Nm

∑

Jm

sgn

{

Nm

Jm

} m

∑

r=1

x

r− 1

jr.

Proof. The proof follows from the definition of a determinant in

Section 1.2 withaij→x

j− 1

i

.

Lemma 4.6.

V

(

xj

1

,xj

2

,...,xj

m

)

= sgn

{

Nm

Jm

}

V(x 1 ,x 2 ,...,xm).

This is Lemma (f) in Section 4.1.8 expressed in the present notation with

n→m.

Lemma 4.7.

Km

∑

Jm

F

(

xj 1 ,xj 2 ,...,xjm

)

=

{

Nm

Jm

}Nm

∑

Jm

F

(

xj 1 ,xj 2 ,...,xjm

)

.

In this lemma, the permutation symbol is used as a substitution operator.

The number of terms on each side ism
2
.

Illustration.Putm=2,F(xj
1
,xj
2
)=xj
1
+x

2

j 2

and denote the left and

right sides of the lemma byPandQrespectively. Then,

P=xk 1 +x

2

k 1 +xk 2 +x

2

k 2

Q=

{

12

k 1 k 2

}

(x 1 +x

2

1 +x^2 +x

2

2 )

=P.

Theorem.

a.

Nm

∑

Jm

(

m

∏

r=1

x

r− 1

jr

)

V

(

xj

1

,xj

2

,...,xj

m

)

=[V(x 1 ,x 2 ,...,xm)]

2

,

b.

Km

∑

Jm

(

m

∏

r=1

x

r− 1

jr

)

V

(

xj

1

,xj

2

,...,xj

m

)

=

[

V

(

xk

1

,xk

2

,...,xk

m

)] 2

.