Page 78 - Determinants and Their Applications in Mathematical Physics

4.1 Alternants 63

4.1.9 Further Vandermondian Identities
The notation
N m = {12 ··· m},
J m = {j 1 j 2 ··· j m },
K m = {k 1 k 2 ··· k m },

where J m and K m are permutations of N m , is used to simplify the following
lemmas.
Lemma 4.5.

N m m
r−1
V (x 1 ,x 2 ,...,x m )= sgn N m x .
j r
J m J m r=1
Proof. The proof follows from the definition of a determinant in
Section 1.2 with a ij → x j−1 .
i
Lemma 4.6.

= sgn N m V (x 1 ,x 2 ,...,x m ).
V x j 1 ,x j 2 ,...,x j m J m
This is Lemma (f) in Section 4.1.8 expressed in the present notation with
n → m.

Lemma 4.7.
K m N m

= N m .
F x j 1 ,x j 2 ,...,x j m F x j 1 ,x j 2 ,...,x j m
J m J m J m
In this lemma, the permutation symbol is used as a substitution operator.
2
The number of terms on each side is m .
+ x 2 and denote the left and
Illustration. Put m =2, F(x j 1 ,x j 2 )= x j 1
j 2
right sides of the lemma by P and Q respectively. Then,
+ x 2 + x 2
P = x k 1 + x k 2
k 1 k 2

1 2 2 2
Q = (x 1 + x + x 2 + x )
1
2
k 1 k 2
= P.
Theorem.

N m m
r−1 2
a. x V x j 1 ,x j 2 ,...,x j m =[V (x 1 ,x 2 ,...,x m )] ,
j r
J m r=1

K m m
r−1 2
b. x V x j 1 ,x j 2 ,...,x j m = V x k 1 ,x k 2 ,...,x k m .
j r
r=1
J m