Page 72 - Determinants and Their Applications in Mathematical Physics

4.1 Alternants 57

4.1.5 The Cauchy Double Alternant
The Cauchy double alternant is the determinant

1

A n = ,
x i − y j n

which can be evaluated in terms of the Vandermondians X n and Y n as
follows.
Perform the column operations
C = C j − C n , 1 ≤ j ≤ n − 1,

j
and then remove all common factors from the elements of rows and columns.
The result is

n−1

(y r − y n )
A n = r=1 B n , (4.1.5)
n
(x r − y n )
r=1
where B n is a determinant in which the last column is

T
111 ... 1
n
and all the other columns are identical with the corresponding columns of
A n .
Perform the row operations

R = R i − R n , 1 ≤ i ≤ n − 1,
i
on B n , which then degenerates into a determinant of order (n − 1). After
removing all common factors from the elements of rows and columns, the
result is
n−1
(x n − x r )

r=1
B n = A n−1 . (4.1.6)
n−1

(x n − y r )
r=1
Eliminating B n from (4.1.5) and (4.1.6) yields a reduction formula for A n ,
which, when applied, gives the formula

(−1) n(n−1)/2
A n = X n Y n .
(x r − y s )
n
r,s=1