4.1 Alternants 57
4.1.5 The Cauchy Double Alternant............
The Cauchy double alternant is the determinant
An=
∣
∣
∣
∣
1
xi−yj
∣
∣
∣
∣
n
,
which can be evaluated in terms of the VandermondiansXnandYnas
follows.
Perform the column operations
C
′
j=Cj−Cn,^1 ≤j≤n−^1 ,
and then remove all common factors from the elements of rows and columns.
The result is
An=
n∏− 1
r=1
(yr−yn)
∏n
r=1
(xr−yn)
Bn, (4.1.5)
whereBnis a determinant in which the last column is
[
111 ... 1
]T
n
and all the other columns are identical with the corresponding columns of
An.
Perform the row operations
R
′
i=Ri−Rn,^1 ≤i≤n−^1 ,
onBn, 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
Bn=
n∏− 1
r=1
(xn−xr)
n∏− 1
r=1
(xn−yr)
An− 1. (4.1.6)
EliminatingBnfrom (4.1.5) and (4.1.6) yields a reduction formula forAn,
which, when applied, gives the formula
An=
(−1)
n(n−1)/ 2
XnYn
∏n
r,s=1
(xr−ys)