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

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)

.