Page 74 - Determinants and Their Applications in Mathematical Physics

4.1 Alternants 59

n

n(n+1)/2
W n =(−1) X n Y n (x i + 1)(y i − 1).
i=1
Removing f(x 1 ),f(x 2 ),...,f(x n ), from the first n rows in V n and W n ,
and expanding each determinant by the last row and column, deduce
that

n
1
1
= (x i + 1)(y i − 1)
1 − x i y j

i=1
x i − y j
n
2 x i − y j n

n

+ (x i − 1)(y i +1) .
i=1
4.1.6 A Determinant Related to a Vandermondian
Let P r (x) be a polynomial defined as
r
s−1
P r (x)= a sr x , r ≥ 1.
s=1
Note that the coefficient is a sr , not the usual a rs .
Let
X n = |x i−1 | n .
j
Theorem.

|P i (x j )| n =(a 11 a 22 ··· a nn )X n .
Proof. Define an upper triangular determinant U n as follows:

U n = |a ij | n , a ij =0, i>j,
= a 11 a 22 ··· a nn . (4.1.7)
Some of the cofactors of U i are given by

(i) 0, j > i,
U =
ij U i−1 ,j = i, U 0 =1.
Those cofactors for which j< i are not required in the analysis which
(i)
follows. Hence, |U | n is also upper triangular and
ij
(1) (2) (n) (1)
(i) U U ··· U nn ,U =1,
|U | n = 11 22 11 (4.1.8)
U 1 U 2 ··· U n−1 .
ij
Applying the formula for the product of two determinants in Section 1.4,
(j)
|U | n |P i (x j )| n = |q ij | n , (4.1.9)
ij