Determinants and Their Applications in Mathematical Physics

4.1 Alternants 59

Wn=(−1)

n(n+1)/ 2

XnYn

n

∏

i=1

(xi+ 1)(yi−1).

Removingf(x 1 ),f(x 2 ),...,f(xn), from the firstnrows inVnandWn,

and expanding each determinant by the last row and column, deduce

that

∣

∣

∣

∣

1 −xiyj

xi−yj

∣

∣

∣

∣

n

=

1

2

∣

∣

∣

∣

1

xi−yj

∣

∣

∣

∣

n

{

n

∏

i=1

(xi+ 1)(yi−1)

+

n

∏

i=1

(xi−1)(yi+1)

}

4.1.6 A Determinant Related to a Vandermondian

LetPr(x) be a polynomial defined as

Pr(x)=

r

∑

s=1

asrx

s− 1

,r≥ 1.

Note that the coefficient isasr, not the usualars.

Let

Xn=|x

i− 1

j

|n.

Theorem.

|Pi(xj)|n=(a 11 a 22 ···ann)Xn.

Proof. Define an upper triangular determinantUnas follows:

Un=|aij|n,aij=0, i>j,

=a 11 a 22 ···ann. (4.1.7)

Some of the cofactors ofUiare given by

U

(i)

ij

=

{

0 ,j>i,

Ui− 1 ,j=i,U 0 =1.

Those cofactors for whichj<iare not required in the analysis which

follows. Hence,|U

(i)

ij

|nis also upper triangular and

|U

(i)

ij

|n=

{

U

(1)

11 U

(2)

22 ···U

(n)

nn,U

(1)

11 =1,

U 1 U 2 ···Un− 1.

(4.1.8)

Applying the formula for the product of two determinants in Section 1.4,

|U

(j)

ij

|n|Pi(xj)|n=|qij|n, (4.1.9)