Determinants and Their Applications in Mathematical Physics

5.3 The Matsuno Identities 191

Hence, if

aij=

{

uij,j=i

x−

xi

1 −x

2

i

,j=i,

then

An=|aij|n=x

n

. (5.3.13)

Exercises

1.LetAndenote the determinant defined in (5.3.9) and let

Bn=|bij|n,

where

bij=

{

2

xi−xj

,j=i

x+

1

xi

,j=i,

where, as forAn(x), thexidenote the zeros of the Laguerre polynomial.

Prove that

Bn(x−1)=2

n

An

(

x

2

)

and, hence, prove that

Bn(x)=(x+1)

n

.

2.Let

A

(p)

n

=|a

(p)

ij

|n,

where

a

(p)

ij

=











u

p

ij

,j=i

x−

∑n

r=1

r=i

u

p

ir,j=i,

uij=

1

xi−xj

=−uji

and thexiare the zeros of the Hermite polynomialHn(x). Prove that

A

(2)

n

=

n

∏

r=1

[x−(r−1)],

A

(4)

n =

n

∏

r=1

[

x−

1

6

(r

2

−1)

]

.