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)