Page 206 - Determinants and Their Applications in Mathematical Physics

5.3 The Matsuno Identities 191

Hence, if
u ij , j = i

a ij = x i
x − 2 ,j = i,
i
1−x
then
A n = |a ij | n = x . (5.3.13)
n

Exercises

1. Let A n denote the determinant defined in (5.3.9) and let
B n = |b ij | n ,

where

2 , j = i
b ij = x i −x j
x + 1 ,j = i,
x i
where, as for A n (x), the x i denote the zeros of the Laguerre polynomial.
Prove that
x
1 2
n
2
B n (x − 1)=2 A n
and, hence, prove that
B n (x)=(x +1) .
n
2. Let

A (p) = |a (p) | n ,
n ij
where
 u , j = i
 p
ij

(p) n !
a = x − u ,j = i,
p
ij
ir
r=1


r =i
1
u ij = x i − x j = −u ji
and the x i are the zeros of the Hermite polynomial H n (x). Prove that
n

A (2) = [x − (r − 1)],
n
r=1
n
1 2
(4)
A = x − (r − 1) .
6
n
r=1