Page 204 - Determinants and Their Applications in Mathematical Physics

5.3 The Matsuno Identities 189

Hence,
A n = xA n−1 .
2
But A 2 = x . The theorem follows.

5.3.2 Particular Identities

n
It is shown in the previous section that A n = x provided only that the x i
are distinct. It will now be shown that the diagonal elements of A n can be
modified in such a way that A n = x as before, but only if the x i are the
n
zeros of certain orthogonal polynomials. These identities supplement those
given by Matsuno.
It is well known that the zeros of the Laguerre polynomial L n (x), the
Hermite polynomial H n (x), and the Legendre polynomial P n (x) are dis-
tinct. Let p n (x) represent any one of these polynomials and let its zeros be
denoted by x i ,1 ≤ i ≤ n. Then,
n

p n (x)= k (x − x r ), (5.3.3)
r=1
where k is a constant. Hence,

n

log p n (x) = log k + log(x − x r ),
r=1
p (x) 1

n
= . (5.3.4)
n
p n (x)
r=1 x − x r
It follows that
1 (x − x i )p (x) − p n (x)
n

= n . (5.3.5)
(x − x i )p n (x)
r=1 x − x r
r =i
Hence, applying the l’Hopital limit theorem twice,

1 (x − x i )p (x) − p n (x)
n

= lim n
x→x i (x − x i )p n (x)
r=1 x i − x r
r =i
(x − x i )p (x)+ p (x)

= lim n n
x→x i (x − x i )p (x)+2p (x)

n n
p (x i )

= n . (5.3.6)
2p (x i )

n
The sum on the left appears in the diagonal elements of A n . Now redefine
A n as follows:
A n = |a ij | n ,