Determinants and Their Applications in Mathematical Physics

5.3 The Matsuno Identities 189

Hence,

An=xAn− 1.

ButA 2 =x

2

. The theorem follows.

5.3.2 Particular Identities..................

It is shown in the previous section thatAn=x

n

provided only that thexi

are distinct. It will now be shown that the diagonal elements ofAncan be

modified in such a way thatAn=x

n

as before, but only if thexiare the

zeros of certain orthogonal polynomials. These identities supplement those

given by Matsuno.

It is well known that the zeros of the Laguerre polynomialLn(x), the

Hermite polynomialHn(x), and the Legendre polynomialPn(x) are dis-

tinct. Letpn(x) represent any one of these polynomials and let its zeros be

denoted byxi,1≤i≤n. Then,

pn(x)=k

n

∏

r=1

(x−xr), (5.3.3)

wherekis a constant. Hence,

logpn(x) = logk+

n

∑

r=1

log(x−xr),

p

′

n

(x)

pn(x)

=

n

∑

r=1

1

x−xr

. (5.3.4)

It follows that

n

∑

r=1

r=i

1

x−xr

=

(x−xi)p

′

n

(x)−pn(x)

(x−xi)pn(x)

. (5.3.5)

Hence, applying the l’Hopital limit theorem twice,

n

∑

r=1

r=i

1

xi−xr

= lim

x→xi

[

(x−xi)p

′

n

(x)−pn(x)

(x−xi)pn(x)

]

= lim

x→xi

[

(x−xi)p

′′′

n(x)+p

′′

n(x)

(x−xi)p

′′

n

(x)+2p

′

n

(x)

]

=

p

′′

n

(xi)

2 p

′

n

(xi)

. (5.3.6)

The sum on the left appears in the diagonal elements ofAn. Now redefine

Anas follows:

An=|aij|n,