Determinants and Their Applications in Mathematical Physics

190 5. Further Determinant Theory

where

aij=

{

uij,j=i

x−

p

′′

n(xi)

2 p′n(xi)

,j=i.

(5.3.7)

This An clearly has the same value as the original An since the left-

hand side of (5.3.6) has been replaced by the right-hand side, its algebraic

equivalent.

The right-hand side of (5.3.6) will now be evaluated for each of the three

particular polynomials mentioned above with the aid of their differential

equations (Appendix A.5).

Laguerre Polynomials.

xL

′′

n

(x)+(1−x)L

′

n

(x)+nLn(x)=0,

Ln(xi)=0, 1 ≤i≤n,

L

′′

n

(xi)

2 L

′

n(xi)

=

xi− 1

xi

. (5.3.8)

Hence, if

aij=

{

uij,j=i

x−

xi− 1

2 xi

,j=i,

then

An=|aij|n=x

n

. (5.3.9)

Hermite Polynomials.

H

′′

n(x)−^2 xH

′

n(x)+2nHn(x)=0,

Hn(xi)=0, 1 ≤i≤n,

H

′′

n

(xi)

2 H

′

n(xi)

=xi. (5.3.10)

Hence if,

aij=

{

uij,j=i

x−xi,j=i,

then

An=|aij|n=x

n

. (5.3.11)

Legendre Polynomials.

(1−x

2

)P

′′

n

(x)− 2 xP

′

n

(x)+n(n+1)Pn(x)=0,

Pn(xi)=0, 1 ≤i≤n,

P

′′

n(xi)

2 P

′

n

(xi)

=

xi

1 −x

2

i

. (5.3.12)