Page 205 - Determinants and Their Applications in Mathematical Physics

190 5. Further Determinant Theory

where

u ij , j = i
a ij = p n (x i ) (5.3.7)

x − ,j = i.
2p (x i )
n
This A n clearly has the same value as the original A n 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hand side of (5.3.6) has been replaced by the right-hand side, its algebraic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 (x)+(1 − x)L (x)+ nL n (x)=0,
n n
L n (x i )=0, 1 ≤ i ≤ n,

L (x i ) x i − 1
= . (5.3.8)
n
2L (x i )

n x i
Hence, if
u ij , j = i

a ij = x i −1
x − ,j = i,
2x i
then
A n = |a ij | n = x . (5.3.9)
n
Hermite Polynomials.
H (x) − 2xH (x)+2nH n (x)=0,

n n
H n (x i )=0, 1 ≤ i ≤ n,
H (x i )

= x i . (5.3.10)
n
2H (x i )

n
Hence if,

u ij , j = i
a ij =
x − x i ,j = i,
then
A n = |a ij | n = x . (5.3.11)
n
Legendre Polynomials.
2
(1 − x )P (x) − 2xP (x)+ n(n +1)P n (x)=0,

n n
P n (x i )=0, 1 ≤ i ≤ n,
P (x i )

= 2 . (5.3.12)
n x i
2P (x i ) 1 − x

n i