Page 191 - Determinants and Their Applications in Mathematical Physics

176 5. Further Determinant Theory

yA 1 vy
= D n , A 1 = u =
v y
= D n+1 (y).

Hence,
v n+1 D n+1 (y)
A n+1 = ,
y

which is equivalent to (b).
(α)
The Rodrigues formula for the generalized Laguerre polynomial L n (x)
is
x
x D (e −x n+α )
n
n
(α)
L (x)= . (5.1.16)
n! e −x n+α
x
n
Hence, choosing
v = x,
y = e −x n+α ,
x
so that
u = x − n − α, (5.1.17)

formula (b) becomes
1
(α)
L (x)= × (5.1.18)
n!
n
1

n + α − x
n + α − x − 1 2

−x

n + α − x − 2 3

.
−x

··· ··· ··· ···

n − 1

2+ α − x

−x 1+ α − x n

Exercises
Prove that
n + α − x n + α n + α n + α ···

1 n + α − x n + α n + α

···
1. L (α) (x)= 1 2 n + α − x n + α
···
n
3
n!
n + α − x ···
................

n
(Pandres).