Determinants and Their Applications in Mathematical Physics

176 5. Further Determinant Theory

=D

n

(

yA 1

v

)

,

(

A 1 =u=

vy

′

y

)

=D

n+1

(y).

Hence,

An+1=

v

n+1

D

n+1

(y)

y

,

which is equivalent to (b).

The Rodrigues formula for the generalized Laguerre polynomialL

(α)

n (x)

is

L

(α)

n

(x)=

x

n

D

n

(e

−x

x

n+α

)

n!e

−x

x

n+α

. (5.1.16)

Hence, choosing

v=x,

y=e

−x

x

n+α

,

so that

u=x−n−α, (5.1.17)

formula (b) becomes

L

(α)

n

(x)=

1

n!

× (5.1.18)

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+α−x 1

−xn+α−x− 12

−xn+α−x− 23

··· ··· ··· ···

2+α−xn− 1

−x 1+α−x

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

.

Exercises

Prove that

1.L

(α)

n (x)=

1

n!

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+α−xn+αn+αn+α ···

1 n+α−xn+αn+α ···

2 n+α−xn+α ···

3 n+α−x ···

................

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

(Pandres).