Page 87 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
4.3 Rodrigue s Formula for Laguerre Polynomials L ( )
x
n
To prove

e x d n n x
x
L n ( ) = n x e for n 0, 1, 2, ...(i)
! n dx
Proof: Using Leibnitz s theorem we have

e x d n n x e x n n x n 1 n 1 x n (n 1) n 1 n 2 x x
x e = x ( 1) e n n x ( 1) e ( n n 1)x ( 1) e ! n e
! n dx n ! n 2
e x n n x n 1 n 1 x n (n 1) n 2 n 2 x x
x ( 1) e n n x ( 1) e ( n n 1)x ( 1) e ! n e
! n 2

x
e e x n n n 1 n n ! n 1
= ( 1) x ( 1) x ! n ...(ii)
! n (n 1)!
= ( 1) n ! n x n ( 1) n 1 ! n x n 1 ! n
2
( !) 2 (n 1)! . ! ! n
n
i
n ! n
x
= ( 1) r 2 x r L n ( ) ...(iii)
r
r 0 ( !) (n r )!
e x d n n x
Hence L n ( ) = n x e .
x
! n dx
First few Laguerre Polynomials from Rodrigue s Formula

We have from Rodrigue s formula

e x d n n x
L n ( ) = n x e
x
! n dx
Putting n = 0

e x d 0 0 x
x
L 0 ( ) = 0 (x e ) 1
0! dx
Putting n = 1

e x d x x x x
x
L 1 ( ) = (xe ) e e xe 1 x
1! dx
Putting n = 2

e x d 2 2 x e x d x 2 x
L 2 ( ) = 2 x e 2xe x e
x
2! dx 2! dx
e x x x 2 x
= 2e 4x e x e
2!

80 LOVELY PROFESSIONAL UNIVERSITY