Page 88 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 4: Laguerre Polynomials

Notes
1 2
= 2 4x x
2!
Similarly,
1 2 3
x
L 3 ( ) = 6 18x 9x x
3!
1 2 3 4
x
L 4 ( ) = 24 96x 72x 16x x , etc .
4!
Self Assessment

9. Show that

e x d 2 2 x
x
L 2 ( ) 2 x e
2 dx
3
10. Show that x is given by
x 3 6 L 0 ( ) 3 ( ) 3 ( ) L 3 ( )
L
L
x
x
x
x
1
2
11. From Rodrigue s formula show that
x
dL 2 ( ) L ( ) L ( )
x
x
dx 1 0
x
4.4 Orthogonality Property of Laguerre Polynomials L ( )
n
To prove
0 if m n
x
e L n ( )L m ( )dx = ...(i)
x
x
0 mn 1 if m n
We have from the generating function of Laguerre polynomial, that
1 tx (1 t )
n
x
t L n ( ) = e
n 0 1 t
1 xs /(1 s )
m
x
and s L m ( ) = e
m 0 1 s
sx
1 tx /(1 t ) (1 s )
x
x n m
x
e t s L ( ) L ( ) = e e
x
n m (1 t )(1 s ) ...(ii)
m , n 0
Thus
x
m
e L n ( )L m ( )dx = Coeff. of s t in the expansion of e x 1 e tx /(1 t )e sx /(1 s ) dx
n
x
x
0 0 (1 t )(1 s )
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