Page 85 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes n
x
L ( ) = ( 1) r ! n x r ...(vi)
n 2
r
r 0 (n r )!( !)
The first few Laguerre polynomials are:
L ( ) = 1, L ( ) 1 x
x
x
0 1
1 2
x
L 2 ( ) = 2 4x x
2
1 2 3
L 3 ( ) = 6 18x 9x x
x
6
Self Assessment

1. The value of L n (0) is

(a) 0 (b) 1
(c) 1 (d) None of these
2. L 2 ( ) satisfies Laguerre s differential equation for equal to
x
(a) 1 (b) 3
(c) 2 (d) 1

3. Fill in the blanks:
The Laguerre polynomial tends to infinity as a ............... power of x as x .
x
4. Laguerre polynomial L n ( ) is a polynomial having a leading power of x equal to
(a) n (b) Zero
(c) One (d) None of the above

4.2 Generating Function for Laguerre Polynomials L n ( )
x

1
n
x
To prove e tx /(1 t ) t L n ( ).
1 t
r 0
We have
r
1 tx /(1 ) t 1 1 xt
e =
1 t 1 t ! r 1 t
r 0
r r
( 1) r x t
=
r ! (1 t ) r 1
r 0
( 1) r
r r
= x t (1 t ) (r 1)
! r
r 0

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