Page 94 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 4: Laguerre Polynomials

4.6 Summary Notes

 Laguerre differential equation has x 0 as a regular singular point. Thus Frobenius method
is applied to get a power series.

 For , n n being a positive integer we obtain a finite power series solution known as
n
Laguerre polynomials L n ( ). The highest power of L n ( ) is .
x
x
x
 Like in the previous units here we show a generating function, Rodrigue formula for
x
L ( ).
n
x
 L n ( ) for n 0,1,2 form an orthogonal set of functions and satisfy orthogonality
property.
 Various recurrence relations are obtained that help in understanding Laguerre polynomials.
4.7 Keywords

Laguerre Polynomials are a finite power series in x.

Frobenius Method: Laguerre differential equation has x 0 as regular singular point. So
Frobenius method on application gives a power series solution.

Orthogonal Relations of Laguerre polynomials are relations involving integrals of two Hermite
polynomials. Due to these relations L n ( ) for n 0,1,2, form an orthogonal set of functions.
x
4.8 Review Questions

1. Discuss the nature of singularities of the differential equation

xy y xy 0
2. Find all the singular points of the differential equation

2
2
1 x y xy x y 0
3. Show from recurrence relation III

n 1
x
x
L n ( ) L r ( )
r 0
Prove that
x
x dL ( )
e n L n ( )dx 0, for n 1,2,
x
0 dx
x
L
x
x
4. Show that L 3 ( ), L 2 ( ) and ( ) are related as
1
3 ( ) (5 x ) ( ) L 1 ( )
x
L
x
L
x
3
2
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