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Richa Nandra, Lovely Professional University Unit 4: Laguerre Polynomials

Unit 4: Laguerre Polynomials Notes

CONTENTS
Objectives
Introduction
4.1 Solution of Laguerre s Differential Equation

x
4.2 Generating Function for Laguerre Polynomials L n ( )
4.3 Rodrigue s Formula for Laguerre Polynomials L n ( )
x
x
4.4 Orthogonality Property of Laguerre Polynomials L n ( )
4.5 Recurrence Formulae for Laguerre Polynomials L n ( )
x
4.6 Summary
4.7 Keywords
4.8 Review Questions
4.9 Further Readings

Objectives

After studying this unit, you should be able to:

 Use generating function which helps you to familiarise with more properties of Laguerre
polynomials.
 Use Rodrigue formula which is quite helpful in making you more familiar with properties
of Laguerre polynomials.
 Employ of orthogonal properties to evaluate certain integrals.
 Use recurrence relations to correct one set of polynomials into another.

Introduction

Laguerre polynomials are shown to satisfy Laguerre differential equation. This equation has
x = 0 as regular singular point whereas x is an irregular singular point. A power series
solution is obtained by Frobenius method.
Generating function is obtained wherein it will be seen that most properties of Laguerre
polynomials are obtained orthogonal properties, recurrence relations Rodrigue s formula for
Laguerre polynomials are very important and almost all properties of L (x) are obtained from
n
the above relations.

4.1 Solution of Laguerre s Differential Equation

Consider the following differential equation containing a parameter .

x
x
x e y e y 0

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