Page 64 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Richa Nandra, Lovely Professional University Unit 3: Hermite Polynomials

Unit 3: Hermite Polynomials Notes

CONTENTS
Objectives
Introduction
3.1 Power Series Solution of Hermite Polynomials

3.2 Generating Functions of Hermite Polynomials H n ( )
z
3.3 The Rodrigue s Formula for H n ( )
x
3.4 Orthogonal Properties of Hermite Polynomials

3.5 Recurrence Formula for Hermite Polynomials
3.6 Summary
3.7 Keywords
3.8 Review Questions
3.9 Further Readings

Objectives

After studying this unit, you should be able to:
 Solve second order differential equation like Hermite equation.

 Familiarize yourself with the properties of Hermite Polynomials through generating
function.
 Obtain certain relations involving Hermite polynomials with the help of Rodrigue formula.

 Solve certain integrals. You can express any function ( )f x in terms of Hermite polynomials
H (x).
n
 Relate some Hermite polynomials in terms of others with the help of recurrence relations.

Introduction

In the previous two units you have learnt the method of Frobenius in solving second order
differential equations in power series. This method will help us to solve Hermite differential
equation. In this unit we will be able to solve the equation for x range.
Just as the generating functions were introduced in the previous chapter, here in this chapter
also it will be introduced for Hermite polynomials. Also orthogonal properties and recurrence
relations are very important in understanding the properties of Hermite polynomials.

3.1 Power Series Solution of Hermite Polynomials

Consider the following equation, containing a parameter ,

d x 2 dy x 2
e 2 e y = 0 ...(A)
dx dx

LOVELY PROFESSIONAL UNIVERSITY 57