Page 69 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes This value of y is known as the Hermite s polynomial of degree n and is written as
n
( n n 1) n 2 ( n n 1)(n 2)(n 3) n 4 n /2 ! n
n
x
x
x
H n ( ) = (2 )x (2 ) (2 ) ( 1)
1! 2! ( /2)!
n
n
( /2) ! n
x
or H n ( ) = ( 1) r r !(n 2 )! (2 ) n 2r
x
r
r 0
n
n /2 if is even
n
where = 1
2 (n 1) if is odd
n
2
A first few H n ( ) are given as follows
x
H 0 ( ) = 1, H 1 ( ) 2x
x
x
H 2 ( ) = (2 )x 2 2 4x 2 2
x
3.2 2
3
x
H 3 ( ) = (2 )x (2 ) 4x 2x 3
x
1
12 2 4.3.2.1
4
x
H 4 ( ) = (2 )x (2 ) (1)
x
1 2
= 16x 4 48x 2 12
Self Assessment
Fill in the blanks:
x
1. Hermite polynomial H n ( ) is a ............... series.
x
2. As x , H 4 ( ) tends to infinity of an order not greater than ............... power of x.
x
3. H 3 ( ) satisfies equation (i) for = ...............
4. The value of H 4 ( ) is ...............
o
We now give some of the properties of Hermite polynomials like generating functions, Rodrigue
formula, orthogonality relations and the recurrence formulae.

3.2 G en eratin g F u n ctio n s o f H erm ite P o ly n o m ials H (x)
n

To prove that

2 t n
2xt t
e = H n ( )
x
n
n 0
or
H ( ) n 2xt t 2
x
show that n are the coefficients of t in the expansion of the function e (known as
n
x
generating function for H n ( ) ),

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