Page 73 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes First few Hermite Polynomials from Rodrigue s Formula

x
From Rodrigue s Formula for H n ( )
2 d n x 2
n x
x
H n ( ) = ( 1) e . n e
dx
Putting n = 0, 1, 2, 3, ... we get

x
H ( ) = e x 2 . e x 2 1
0
2 d x 2
x
x
H 1 ( ) = ( 1)e e 2x
dx
2 d 2 2 2 d 2
2 x
x
H 2 ( ) = ( 1) e 2 e x e x 2xe x
dx dx
2 2 x 2 x 2
x
= e 4x e 2e
= 4x 2 2 .

2 d 3 2
3 x
H 3 ( ) = ( 1) e e x
x
dx 3
2 d 2 x 2
x
= e 4x 2 e
dx
= e x 2 2x 4x 2 2 e x 2 8xe x 2

x
= e x 2 8x 3 12x e x 2 8x 3 12 .

x
Similarly, H 4 ( ) = 16x 4 48x 2 12 etc.
3.4 Orthogonal Properties of Hermite Polynomials

Prove

x
e x 2 H n ( )H m ( )dx = 0, if m n
x
2
2 ( )! if m n
n
We have e t 2 2tx = H ( ) t n
x
n
n 0 ! n
x
and e s 2 2sx = H m ( ) s m
m 0 m !
t n s m
2 2tx s 2 2sx
x
x
t
e e = H n ( ) H m ( )
n 0 ! n m 0 m !
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