Page 74 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 3: Hermite Polynomials

Notes
1 t 2 2tx s 2 2sx
x
H n ( )H m ( ) = Coeff. of t s in the expansion of e e
x
n m
m
n ! !
2 n m
x
e H n ( )H m ( )dx is equal to n! m! times the coefficient of t s in the expansion of
x
x
2 t 2 2tx s 2 2sx
x
e .e .e dx
2 t 2 2tx s 2 2sx
x
Now, e .e .e dx
2 s 2 x 2 2tx 2sx
t
= e e dx
2 2 2
2 s 2 x (t s ) (t s ) dx
t
= e e
= e 2ts e x (t x ) 2dx

2
2ts
u
= e e du , putting x (t s ) u
= e 2ts , since e u 2 du

ts
(2 ) 2 (2 ) n
ts
= 1 2ts
2! ! n
n m
Coefficient of t s in the expansion of

e x 2 e t 2 2tx e s 2 2sx dx

is 0 if m n

2 n
and . if m n .
! n
We can also write it as follows

n
x
x
e x 2 H n ( )H m ( )dx 2 n ! mn ,
where is Kronecker delta defined as
mn
0, if m n
mn
1, if m n

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