Hermite polynomials

Hermite polynomials

The

polynomial

solutions of the

Hermite differential equation

, with

n

a non-negative integer, are usually normed so that the highest degree (

http://planetmath.org/PolynomialRing

) is

(

2

⁢

z

)

n

and called the

Hermite polynomials

H

n

⁢

(

z

)

.  The Hermite polynomials may be defined explicitly by

H

n

⁢

(

z

)

:=

(

-

1

)

n

⁢

e

z

2

⁢

d

n

d

⁢

z

n

⁢

e

-

z

2

,

(1)

since this is a polynomial having the highest

(

2

⁢

z

)

n

and satisfying the Hermite equation.  The equation (1) is the Rodrigues’s formula for Hermite polynomials.  Using the Faà di Bruno’s formula, one gets from (1) also

H

n

⁢

(

x

)

=

(

-

1

)

n

⁢

∑

m

1

+

2

⁢

m

2

=

n

n

!

m

1

!

⁢

m

2

!

⁢

(

-

1

)

m

1

+

m

2

⁢

(

2

⁢

x

)

m

1

.

The first six Hermite polynomials are

H

0

⁢

(

z

)

≡

1

,

H

1

⁢

(

z

)

≡

2

⁢

z

,

H

2

⁢

(

z

)

≡

4

⁢

z

2

-

2

,

H

3

⁢

(

z

)

≡

8

⁢

z

3

-

12

⁢

z

,

H

4

⁢

(

z

)

≡

16

⁢

z

4

-

48

⁢

z

2

+

12

,

H

5

⁢

(

z

)

≡

32

⁢

z

5

-

160

⁢

z

3

+

120

⁢

z

,

and the general is

H

n

(

z

)

≡

(

2

z

)

n

-

n

⁢

(

n

-

1

)

1

!

(

2

z

)

n

-

2

+

n

⁢

(

n

-

1

)

⁢

(

n

-

2

)

⁢

(

n

-

3

)

2

!

(

2

z

)

n

-

4

-

+

…

Differentiating this termwise gives

H

n

′

(

z

)

=

2

n

[

(

2

z

)

n

-

1

-

(

n

-

1

)

⁢

(

n

-

2

)

1

!

(

2

z

)

n

-

3

+

(

n

-

1

)

⁢

(

n

-

2

)

⁢

(

n

-

3

)

⁢

(

n

-

4

)

2

!

(

2

z

)

n

-

5

-

+

…

]

,

i.e.

H

n

′

⁢

(

z

)

=

2

⁢

n

⁢

H

n

-

1

⁢

(

z

)

.

(2)

The Hermite polynomials are sometimes scaled to such ones

He

n

which obey the differentiation rule

He

n

′

⁢

(

z

)

=

n

⁢

He

n

-

1

⁢

(

z

)

.

(3)

Such Hermite polynomials form an Appell sequence.

We shall now show that the Hermite polynomials form an orthogonal set (

http://planetmath.org/OrthogonalPolynomials

) on the

interval

(

-

∞

,

∞

)

with the weight factor (

http://planetmath.org/OrthogonalPolynomials

)

e

-

x

2

.  Let

m

<

n

;  using (1) and integrating by parts (

http://planetmath.org/IntegrationByParts

) we get

(

-

1

)

n

⁢

∫

-

∞

∞

H

m

⁢

(

x

)

⁢

H

n

⁢

(

x

)

⁢

e

-

x

2

⁢

𝑑

x

=

∫

-

∞

∞

H

m

⁢

(

x

)

⁢

d

n

⁢

e

-

x

2

d

⁢

x

n

⁢

𝑑

x

=

/

-

∞

∞

⁡

H

m

⁢

(

x

)

⁢

d

n

-

1

⁢

e

-

x

2

d

⁢

x

n

-

1

-

∫

-

∞

∞

H

m

′

⁢

(

x

)

⁢

d

n

-

1

⁢

e

-

x

2

d

⁢

x

n

-

1

⁢

𝑑

x

.

The substitution portion here equals to zero because

e

-

x

2

and its

derivatives

vanish at

±

∞

.  Using then (2) we obtain

∫

-

∞

∞

H

m

⁢

(

x

)

⁢

H

n

⁢

(

x

)

⁢

e

-

x

2

⁢

𝑑

x

=

2

⁢

(

-

1

)

1

+

n

⁢

m

⁢

∫

-

∞

∞

H

m

-

1

⁢

(

x

)

⁢

d

n

-

1

⁢

e

-

x

2

d

⁢

x

n

-

1

⁢

𝑑

x

.

Repeating the

integration by parts

gives the result

∫

-

∞

∞

H

m

⁢

(

x

)

⁢

H

n

⁢

(

x

)

⁢

e

-

x

2

⁢

𝑑

x

=

2

m

⁢

(

-

1

)

m

+

n

⁢

m

!

⁢

∫

-

∞

∞

H

0

⁢

(

x

)

⁢

d

n

-

m

⁢

e

-

x

2

d

⁢

x

n

-

m

⁢

𝑑

x

=

2

m

⁢

(

-

1

)

m

+

n

⁢

m

!

⁢

/

-

∞

∞

⁡

d

n

-

m

-

1

⁢

e

-

x

2

d

⁢

x

n

-

m

-

1

=

0

,

whereas in the case

m

=

n

the result

∫

-

∞

∞

(

H

n

⁢

(

x

)

)

2

⁢

e

-

x

2

⁢

𝑑

x

=

2

n

⁢

(

-

1

)

2

⁢

n

⁢

n

!

⁢

∫

-

∞

∞

e

-

x

2

⁢

𝑑

x

=

2

n

⁢

n

!

⁢

π

(see

area under Gaussian curve

).
The results that the

functions

x

↦

H

n

⁢

(

x

)

2

n

⁢

n

!

⁢

π

⁢

e

-

x

2

2

form an orthonormal set on

(

-

∞

,

∞

)

.

The Hermite polynomials are used in the quantum mechanical treatment of a

harmonic oscillator

, the wave functions of which have the form

ξ

↦

Ψ

n

⁢

(

ξ

)

=

C

n

⁢

H

n

⁢

(

ξ

)

⁢

e

-

ξ

2

2

.

Title

Hermite polynomials

Canonical name

HermitePolynomials

Date of creation

2013-03-22 15:16:25

Last modified on

2013-03-22 15:16:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

28

Author

pahio (2872)

Entry type

Definition

Classification

msc 33E30

Classification

msc 33B99

Classification

msc 26C05

Classification

msc 26A09

Classification

msc 12D99

Related topic

SubstitutionNotation

Related topic

AppellSequence

Related topic

LaguerrePolynomial

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