applying generating function

applying generating function

The

generating function

of a

function

sequence carries information common to the members of the sequence.  It may be utilised for deriving various properties, such as recurrence relations, orthogonality properties etc.  We take as example

e

2

⁢

z

⁢

t

-

t

2

=

∑

n

=

0

∞

H

n

⁢

(

z

)

n

!

⁢

t

n

,

(1)

the

http://planetmath.org/node/11980

generating function of the of

Hermite polynomials

, and derive from it a recurrence relation and the orthonormality (

http://planetmath.org/Orthonormal

) formula.

1.  First we form the

partial derivative

with respect to

t

of both of (1):

(

2

⁢

z

-

2

⁢

t

)

⁢

e

2

⁢

z

⁢

t

-

t

2

=

∑

m

=

1

∞

H

m

⁢

(

z

)

(

m

-

1

)

!

⁢

t

m

-

1

Here we substitute (1) to the

left hand side

and rewrite the right hand side, getting

∑

n

=

0

∞

2

⁢

z

⁢

H

n

⁢

(

z

)

n

!

⁢

t

n

-

∑

n

=

1

∞

2

⁢

H

n

-

1

⁢

(

z

)

(

n

-

1

)

!

⁢

t

n

=

∑

n

=

0

∞

H

n

+

1

⁢

(

z

)

n

!

⁢

t

n

,

where we can compare the

coefficients

of

t

n

:

2

⁢

z

⁢

H

n

n

!

-

2

⁢

H

n

-

1

(

n

-

1

)

!

=

H

n

+

1

n

!

(

n

=

1

,

2

,

…

)

Thus we have gotten the recurrence relation

H

n

+

1

(

z

)

=

2

z

H

n

(

z

)

-

2

n

H

n

-

1

(

z

)

(

n

=

1

,

2

,

…

)

.

(2)

Differentiating (1) partially with respect to

z

enables respectively to find a formula expressing the

derivative

H

n

′

⁢

(

z

)

via the themselves.

2.  We copy the equation (1) twice in the forms

∑

n

=

0

∞

H

n

⁢

(

x

)

n

!

⁢

t

n

=

e

2

⁢

x

⁢

t

-

t

2

,

∑

n

=

0

∞

H

n

⁢

(

x

)

n

!

⁢

u

n

=

e

2

⁢

x

⁢

u

-

u

2

,

multiply these with each other and by

e

-

x

2

and then integrate the obtained equation termwise over

ℝ

:

∑

m

=

0

∞

∑

n

=

0

∞

(

∫

-

∞

∞

e

-

x

2

⁢

H

m

⁢

(

x

)

⁢

H

n

⁢

(

x

)

⁢

𝑑

x

)

⁢

t

m

⁢

u

n

m

!

⁢

n

!

=

∫

-

∞

∞

e

-

x

2

⁢

e

2

⁢

x

⁢

t

-

t

2

⁢

e

2

⁢

x

⁢

u

-

u

2

⁢

𝑑

x

=

∫

-

∞

∞

e

2

⁢

x

⁢

(

t

+

u

)

-

t

2

-

u

2

-

x

2

⁢

𝑑

x

=

∫

-

∞

∞

e

-

[

(

t

+

u

)

2

-

2

⁢

(

t

+

u

)

⁢

x

+

x

2

]

+

2

⁢

t

⁢

u

⁢

𝑑

x

=

e

2

⁢

t

⁢

u

⁢

∫

-

∞

∞

e

-

[

x

-

(

t

+

u

)

]

2

⁢

𝑑

x

=

e

2

⁢

t

⁢

u

⁢

∫

-

∞

∞

e

-

y

2

⁢

𝑑

y

=

e

2

⁢

t

⁢

u

⁢

π

=

∑

ȷ

=

0

∞

π

⁢

2

j

⁢

t

j

⁢

u

j

j

!

=

∑

m

=

0

∞

∑

n

=

0

∞

(

π

n

!

⋅

2

n

⁢

δ

m

⁢

n

)

⁢

t

m

⁢

u

n

Thus we can infer that

∫

-

∞

∞

e

-

x

2

⁢

H

m

⁢

(

x

)

⁢

H

n

⁢

(

x

)

⁢

𝑑

x

m

!

⁢

n

!

=

π

n

!

⋅

2

n

⁢

δ

m

⁢

n

,

which implies the orthonormality

relation

∫

-

∞

∞

e

-

x

2

⁢

H

m

⁢

(

x

)

⁢

H

n

⁢

(

x

)

⁢

𝑑

x

=

2

m

⁢

m

!

⁢

δ

m

⁢

n

⁢

π

.

(3)

Cf. Hermite polynomials.

Title

applying generating function

Canonical name

ApplyingGeneratingFunction

Date of creation

2013-03-22 19:06:58

Last modified on

2013-03-22 19:06:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Example

Classification

msc 33B99

Classification

msc 33C45

Classification

msc 26C05

Classification

msc 26A42

Related topic

AreaUnderGaussianCurve

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