Laguerre polynomial

Laguerre polynomial

1

Definition

The

Laguerre polynomials

are

orthogonal polynomials

with respect to the

weighting function

e

-

x

on the half-line

[

0

,

∞

)

. They are denoted
by the letter “

L

” with the order as subscript and are normalized by the condition that the

coefficient

of the highest order term of

L

n

is

(

-

1

)

n

/

n

!

.

The first few Laguerre poynomials are as follows:

L

0

⁢

(

x

)

=

1

L

1

⁢

(

x

)

=

-

x

+

1

L

2

⁢

(

x

)

=

1

2

⁢

x

2

-

2

⁢

x

+

1

L

3

⁢

(

x

)

=

-

1

6

⁢

x

3

+

3

2

⁢

x

2

-

3

⁢

x

+

1

A generalization is given by the

associated Laguerre polynomials

which depends on a

parameter

(traditionally denoted “

α

”). As it
turns out, they are

polynomials

of the

argument

α

as as well, so
they are polynomials of two variables. They are defined over the same

interval

with the same normalization condition, but the weight function
is generalized to

x

α

⁢

e

-

x

. They are notated by including the
parameter as a parenthesized superscript (not all authors use the
parentheses).

The ordinary Laguerre polynomials are the special case of the generalized
Laguerre polynomials when the parameter goes to zero. When some result
holds for generalized Laguerre polynomials which is not more complicated
than that for ordinary Laguerre polynomials, we shall only provide the more
general result and leave it to the reader to send the parameter to zero
to recover the more specific result.

The first few generalized Laguerre polynomials are as follows:

L

0

(

α

)

⁢

(

x

)

=

1

L

0

(

α

)

⁢

(

x

)

=

-

x

+

α

+

1

L

0

(

α

)

⁢

(

x

)

=

1

2

⁢

x

2

-

(

α

+

2

)

⁢

x

+

1

2

⁢

(

α

+

2

)

⁢

(

α

+

1

)

L

0

(

α

)

⁢

(

x

)

=

-

1

6

⁢

x

3

+

1

2

⁢

(

α

+

3

)

⁢

x

2

-

1

2

⁢

(

α

+

2

)

⁢

(

α

+

3

)

⁢

x

+

1

6

⁢

(

α

+

1

)

⁢

(

α

+

2

)

⁢

(

α

+

3

)

2

Formulae for these polynomials

The Laguerre polynomials may be exhibited explicitly as a sum in terms
of

factorials

, which may also be written using

binomial coefficients

:

L

n

⁢

(

x

)

=

∑

k

=

0

n

n

!

(

k

!

)

2

⁢

(

n

-

k

)

!

⁢

(

-

x

)

k

=

∑

k

=

0

n

(

n

k

)

⁢

(

-

x

)

k

k

!

The generalization may be expressed in terms of

gamma functions

or

falling factorials

:

L

n

(

α

)

=

∑

k

=

0

n

Γ

⁢

(

n

+

α

+

1

)

Γ

⁢

(

k

+

α

+

1

)

⁢

(

-

x

)

k

k

!

⁢

(

n

-

k

)

!

=

∑

k

=

0

n

(

n

+

α

)

n

-

k

¯

k

!

⁢

(

n

-

k

)

!

⁢

(

-

x

)

k

They can be computed from a

Rodrigues formula

:

L

n

(

α

)

⁢

(

x

)

=

1

n

!

⁢

x

-

α

⁢

e

x

⁢

d

n

d

⁢

x

n

⁢

(

e

-

x

⁢

x

n

+

α

)

They have several

integral

representations

. They can be expressed
in terms of a countour integral

L

n

⁢

(

x

)

=

1

2

⁢

π

⁢

i

⁢

∮

e

-

x

⁢

t

1

-

t

(

1

-

t

)

⁢

t

n

+

1

⁢

𝑑𝑡

,

where the origin is enclosed by the

contour

, but not

z

=

1

.

3

Equations they satisfy

The Laguerre polynomials satisfy the

orthogonality relation

∫

0

∞

e

-

x

⁢

x

α

⁢

L

n

(

α

)

⁢

(

x

)

⁢

L

m

(

α

)

⁢

(

x

)

⁢

𝑑

x

=

(

n

+

α

)

!

n

!

⁢

δ

n

⁢

m

.

The Laguerre polynomials satisfy the differential equation

x

⁢

d

2

d

⁢

x

2

⁢

L

n

(

α

)

⁢

(

x

)

+

(

α

+

1

-

x

)

⁢

d

d

⁢

x

⁢

L

n

(

α

)

⁢

(

x

)

+

(

n

-

α

)

⁢

L

n

(

α

)

⁢

(

x

)

=

0

This

equation

arises in many contexts such as in the quantum
mechanical treatment of the hydrogen atom.

Title

Laguerre polynomial

Canonical name

LaguerrePolynomial

Date of creation

2013-03-22 12:30:56

Last modified on

2013-03-22 12:30:56

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

22

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 26C99

Related topic

HermitePolynomials

Related topic

OrthogonalityOfChebyshevPolynomials

Defines

associated Laguerre polynomial

Defines

Laguerre’s equation

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