LAGUERRE POLYNOMIALS

In mathematics, the 'Laguerre polynomials', named after Edmond Laguerre (1834 - 1886),
are the canonical solutions of 'Laguerre's equation':
:$$
x,y'' + (1 - x),y' + n,y = 0,

which is a second-order linear differential equation.
This equation has nonsingular solutions only if ''n'' is a non-negative integer.
These polynomials, usually denoted $L\_0,\; L\_1,\; dots$, are a polynomial sequence which may be defined by the Rodrigues formula
:$$
L_n(x)=rac{e^x}{n!}rac{d^n}{dx^n}left(e^{-x} x^n
ight).

They are orthogonal to each other with respect to the inner product given by
:$langle\; f,g$
angle = int_0^infty f(x) g(x) e^{-x},dx.
The sequence of Laguerre polynomials is a Sheffer sequence.
The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution
of the Schrödinger equation for a one-electron atom.
Physicists often use a definition for the Laguerre polynomials that is larger,
by a factor of $(n!)$, than the definition used here.

Contents

The first few polynomials

As contour integral

Recursive definition

Generalized Laguerre polynomials

Explicit examples of generalized Laguerre polynomials

Derivatives of generalized Laguerre polynomials

Relation to Hermite polynomials

Relation to hypergeometric functions

External links

References

The first few polynomials

These are the first few Laguerre polynomials:

'n'	$L\_n(x),$
0	$1,$
1	$-x+1,$
2
3
4
5
6

As contour integral

The polynomials may be expressed in terms of a contour integral
:
where the contour circles the origin once in a counterclockwise direction.

Recursive definition

We can also define the Laguerre polynomials recursively, defining the first two polynomials as
:$L\_0(x)\; =\; 1,$
:$L\_1(x)\; =\; 1\; -\; x,$
and then using the recurrence relation for any $k\; geq\; 1$:
:

Generalized Laguerre polynomials

The orthogonality property stated above is equivalent to saying that if ''X'' is an exponentially distributed random variable with probability density function
:
ight.
then
:$E(L\_n(X)L\_m(X))=0\; mbox\{whenever\}\; n$
eq m.
The exponential distribution is not the only gamma distribution. A polynomial sequence orthogonal with respect to the gamma distribution whose probability density function is, for ,
:
ight.
(see gamma function) is given by the defining Rodrigues equation for the 'generalized Laguerre polynomials':
:
{x^{-lpha} e^x over n!}{d^n over dx^n} left(e^{-x} x^{n+lpha}
ight) .
These are also sometimes called the 'associated Laguerre polynomials'. The simple Laguerre polynomials are recovered from the generalized polynomials by setting α = 0:
:$L^\{(0)\}\_n(x)=L\_n(x).$
The associated Laguerre polynomials are orthogonal over $[0,infty)$ with respect to the weighting function :
:
The following integral is needed in the quantum mechanical treatment of the hydrogen atom,
:
ight]^2 dx=
rac{(n+lpha)!}{n!}(2n+lpha+1).
The associated Laguerre polynomials obey the following differential equation:
:$$
x L_n^{(lpha) primeprime}(x) + (lpha+1-x)L_n^{(lpha)prime}(x) + n L_n^{(lpha)}(x)=0.,

They obey the following recurrence relation for $n\; geq\; 1$:
:
Two other recurrence relations which are sometimes useful are
:
:

Explicit examples of generalized Laguerre polynomials

The generalized Laguerre polynomial of degree $n$ is (as follows from applying Leibniz's theorem for differentiation of a product to the defining Rodrigues formula)
:$$
L_n^{(lpha)} (x) = sum_{m=0}^n {n+lpha choose n-m} rac{(-x)^m}{m!}

from which we see that the coefficient of the leading term is $(-1)^n/n!$ and the constant term (which is also the value at the origin) is
The first few generalized Laguerre polynomials are:
:
:
:
:
+ rac{(lpha+1)(lpha+2)(lpha+3)}{6}

Derivatives of generalized Laguerre polynomials

Differentiating the power series representation of a generalized Laguerre polynomial $k$ times leads to
:$$
rac{mathrm d^k}{mathrm d x^k} L_n^{(lpha)} (x)
=
(-1)^k L_{n-k}^{(lpha+k)} (x),.

Relation to Hermite polynomials

The generalized Laguerre polynomials are related to the Hermite polynomials:
:$H\_\{2n\}(x)\; =\; (-1)^n\; 2^\{2n\}\; n!\; L\_n^\{(-1/2)\}\; (x^2)$
and
:$H\_\{2n+1\}(x)\; =\; (-1)^n\; 2^\{2n+1\}\; n!\; x\; L\_n^\{(1/2)\}\; (x^2)$
where the $H\_n(x)$ are the Hermite polynomials based on the weighting function $exp\{(-x^2)\}$, the so-called "physicist's version."
Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.

Relation to hypergeometric functions

The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as
:
where $(a)\_n$ is the Pochhammer symbol (which in ''this'' case represents the ''rising factorial'').

External links

★ A quick informal derivation of the Laguerre polynomial in the context of the quantum mechanics of hydrogen

References

★

★ Eric W. Weisstein, "Laguerre Polynomial", From MathWorld--A Wolfram Web Resource.

★ Mathematical Methods for Physicists, George Arfken and Hans Weber, , , Academic Press, 2000, ISBN 0-12-059825-6