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Laguerre polynomials

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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)=\frac\frac\left(e^ x^n\right).]
They are orthogonal to each other with respect to the inner product given by

[\langle f,g \rangle = \int_0^\infty f(x) g(x) e^\,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

These are the first few Laguerre polynomials:

n [L_n(x)\,]
0 [1\,]
1 [-x+1\,]
2 [\begin\frac12\end (x^2-4x+2) \,]
3 [\begin\frac16\end (-x^3+9x^2-18x+6) \,]
4 [\begin\frac1\end (x^4-16x^3+72x^2-96x+24) \,]
5 [\begin\frac1\end (-x^5+25x^4-200x^3+600x^2-600x+120) \,]
6 [\begin\frac1\end (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,]

As contour integral

The polynomials may be expressed in terms of a contour integral

[L_n(x)=\frac\oint\frac}} \; dt]
where the contour circles the origin once in a counterclockwise direction.

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

[f(x)=\left\ e^ & \mbox\ x>0, \\ 0 & \mbox\ x<0, \end\right.]
then

[E(L_n(X)L_m(X))=0\ \mbox\ n\neq 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 [\alpha>-1],

[f(x)=\left\ x^\alpha e^/\Gamma(1+\alpha) & \mbox\ x>0, \\ 0 & \mbox\ x<0, \end\right.]
(see gamma function) is given by the defining Rodrigues equation for the generalized Laguerre polynomials:

[L_n^(x)= e^x \over n!} \left(e^ x^\right) .]
These are also sometimes called the associated Laguerre polynomials. The simple Laguerre polynomials are recovered from the generalized polynomials by setting α=0:

[L^_n(x)=L_n(x).]
The associated Laguerre polynomials are orthogonal over [[0,\infty)] with respect to the weighting function [x^\alpha e^]:

[\int_0^e^x^\alpha L_n^(x)L_m^(x)dx=\frac\delta_.]
The associated Laguerre polynomials obey the following differential equation

[x L_n^(x) + (\alpha+1-x)L_n^(x) + n L_n^(x)=0.\,]

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^ (x) = \sum_^n \frac]
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

[ L_0^ (x) = 1 ]
[ L_1^(x) = -x + \alpha +1]
[ L_2^(x) = \frac - (\alpha + 2)x + \frac]
[ L_3^(x) = \frac + \frac - \frac+ \frac]

Derivatives of generalized Laguerre polynomials

Differentiating the power series representation of a generalized Laguerre polynomial [k] times leads to

[\frac L_n^ (x)=(-1)^k L_^ (x)\,.]

Relation to Hermite polynomials

The generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator, due to their relation to the Hermite polynomials, which can be expressed as

[H_(x) = (-1)^n 2^ n! L_n^ (x^2)]
and

[H_(x) = (-1)^n 2^ n! x L_n^ (x^2)]
where the [H_n(x)] are the Hermite polynomials.

Relation to hypergeometric functions

The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as

[L^_n(x) = M(-n,\alpha+1,x) =\frac \,_1F_1(-n,\alpha+1,x)]
where [(a)_n] is the Pochhammer symbol (which in this case represents the rising factorial).

External links

References

 

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
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