Laguerre polynomial

1 Definition

The Laguerre polynomials are orthogonal polynomials with respect to theweighting function $e^{-x}$ on the half-line $[0,\\infty)$ . They are denotedby 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:

	$\\displaystyle L_{0}(x)$	$\\displaystyle=1$
	$\\displaystyle L_{1}(x)$	$\\displaystyle=-x+1$
	$\\displaystyle L_{2}(x)$	$\\displaystyle=\\frac{1}{2}x^{2}-2x+1$
	$\\displaystyle L_{3}(x)$	$\\displaystyle=-\\frac{1}{6}x^{3}+\\frac{3}{2}x^{2}-3x+1$

A generalization is given by the associated Laguerre polynomialswhich depends on a parameter (traditionally denoted “ $\\alpha$ ”). As itturns out, they are polynomials of the argument $\\alpha$ as as well, sothey are polynomials of two variables. They are defined over the sameinterval with the same normalization condition, but the weight functionis generalized to $x^{\\alpha}e^{-x}$ . They are notated by including theparameter as a parenthesized superscript (not all authors use theparentheses).

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

The first few generalized Laguerre polynomials are as follows:

	$\\displaystyle L_{0}^{(\\alpha)}(x)$	$\\displaystyle=1$
	$\\displaystyle L_{0}^{(\\alpha)}(x)$	$\\displaystyle=-x+\\alpha+1$
	$\\displaystyle L_{0}^{(\\alpha)}(x)$	$\\displaystyle=\\frac{1}{2}x^{2}-(\\alpha+2)x+\\frac{1}{2}(\\alpha+2)(\\alpha+1)$
	$\\displaystyle L_{0}^{(\\alpha)}(x)$	$\\displaystyle=-\\frac{1}{6}x^{3}+\\frac{1}{2}(\\alpha+3)x^{2}-\\frac{1}{2}(\\alpha+%2)(\\alpha+3)x+\\frac{1}{6}(\\alpha+1)(\\alpha+2)(\\alpha+3)$

2 Formulae for these polynomials

The Laguerre polynomials may be exhibited explicitly as a sum in termsof factorials, which may also be written using binomial coefficients:

L_{n}(x)=\\sum_{k=0}^{n}{n!\\over(k!)^{2}(n-k)!}(-x)^{k}=\\sum_{k=0}^{n}{n\\choosek%}{(-x)^{k}\\over k!}

The generalization may be expressed in terms of gamma functions orfalling factorials:

L_{n}^{(\\alpha)}=\\sum_{k=0}^{n}{\\Gamma(n+\\alpha+1)\\over\\Gamma(k+\\alpha+1)}{(-x%)^{k}\\over k!(n-k)!}=\\sum_{k=0}^{n}{(n+\\alpha)^{\\underline{n-k}}\\over k!(n-k)!%}(-x)^{k}

They can be computed from a Rodrigues formula:

L_{n}^{(\\alpha)}(x)=\\frac{1}{n!}x^{-\\alpha}e^{x}\\frac{d^{n}}{dx^{n}}\\left(e^{-%x}x^{n+\\alpha}\\right)

They have several integral representations. They can be expressedin terms of a countour integral

L_{n}(x)=\\frac{1}{2\\pi i}\\oint\\frac{e^{-\\frac{xt}{1-t}}}{(1-t)t^{n+1}}\\,%\\mathit{dt},

where the origin is enclosed by the contour, but not $z=1$ .

3 Equations they satisfy

The Laguerre polynomials satisfy the orthogonality relation

\\int_{0}^{\\infty}e^{-x}x^{\\alpha}L_{n}^{(\\alpha)}(x)L_{m}^{(\\alpha)}(x)\\,dx=%\\frac{(n+\\alpha)!}{n!}\\delta_{nm}.

The Laguerre polynomials satisfy the differential equation

x\\frac{d^{2}}{dx^{2}}L_{n}^{(\\alpha)}(x)+(\\alpha+1-x)\\frac{d}{dx}L_{n}^{(%\\alpha)}(x)+(n-\\alpha)L_{n}^{(\\alpha)}(x)=0

This equation arises in many contexts such as in the quantummechanical treatment of the hydrogen atom.