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Laguerre polynomial

(Definition)

Definiton

The Laguerre polynomials are orthogonal polynomials with respect to the weighting function $e^{-x}$ on the half-line $[0,\infty)$ . They are denoted by the letter “” with the order as subscript and are normalized by the condition that the coefficient of the highest order term of is .

The first few Laguerre poynomials are as follows:

$\displaystyle L_0 (x)$	$\displaystyle = 1$
$\displaystyle L_1 (x)$	$\displaystyle = -x + 1$
$\displaystyle L_2 (x)$	$\displaystyle = \frac{1}{2} x^2 - 2x + 1$
$\displaystyle L_3 (x)$	$\displaystyle = -\frac{1}{6} x^3 + \frac{3}{2} x^2 - 3 x + 1$

A generalization is given by the associated Laguerre polynomials which depends on a parameter (traditionally denoted “ $\alpha$ ”). As it turns out, they are polynomials of the argument 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 $x^\alpha e^{-x}$ . 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 poynomials are as follows:

$\displaystyle L_0^{(\alpha)} (x)$	$\displaystyle = 1$
$\displaystyle L_0^{(\alpha)} (x)$	$\displaystyle = -x + \alpha + 1$
$\displaystyle L_0^{(\alpha)} (x)$	$\displaystyle = \frac{1}{2} x^2 - (\alpha + 2) x + \frac{1}{2} (\alpha + 2) (\alpha + 1)$
$\displaystyle L_0^{(\alpha)} (x)$	$\displaystyle = - \frac{1}{6} x^3 + \frac{1}{2} (\alpha + 3) x^2 - \frac{1}{2} (\alpha + 2) (\alpha + 3) x + \frac{1}{6} (\alpha + 1) (\alpha + 2) (\alpha + 3)$

Formulae for these polynomials

The Laguerre poynomials may be exhibited explicitly as a sum in terms of factorials, which may also be written using binomial coefficients:

$\displaystyle L_n (x) = \sum_{k=0}^n {n! \over (k!)^2 (n-k)!} (-x)^k = \sum_{k=0}^n {n \choose k} {(-x)^k \over k!}$

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

$\displaystyle L_n^{(\alpha)} = \sum_{k=0}^n {\Gamma (n + \alpha + 1) \over \Gam... ...(n-k)!} = \sum_{k=0}^n {(n + \alpha)^{\underline{n-k}} \over k! (n-k)!} (-x)^k$

They can be computed from a Rodrigues formula:

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

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

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

where the origin is enclosed by the contour, but not

Equations they satisfy

The Laguerre polynomials satisfy the orthogonality relation

$\displaystyle \int_0^{\infty}e^{-x} x^\alpha L_n^{(\alpha)} (x) L_m^{(\alpha)} (x) \, dx = \frac {(n+\alpha)!} {n!} \delta_{nm}.$

The Laguerre polynomials satisfy the differential equation

$\displaystyle x \frac{d^2}{dx^2} L_n^{(\alpha)} (x) + (\alpha+1-x) \frac{d}{dx} L_n^{(\alpha)} (x) + (n-\alpha) L_n^{(\alpha)} (x) = 0$

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

"Laguerre polynomial" is owned by rspuzio. [ full author list (2) | owner history (1) ]

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