Laguerre polynomials
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2008-06-27T21:46:51Z
A. Pichler
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In [[mathematics]], the '''Laguerre polynomials''', named after [[Edmond Laguerre]] (1834 - 1886),
are the [[canonical]] solutions of '''Laguerre's equation''':
:<math>
x\,y'' + (1 - x)\,y' + n\,y = 0\,
</math>
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 <math>L_0, L_1, \dots</math>, are a [[polynomial sequence]] which may be defined by the [[Rodrigues formula#Rodrigues formula|Rodrigues formula]]
:<math>
L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right).
</math>
They are [[orthogonal polynomials|orthogonal]] to each other with respect to the [[inner product]] given by
:<math>\langle f,g \rangle = \int_0^\infty f(x) g(x) e^{-x}\,dx.</math>
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 <math>\, (n!)</math>, than the definition used here.
==The first few polynomials==
These are the first few Laguerre polynomials:
<center><table class="wikitable">
<tr>
<td width="20%" align="center">'''n'''</td>
<td align="center"><math>L_n(x)\,</math></td>
</tr>
<tr>
<td align="center">0</td>
<td><math>1\,</math></td>
</tr>
<tr>
<td align="center">1</td>
<td><math>-x+1\,</math></td>
</tr>
<tr>
<td align="center">2</td>
<td><math>{\scriptstyle\frac{1}{2}} (x^2-4x+2) \,</math></td>
</tr>
<tr>
<td align="center">3</td>
<td><math>{\scriptstyle\frac{1}{6}} (-x^3+9x^2-18x+6) \,</math></td>
</tr>
<tr>
<td align="center">4</td>
<td><math>{\scriptstyle\frac{1}{24}} (x^4-16x^3+72x^2-96x+24) \,</math>
</tr>
<tr>
<td align="center">5</td>
<td><math>{\scriptstyle\frac{1}{120}} (-x^5+25x^4-200x^3+600x^2-600x+120) \,</math>
</tr>
<tr>
<td align="center">6</td>
<td><math>{\scriptstyle\frac{1}{720}} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,</math>
</tr>
</table>
</center>
[[Image:Laguerre_poly.svg|thumb|center|400px|The first six Laguerre polynomials.]]
==As contour integral==
The polynomials may be expressed in terms of a [[contour integral]]
:<math>L_n(x)=\frac{1}{2\pi i}\oint\frac{e^{-xt/(1-t)}}{(1-t)\,t^{n+1}} \; dt</math>
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
:<math>L_0(x) = 1\,</math>
:<math>L_1(x) = 1 - x\,</math>
and then using the [[Orthogonal_polynomials#Recurrence_relations|recurrence relation]] for any <math>k \geq 1</math>:
:<math>L_{k + 1}(x) = \frac{1}{k + 1} \bigg( (2k + 1 - x)L_k(x) - k L_{k - 1}(x)\bigg). </math>
==Generalized Laguerre polynomials==
The orthogonality property stated above is equivalent to saying that if ''X'' is an [[exponential distribution|exponentially distributed]] [[random variable]] with [[probability density function]]
:<math>f(x)=\left\{\begin{matrix} e^{-x} & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right.</math>
then
:<math>E \left[ L_n(X)L_m(X) \right]=0\ \mbox{whenever}\ n\neq m.</math>
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 <math>\alpha>-1</math>,
:<math>f(x)=\left\{\begin{matrix} x^\alpha e^{-x}/\Gamma(1+\alpha) & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right.</math>
(see [[gamma function]]) is given by the defining [[Rodrigues formula#Rodrigues formula|Rodrigues]] equation for the '''generalized Laguerre polynomials''':
:<math>L_n^{(\alpha)}(x)=
{x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right) .</math>
These are also sometimes called the '''associated Laguerre polynomials'''. The simple Laguerre polynomials are recovered from the generalized polynomials by setting α = 0:
:<math>L^{(0)}_n(x)=L_n(x).</math>
The associated Laguerre polynomials are orthogonal over <math>[0,\infty)</math> with respect to the weighting function <math>x^\alpha e^{-x}</math>:
:<math>\int_0^{\infty}e^{-x}x^\alpha L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)dx=\frac{\Gamma(n+\alpha+1)}{n!}\delta_{nm}.</math>
The following integral is needed in the quantum mechanical treatment of the hydrogen atom,
:<math>\int_0^{\infty}e^{-x}x^{\alpha+1} \left[L_n^{(\alpha)}\right]^2 dx=
\frac{(n+\alpha)!}{n!}(2n+\alpha+1).</math>
===Explicit examples of generalized Laguerre polynomials===
The generalized Laguerre polynomial of degree <math>n</math> is (as follows from applying [[Leibniz rule (generalized product rule)|Leibniz's theorem for differentiation of a product]] to the defining Rodrigues formula)
:<math>
L_n^{(\alpha)} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!}
</math>
from which we see that the [[coefficient]] of the leading term is <math>(-1)^n/n!</math> and the [[constant term]] (which is also the value at the origin) is <math>{n+\alpha\choose n}.</math>
The first few generalized Laguerre polynomials are:
:<math> L_0^{(\alpha)} (x) = 1 </math>
:<math> L_1^{(\alpha)}(x) = -x + \alpha +1</math>
:<math> L_2^{(\alpha)}(x) = \frac{x^2}{2} - (\alpha + 2)x + \frac{(\alpha+2)(\alpha+1)}{2}</math>
:<math> L_3^{(\alpha)}(x) = \frac{-x^3}{6} + \frac{(\alpha+3)x^2}{2} - \frac{(\alpha+2)(\alpha+3)x}{2}
+ \frac{(\alpha+1)(\alpha+2)(\alpha+3)}{6}</math>
===Recurrence Relations===
Laguerre's polynomials satisfy the recurrence relations
:<math>L_n^{(\alpha+1)}(x)= \sum_{i=0}^n L_i^{(\alpha)}(x) </math> and
:<math>L_n^{(\alpha)}(x)= {n+\alpha \choose n} - \frac{x}{n} \sum_{i=0}^{n-1} \frac{{n+\alpha \choose n-1-i}}{{n-1 \choose i}}L_i^{(\alpha+1)}(x). </math>
They can be used to derive
:<math>L_n^{(\alpha)}(x) = L_n^{(\alpha+1)}(x) - L_{n-1}^{(\alpha+1)}(x)</math> and
:<math>n L_n^{(\alpha)}(x) = (n + \alpha )L_{n-1}^{(\alpha)}(x) - x L_{n-1}^{(\alpha+1)}(x);</math>
combined they give this additional, popular recurrence relation
:<math>L_{n + 1}^{(\alpha)}(x) = \frac{1}{n + 1} \bigg( (2n + 1 + \alpha - x)L_n^{(\alpha)}(x) - (n + \alpha) L_{n - 1}^{(\alpha)}(x)\bigg).</math>
===Derivatives of generalized Laguerre polynomials===
Differentiating the power series representation of a generalized Laguerre polynomial <math>k</math> times leads to
:<math>
\frac{\mathrm d^k}{\mathrm d x^k} L_n^{(\alpha)} (x)
= (-1)^k L_{n-k}^{(\alpha+k)} (x)\,;
</math>
moreover, this following equation holds
:<math>\frac{1}{k!} \frac{\mathrm d^k}{\mathrm d x^k} x^\alpha L_n^{(\alpha)} (x)
= {n+\alpha \choose k} x^{\alpha-k} L_n^{(\alpha-k)}(x) \,.</math>
The generalized associated Laguerre polynomials obey the differential equation
:<math>
x L_n^{(\alpha) \prime\prime}(x) + (\alpha+1-x)L_n^{(\alpha)\prime}(x) + n L_n^{(\alpha)}(x)=0.\,
</math>
===Series Expansions===
[[Monomial|Monomials]] are representated as
:<math>\frac{x^n}{n!}= \sum_{i=0}^n (-1)^i {n+ \alpha \choose n-i} L_i^{(\alpha)}(x).</math>
This leads directly to
:<math>e^{-\gamma x}= \sum_{i=0} \frac{\gamma^i}{(1+\gamma)^{i+\alpha+1}} L_i^{(\alpha)}(x)</math>
and, even more generally,
:<math> \frac{x^\beta e^{-\gamma x}}{\Gamma(\beta+1)}= {\alpha+\beta \choose \alpha} \sum_{i=0} \frac{L_i^{(\alpha)}(x)}{ {\alpha+i \choose i}} \sum_{j=0}^i \frac{(-1)^j}{(1+\gamma)^{\alpha+ \beta+ j+ 1}} {\alpha+\beta+j \choose j} {\alpha+i \choose i-j}.</math>
For <math>\beta</math> a non-negative integer this simplifies to
:<math>\frac{x^n e^{-\gamma x}}{n!}= \sum_{i=0} \frac{\gamma^i L_i^{(\alpha)}(x)}{(1+\gamma)^{i+n+\alpha+1}} \sum_{j=0}^n (-1)^{n-j} \gamma^j {n+\alpha \choose j} {i \choose n-j}.</math>
The [[Taylor series]] expansion
:<math>\frac{e^{-\frac{x t}{1-t}}}{(1-t)^{\alpha+1}} = \sum_{i=0} t^i L_i^{(\alpha)}(x)</math>
is an immediate consequence of the definition of Laguerre's polynomials.
Derived from that are the identities
:<math>\frac{e^{-\frac{x t}{1-t}}}{(1-t)^{n+a+1}}L_n^{(\alpha)}\left( \frac{x}{1-t} \right)= \sum_{i=n} {i \choose n} t^{i-n} L_i^{(\alpha)} (x)</math>
and this surprising [[fourier series]] expansion of <math>\frac{1}{x^\beta}</math>,
:<math>\frac{\Gamma(\beta)}{x^\beta}= \sum_{i=0} \frac{L_i^{(\alpha)} (x)}{(\alpha- \beta+ 1) {\alpha+i \choose \alpha- \beta+ 1}} ,</math>
which involves the [[Gamma function]].
==Relation to Hermite polynomials==
The generalized Laguerre polynomials are related to the [[Hermite polynomial]]s:
:<math>H_{2n}(x) = (-1)^n\ 2^{2n}\ n!\ L_n^{(-1/2)} (x^2)</math>
and
:<math>H_{2n+1}(x) = (-1)^n\ 2^{2n+1}\ n!\ x\ L_n^{(1/2)} (x^2)</math>
where the <math>H_n(x)</math> are the [[Hermite polynomial]]s based on the weighting function <math>\exp{(-x^2)}</math>, 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 function]]s, specifically the [[confluent hypergeometric function]]s, as
:<math>L^{(\alpha)}_n(x) = {n+\alpha \choose n} M(-n,\alpha+1,x) =\frac{(\alpha+1)_n} {n!} \,_1F_1(-n,\alpha+1,x)</math>
where <math>(a)_n</math> is the [[Pochhammer symbol]] (which in ''this'' case represents the ''rising factorial'').
==External links==
*[http://www.physics.drexel.edu/~tim/open/hydrofin A quick informal derivation of the Laguerre polynomial in the context of the quantum mechanics of hydrogen]
==References==
* {{Abramowitz_Stegun_ref|22|773}}
* Eric W. Weisstein, "[http://mathworld.wolfram.com/LaguerrePolynomial.html Laguerre Polynomial]", From MathWorld--A Wolfram Web Resource.
* {{cite book | author=George Arfken and Hans Weber| title= Mathematical Methods for Physicists| publisher=Academic Press| year=2000| id = ISBN 0-12-059825-6 }}
[[Category:Polynomials]]
[[Category:Orthogonal polynomials]]
[[Category:Special hypergeometric functions]]
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