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In [[mathematics]], the '''Laguerre polynomials''', named after [[Edmond Laguerre]] (1834 – 1886),
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are  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. 


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The '''associated Laguerre polynomials''' (alternatively, but rarely, named '''Sonin polynomials''', after their inventor<ref>Sonine, N. Y.  (1880): "Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries", ''Math Ann.'' '''16''' (1880) 1.</ref>  [[Nikolay Yakovlevich Sonin| N. Y. Sonin]]) are  solutions of
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: <math>
x\,y'' + (\alpha+1 - x)\,y' + n\,y = 0~.</math>


The Laguerre polynomials are also used for [[Gaussian quadrature]] to numerically compute integrals of the form
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: <math>\int_0^\infty f(x) e^{-x} \, dx.</math>
 
 
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These polynomials, usually denoted ''L''<sub>0</sub>,&nbsp;''L''<sub>1</sub>,&nbsp;..., are a [[polynomial sequence]] which may be defined by the [[Rodrigues formula#Rodrigues formula|Rodrigues formula]],
 
:<math>
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L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right) =\frac{( \frac{d}{dx} -1 ) ^n}{n!}  x^n    ,</math>
 
reducing to the closed form of a following section.
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They are [[orthogonal polynomials]] with respect to an [[inner product]]
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: <math>\langle f,g \rangle = \int_0^\infty f(x) g(x) e^{-x}\,dx.</math>
 
 
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The sequence of Laguerre polynomials {{math|''n''! L<sub>''n''</sub>}} is a [[Sheffer sequence]],  {{math|<sup>''d''</sup>&frasl;<sub>''dx''</sub>  L<sub>''n''</sub>  {{=}} (<sup>''d''</sup>&frasl;<sub>''dx''</sub>−1) L<sub>''n''−1</sub>}}.
 
 
</ul>
The [[Rook polynomial]]s in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables.
 
The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution
of the [[Schrödinger equation]] for a one-electron atom. They also describe the static Wigner functions of oscillator systems in [[Phase_space_formulation#Simple_harmonic_oscillator|quantum mechanics in phase space]]. They further enter in the quantum mechanics of the [[Quantum_harmonic_oscillator#Example:_3D_isotropic_harmonic_oscillator|3D isotropic harmonic oscillator]].
 
Physicists sometimes use a definition for the Laguerre polynomials which is larger by a factor of ''n''<nowiki>!</nowiki> than the definition used here. (Likewise, some physicist may use somewhat different definitions of the so-called associated Laguerre polynomials.)
 
== 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.]]
 
== Recursive definition, closed form, and generating function ==
 
One 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 following [[Orthogonal polynomials#Recurrence relations|recurrence relation]] for any ''k''&nbsp;≥&nbsp;1:
: <math>L_{k + 1}(x) = \frac{1}{k + 1} \left( (2k + 1 - x)L_k(x) - k L_{k - 1}(x)\right). </math>
 
The '''closed form''' is
:<math>L_n(x)=\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!}    x^k .</math>
 
The [[generating function]] for them likewise follows,
:<math>\sum_n^\infty  t^n L_n(x)=  \frac{1}{1-t} ~ e^{\frac{-tx}{1-t}} ~ .</math>
 
== Generalized Laguerre polynomials ==
For arbitrary real α the polynomial solutions of the differential equation <ref>A&S p. 781</ref>
:<math>
x\,y'' + (\alpha +1 - x)\,y' + n\,y = 0</math>
are called '''generalized Laguerre polynomials''', or '''associated Laguerre polynomials'''.  
 
 
The simple Laguerre polynomials are included in the associated polynomials, through  ''α'' = 0,
: <math>L^{(0)}_n(x)=L_n(x).</math>
 
 
The [[Rodrigues formula]] for them is
: <math>L_n^{(\alpha)}(x)=
{x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right)      =  x^{-\alpha}  ~\frac{( \frac{d}{dx} -1 ) ^n}{n!} ~ x^{n+\alpha}            .</math>
 
 
The [[generating function]] for them is
:<math>\sum_n^\infty  t^n L^{(\alpha)}_n(x)=  \frac{1}{(1-t)^{\alpha+1}} ~ e^{\frac{-tx}{1-t}} ~ .</math>
 
 
=== Explicit examples and properties of the associated Laguerre polynomials ===
 
* Laguerre functions are defined by [[confluent hypergeometric function]]s and Kummer's transformation as<ref>A&S p.509</ref>
:<math> L_n^{(\alpha)}(x) := {n+ \alpha \choose n} M(-n,\alpha+1,x).</math>
: When ''n'' is an integer the function reduces to a polynomial of degree&nbsp;''n''. It has the alternative expression<ref>A&S p.510</ref>
:: <math>L_n^{(\alpha)}(x)= \frac {(-1)^n}{n!} U(-n,\alpha+1,x)</math>
: in terms of [[confluent hypergeometric function|Kummer's function of the second kind]].
 
* The closed form for these associated Laguerre polynomials of degree ''n'' is<ref>A&S p. 775</ref>
:: <math> L_n^{(\alpha)} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!} </math>
:derived by applying [[Leibniz rule (generalized product rule)|Leibniz's theorem for differentiation of a product]] to Rodrigues' formula.
 
* The first few generalized Laguerre polynomials are:
:: <math>
\begin{align}
L_0^{(\alpha)} (x) & = 1 \\
L_1^{(\alpha)}(x) & = -x + \alpha +1 \\
L_2^{(\alpha)}(x) & = \frac{x^2}{2} - (\alpha + 2)x + \frac{(\alpha+2)(\alpha+1)}{2} \\
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}
\end{align}
</math>
 
* The [[coefficient]] of the leading term is (&minus;1)<sup>''n''</sup>/''n''<nowiki>!</nowiki>;
* The [[constant term]], which is the value at&nbsp;0, is
:: <math>L_n^{(\alpha)}(0)= {n+\alpha\choose n} \approx \frac{n^\alpha}{\Gamma(\alpha+1)};</math>
 
* ''L''<sub>''n''</sup><sup>(''α'')</sup> has ''n'' [[real number|real]], strictly positive [[Root of a function|roots]] (notice that <math>\left((-1)^{n-i} L_{n-i}^{(\alpha)}\right)_{i=0}^n</math> is a [[Sturm chain]]), which are all in the [[Interval (mathematics)|interval]] <math>\left( 0, n+\alpha+ (n-1) \sqrt{n+\alpha} \right].</math>{{citation needed|date=September 2011}}
 
* The polynomials' asymptotic behaviour for large ''n'', but fixed ''&alpha;'' and ''x''&nbsp;>&nbsp;0, is given by<ref>G. Szegő, "Orthogonal polynomials", 4th edition, ''Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI'', 1975, p. 198.</ref><ref>D. Borwein, J. M. Borwein, R. E. Crandall, "Effective Laguerre asymptotics", ''SIAM J. Numer. Anal.'', vol. 46 (2008), no. 6, pp. 3285-3312, http://dx.doi.org/10.1137/07068031X</ref>
:: <math>L_n^{(\alpha)}(x) = \frac{n^{\frac{\alpha}{2}-\frac{1}{4}}}{\sqrt{\pi}} \frac{e^{\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} \cos\left(2 \sqrt{nx}- \frac{\pi}{2}\left(\alpha+\frac{1}{2} \right) \right)+O\left(n^{\frac{\alpha}{2}-\frac{3}{4}}\right),</math>
:: <math>L_n^{(\alpha)}(-x) = \frac{(n+1)^{\frac{\alpha}{2}-\frac{1}{4}}}{2\sqrt{\pi}} \frac{e^{-\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} e^{2 \sqrt{x(n+1)}} \cdot\left(1+O\left(\frac{1}{\sqrt{n+1}}\right)\right),</math>
: and summarizing by
:: <math>\frac{L_n^{(\alpha)}\left(\frac x n\right)}{n^\alpha}\approx e^\frac x {2n}\cdot\frac{J_\alpha\left(2\sqrt x\right)}{\sqrt x^\alpha},</math>
:where <math>J_\alpha</math> is the [[Bessel function#Asymptotic forms|Bessel function]].
 
:Moreover{{citation needed|date=September 2011}}
::<math>L_n^{(\alpha-n)}(x)\approx e^x\cdot {\alpha\choose n}</math>,
:whenever ''n'' tends to infinity.
 
=== As a contour integral ===
 
Given the generating function specified above, the polynomials may be expressed in terms of a [[contour integral]]
 
: <math>L_n^{(\alpha)}(x)=\frac{1}{2\pi i}\oint\frac{e^{-\frac{x t}{1-t}}}{(1-t)^{\alpha+1}\,t^{n+1}} \; dt ~,</math>
where the contour circles the origin once in a counterclockwise direction.
 
=== Recurrence relations ===
The addition formula for Laguerre polynomials:<ref>A&S equation (22.12.6), p. 785</ref>
 
: <math>L_n^{(\alpha+\beta+1)}(x+y)= \sum_{i=0}^n L_i^{(\alpha)}(x) L_{n-i}^{(\beta)}(y) </math>.
 
Laguerre's polynomials satisfy the recurrence relations
:<math>L_n^{(\alpha)}(x)= \sum_{i=0}^n L_{n-i}^{(\alpha+i)}(y)\frac{(y-x)^i}{i!},</math>
 
in particular
 
: <math>L_n^{(\alpha+1)}(x)= \sum_{i=0}^n L_i^{(\alpha)}(x)</math>
 
and
 
: <math>L_n^{(\alpha)}(x)= \sum_{i=0}^n {\alpha-\beta+n-i-1 \choose n-i} L_i^{(\beta)}(x),</math>
 
or
 
: <math>L_n^{(\alpha)}(x)=\sum_{i=0}^n {\alpha-\beta+n \choose n-i} L_i^{(\beta- i)}(x);</math>
 
moreover
 
: <math>\begin{align}L_n^{(\alpha)}(x)- \sum_{j=0}^{\Delta-1} {n+\alpha \choose n-j} (-1)^j \frac{x^j}{j!}&= (-1)^\Delta\frac{x^\Delta}{(\Delta-1)!} \sum_{i=0}^{n-\Delta} \frac{{n+\alpha \choose n-\Delta-i}}{(n-i){n \choose i}}L_i^{(\alpha+\Delta)}(x)\\[6pt]
 
&=(-1)^\Delta\frac{x^\Delta}{(\Delta-1)!} \sum_{i=0}^{n-\Delta} \frac{{n+\alpha-i-1 \choose n-\Delta-i}}{(n-i){n \choose i}}L_i^{(n+\alpha+\Delta-i)}(x).\end{align}</math>
 
They can be used to derive the four 3-point-rules
 
: <math>
 
\begin{align}
L_n^{(\alpha)}(x) & = L_n^{(\alpha+1)}(x) - L_{n-1}^{(\alpha+1)}(x) = \sum_{j=0}^k {k \choose j} L_{n-j}^{(\alpha-k+j)}(x), \\[10pt]
n L_n^{(\alpha)}(x) & = (n + \alpha )L_{n-1}^{(\alpha)}(x) - x L_{n-1}^{(\alpha+1)}(x), \\[10pt]
& \text{or } \frac{x^k}{k!}L_n^{(\alpha)}(x) = \sum_{i=0}^k (-1)^i {n+i \choose i} {n+\alpha \choose k-i} L_{n+i}^{(\alpha-k)}(x), \\[10pt]
n L_n^{(\alpha+1)}(x) & =(n-x) L_{n-1}^{(\alpha+1)}(x) + (n+\alpha)L_{n-1}^{(\alpha)}(x) \\[10pt]
x L_n^{(\alpha+1)}(x) & = (n+\alpha)L_{n-1}^{(\alpha)}(x)-(n-x)L_n^{(\alpha)}(x);
\end{align}
</math>
 
combined they give this additional, useful recurrence relations
 
: <math>\begin{align}L_n^{(\alpha)}(x)&= \left(2+\frac{\alpha-1-x}n \right) L_{n-1}^{(\alpha)}(x)- \left(1+\frac{\alpha-1}n \right) L_{n-2}^{(\alpha)}(x)\\[10pt]
 
&= \frac{\alpha+1-x}n  L_{n-1}^{(\alpha+1)}(x)- \frac x n L_{n-2}^{(\alpha+2)}(x). \end{align}</math>
 
A somewhat curious identity, valid for integer ''i'' and&nbsp;''n'', is
 
: <math> \frac{(-x)^i}{i!} L_n^{(i-n)}(x) = \frac{(-x)^n}{n!} L_i^{(n-i)}(x);</math>
 
it may be used to derive the [[partial fraction decomposition]]
 
: <math>\frac{L_n^{(\alpha)}(x)}{{n+ \alpha \choose n}}= 1- \sum_{j=1}^n \frac{x^j}{\alpha + j} \frac{L_{n-j}^{(j)}(x)}{(j-1)!}=
 
1- \sum_{j=1}^n (-1)^j \frac{j}{\alpha + j} {n \choose j}L_n^{(-j)}(x)
 
= 1-x \sum_{i=1}^n \frac{L_{n-i}^{(-\alpha)}(x)  L_{i-1}^{(\alpha+1)}(-x)}{\alpha +i}.</math>
 
=== Derivatives of generalized Laguerre polynomials ===
 
Differentiating the power series representation of a generalized Laguerre polynomial ''k'' times leads to
 
: <math>
\frac{d^k}{d x^k} L_n^{(\alpha)} (x)
= (-1)^k L_{n-k}^{(\alpha+k)} (x)\,.
</math>
 
This points to a special case (&alpha;&nbsp;=&nbsp;0) of the formula above:
for integer &alpha;&nbsp;=&nbsp;''k'' the generalized polynomial may be written
<math>L_n^{(k)}(x)=(-1)^k\frac{d^kL_{n+k}(x)}{dx^k}\,</math>, the shift by k sometimes causing confusion with the usual parenthesis notation for a
derivative.
 
Moreover, this following equation holds
 
: <math>\frac{1}{k!} \frac{d^k}{d x^k} x^\alpha L_n^{(\alpha)} (x)
 
= {n+\alpha \choose k} x^{\alpha-k} L_n^{(\alpha-k)}(x),</math>
 
which generalizes with [[Antiderivative#Techniques of integration|Cauchy's formula]] to
 
: <math>L_n^{(\alpha')}(x) = (\alpha'-\alpha) {\alpha'+ n \choose \alpha'-\alpha} \int_0^x \frac{t^\alpha (x-t)^{\alpha'-\alpha-1}}{x^{\alpha'}} L_n^{(\alpha)}(t)\,dt.</math>
 
 
 
The derivative with respect to the second variable &alpha; has the  form, <ref>W. Koepf, "[http://www.tandfonline.com/doi/abs/10.1080/10652469708819127 Identities for families of orthogonal polynomials and special functions.]", ''Integral Transforms and Special Functions 5'', (1997) pp.69-102. (Theorem 10)</ref>
: <math>\frac{d}{d \alpha}L_n^{(\alpha)}(x)= \sum_{i=0}^{n-1} \frac{L_i^{(\alpha)}(x)}{n-i}.</math>
This is evident from the contour integral representation below.
 
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>
which may be compared with the equation obeyed by the ''k''th derivative of the ordinary Laguerre polynomial,
 
: <math>
x L_n^{(k) \prime\prime}(x) + (k+1-x)L_n^{(k)\prime}(x) + (n-k) L_n^{(k)}(x)=0,\,
</math>
 
where <math>L_n^{(k)}(x)\equiv\frac{d^kL_n(x)}{dx^k}</math> for this equation only.
 
 
In [[Sturm–Liouville theory|Sturm–Liouville form]] the differential equation is
 
: <math>-\left(x^{\alpha+1} e^{-x}\cdot L_n^{(\alpha)}(x)^\prime\right)^\prime= n\cdot x^\alpha e^{-x}\cdot L_n^{(\alpha)}(x),</math>
 
which shows that ''L''{{su|b=''n''|p=&alpha;}} is an eigenvector for the eigenvalue ''n''.
 
=== Orthogonality ===
 
The associated Laguerre polynomials are orthogonal over <nowiki>[</nowiki>0,&nbsp;∞<nowiki>)</nowiki> with respect to the measure with weighting function ''x''<sup>''α''</sup>&nbsp;''e''<sup>&nbsp;&minus;''x''</sup>:<ref>A&S p. 774</ref>
 
:<math>\int_0^\infty x^\alpha e^{-x} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)dx=\frac{\Gamma(n+\alpha+1)}{n!} \delta_{n,m},</math>
 
which follows from
 
:<math>\int_0^\infty x^{\alpha'-1} e^{-x} L_n^{(\alpha)}(x)dx= {\alpha-\alpha'+n \choose n} \Gamma(\alpha').</math>
 
If <math>\Gamma(x,\alpha+1,1)</math> denoted the Gamma distribution then the orthogonality relation can be written as
 
:<math>\int_0^{\infty} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)\Gamma(x,\alpha+1,1) dx={n+ \alpha \choose n}\delta_{n,m},</math>
 
The associated, symmetric kernel polynomial has the representations ([[Christoffel–Darboux formula]]){{citation needed|date=October 2011}}<!--All of these formulas require citations.-->
 
: <math>\begin{align}
 
K_n^{(\alpha)}(x,y)&{:=}\frac{1}{\Gamma(\alpha+1)} \sum_{i=0}^n \frac{L_i^{(\alpha)}(x) L_i^{(\alpha)}(y)}{{\alpha+i \choose i}}\\
 
&{=}\frac{1}{\Gamma(\alpha+1)} \frac{L_n^{(\alpha)}(x) L_{n+1}^{(\alpha)}(y) - L_{n+1}^{(\alpha)}(x) L_n^{(\alpha)}(y)}{\frac{x-y}{n+1} {n+\alpha \choose n}} \\
 
&{=}\frac{1}{\Gamma(\alpha+1)}\sum_{i=0}^n \frac{x^i}{i!} \frac{L_{n-i}^{(\alpha+i)}(x) L_{n-i}^{(\alpha+i+1)}(y)}{{\alpha+n \choose n}{n \choose i}};\end{align}</math>
 
recursively
 
: <math>K_n^{(\alpha)}(x,y)=\frac{y}{\alpha+1} K_{n-1}^{(\alpha+1)}(x,y)+ \frac{1}{\Gamma(\alpha+1)} \frac{L_n^{(\alpha+1)}(x) L_n^{(\alpha)}(y)}{{\alpha+n \choose n}}.</math>
 
Moreover,
 
: <math>y^\alpha e^{-y} K_n^{(\alpha)}(\cdot, y) \rightarrow \delta(y- \, \cdot),</math>
 
in the associated ''L''<sup>2</sup><nowiki>[</nowiki>0,&nbsp;∞<nowiki>)</nowiki>-space.
 
[[Turán's inequalities]] can be derived here, which is
 
: <math>L_n^{(\alpha)}(x)^2- L_{n-1}^{(\alpha)}(x) L_{n+1}^{(\alpha)}(x)= \sum_{k=0}^{n-1} \frac{{\alpha+n-1\choose n-k}}{n{n\choose k}} L_k^{(\alpha-1)}(x)^2>0.</math>
 
The following integral is needed in the quantum mechanical treatment of the hydrogen atom,
 
: <math>\int_0^{\infty}x^{\alpha+1} e^{-x} \left[L_n^{(\alpha)} (x)\right]^2 dx=
 
\frac{(n+\alpha)!}{n!}(2n+\alpha+1).</math>
 
=== Series expansions ===
 
Let a function have the (formal) series expansion
 
: <math>f(x)= \sum_{i=0}^\infty f_i^{(\alpha)} L_i^{(\alpha)}(x).</math>
 
Then
 
: <math>f_i^{(\alpha)}=\int_0^\infty \frac{L_i^{(\alpha)}(x)}{{i+ \alpha \choose i}} \cdot \frac{x^\alpha e^{-x}}{\Gamma(\alpha+1)} \cdot f(x) \,dx .</math>
 
The series converges in the associated [[Hilbert space]] [[Lp space|<math>L^2[0,\infty)</math>]], [[if and only if|iff]]
 
: <math>\| f \|_{L^2}^2 := \int_0^\infty \frac{x^\alpha e^{-x}}{\Gamma(\alpha+1)} | f(x)|^2 dx = \sum_{i=0}^\infty {i+\alpha \choose i} |f_i^{(\alpha)}|^2 < \infty. </math>
 
==== Further examples of expansions====
[[Monomial]]s are represented as
 
: <math>\frac{x^n}{n!}= \sum_{i=0}^n (-1)^i {n+ \alpha \choose n-i} L_i^{(\alpha)}(x),</math>
 
while [[binomial coefficient|binomials]] have the parametrization
 
: <math>{n+x \choose n}= \sum_{i=0}^n \frac{\alpha^i}{i!} L_{n-i}^{(x+i)}(\alpha).</math>
 
This leads directly to
 
: <math>e^{-\gamma x}= \sum_{i=0}^\infty \frac{\gamma^i}{(1+\gamma)^{i+\alpha+1}} L_i^{(\alpha)}(x) \qquad \left(\text{convergent iff }\operatorname{Re}{(\gamma)} > -\frac{1}{2}\right)</math>
 
for the exponential function. The [[incomplete gamma function]] has the representation
 
: <math>\Gamma(\alpha,x)=x^\alpha e^{-x} \sum_{i=0}^\infty \frac{L_i^{(\alpha)}(x)}{1+i} \qquad \left(\Re(\alpha)>-1 , x > 0\right).</math>
 
==Multiplication theorems==
[[Arthur Erdélyi|Erdélyi]] gives the following two [[multiplication theorem]]s <ref>C. Truesdell, "[http://www.pnas.org/cgi/reprint/36/12/752.pdf On the Addition and Multiplication Theorems for the Special Functions]", ''Proceedings of the National Academy of Sciences, Mathematics'',  (1950) pp.752-757.</ref>
 
* <math>t^{n+1+\alpha} e^{(1-t) z} L_n^{(\alpha)}(z t)=\sum_{k=n} {k \choose n}\left(1-\frac 1 t\right)^{k-n} L_k^{(\alpha)}(z),</math>
* <math>e^{(1-t)z} L_n^{(\alpha)}(z t)=\sum_{k=0} \frac{(1-t)^k z^k}{k!}L_n^{(\alpha+k)}(z).</math>
 
== 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 ''H''<sub>''n''</sub>(''x'') are the [[Hermite polynomial]]s based on the weighting function exp(&minus;''x''<sup>2</sup>), 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).
 
=== Poisson Kernel ===
 
: <math>
 
\sum_{n=0}^{\infty}\frac{n!L_{n}^{(\alpha)}(x)L_{n}^{(\alpha)}(y)r^{n}}{\Gamma\left(1+\alpha+n\right)}=\frac{\exp\left(-\frac{\left(x+y\right)r}{1-r}\right)I_{\alpha}\left(\frac{2\sqrt{xyr}}{1-r}\right)}{\left(xyr\right)^{\frac{\alpha}{2}}\left(1-r\right)},\quad,\alpha>-1,\left|r\right|<1.
 
</math>
 
 
== Notes ==
 
<references/>
 
== References ==
 
* {{Abramowitz_Stegun_ref|22|773}}
* {{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
 
* B. Spain, M.G. Smith, ''Functions of mathematical physics'', Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials.
*{{springer|title=Laguerre polynomials|id=p/l057310}}
* [[Eric W. Weisstein]], "[http://mathworld.wolfram.com/LaguerrePolynomial.html Laguerre Polynomial]", From MathWorld&mdash;A Wolfram Web Resource.
* {{cite book | author=[[George Arfken]] and Hans Weber| title= Mathematical Methods for Physicists| publisher=Academic Press| year=2000| isbn = 0-12-059825-6 }}
 
* S. S. Bayin (2006), ''Mathematical Methods in Science and Engineering'', Wiley, Chapter 3.
 
== External links ==
* {{cite web|author=Timothy Jones|url=http://www.physics.drexel.edu/~tim/open/hydrofin | title=The Legendre and Laguerre Polynomials and the elementary quantum mechanical model of the Hydrogen Atom}}
* {{MathWorld|title=Laguerre polynomial|id=LaguerrePolynomial}}
 
[[Category:Polynomials]]
[[Category:Orthogonal polynomials]]
[[Category:Special hypergeometric functions]]

Latest revision as of 13:40, 13 September 2014

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