Difference between revisions of "Laguerre L"

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Laguerre's equation is
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$$x\dfrac{d^2y}{dx^2}+(1-x)\dfrac{dy}{dx}+ny=0.$$
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The Laguerre polynomial of order $n$, $L_n$, is given by
One of the solutions of this differential equations are the Laguerre polynomials
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$$L_n(x) = \displaystyle\sum_{k=0}^n \dfrac{(-1)^kn!}{(n-k)!(k!)^2}x^k.$$
$$L_n(x) = \displaystyle\sum_{k=0}^n (-1)^k \dfrac{n!}{(n-k)!(k!)^2}x^k.$$
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The first few Laguerre polynomials are given by
 
The first few Laguerre polynomials are given by
 
$$\begin{array}{ll}
 
$$\begin{array}{ll}
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<gallery>
 
<gallery>
File:Laguerrepolynomial.png|Graph of $\mathrm{arccos}$ on $[-2,10]$.
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File:Laguerrepolynomial.png|Graph of $L_n$ for $n=0,1,2,\ldots,5$ on $[-3,10]$.
 
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=Properties=
 
=Properties=
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[[Generating function for Laguerre L]]<br />
<strong>Theorem:</strong> The following formula holds:
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[[L n(x)=(e^x/n!)d^n/dx^n(x^n e^(-x))]]<br />
$$\dfrac{e^{\frac{-xt}{1-t}}}{1-t} = \displaystyle\sum_{k=0}^{\infty} L_k(x)t^k.$$
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[[L n(0)=1]]<br />
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[[L n'(0)=-n]]<br />
<strong>Proof:</strong>
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[[Orthogonality of Laguerre L]]<br />
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[[(n+1)L (n+1)(x) = (2n+1-x)L n(x)-nL (n-1)(x)]]<br />
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[[xL n'(x)=nL n(x)-n L (n-1)(x)]]<br />
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[[L n'(x)=-Sum L k(x)]]<br />
  
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=See also=
<strong>Theorem:</strong> The following formula holds:
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[[Associated Laguerre L]]
$$L_n(x) = \dfrac{e^x}{n!} \dfrac{d^n}{dx^n} (x^n e^{-x}).$$
 
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<strong>Proof:</strong> █
 
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=References=
<strong>Theorem (Orthogonality):</strong> The following formula holds:
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* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=findme|next=Generating function for Laguerre L}}: $(6.3)$
$$\displaystyle\int_0^{\infty} e^{-x}L_n(x)L_m(x)dx = \delta_{nm},$$
 
where $\delta_{mn}=0$ when $m\neq n$ and $\delta_{mn}=1$ when $m=n$.
 
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<strong>Proof:</strong> █
 
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{{:Orthogonal polynomials footer}}
<strong>Theorem:</strong> The following formula holds:
 
$$(n+1)L_{n+1}(x)=(2n+1-x)L_n(x)-nL_{n-1}(x).$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The following formula holds:
 
$$xL_n'(x)=nL_n(x)-nL_{n-1}(x).$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The following formula holds:
 
$$L_n'(x)=-\displaystyle\sum_{k=0}^{n-1} L_k(x).$$
 
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<strong>Proof:</strong> █
 
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{{:Orthogonal polynomials footer}}
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[[Category:SpecialFunction]]

Latest revision as of 14:37, 15 March 2018

The Laguerre polynomial of order $n$, $L_n$, is given by $$L_n(x) = \displaystyle\sum_{k=0}^n \dfrac{(-1)^kn!}{(n-k)!(k!)^2}x^k.$$

The first few Laguerre polynomials are given by $$\begin{array}{ll} L_0(x) &= 1 \\ L_1(x) &= -x+1 \\ L_2(x) &= \dfrac{1}{2}(x^2-4x+2) \\ L_3(x) &= \dfrac{1}{6}(-x^3+9x^2-18x+6) \\ L_4(x) &= \dfrac{1}{24}(x^4-16x^3+72x^2-96x+24)\\ \vdots \end{array}$$

Properties

Generating function for Laguerre L
L n(x)=(e^x/n!)d^n/dx^n(x^n e^(-x))
L n(0)=1
L n'(0)=-n
Orthogonality of Laguerre L
(n+1)L (n+1)(x) = (2n+1-x)L n(x)-nL (n-1)(x)
xL n'(x)=nL n(x)-n L (n-1)(x)
L n'(x)=-Sum L k(x)

See also

Associated Laguerre L

References

Orthogonal polynomials