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| | Let $\lambda \in \mathbb{R}$. The associated Laguerre polynomials, $L_n^{(\lambda)}(x)$ are solutions of the differential equation | | Let $\lambda \in \mathbb{R}$. The associated Laguerre polynomials, $L_n^{(\lambda)}(x)$ are solutions of the differential equation |
| | $$x\dfrac{d^2y}{dx^2} + (\lambda+1-x)\dfrac{dy}{dx} + ny=0.$$ | | $$x\dfrac{d^2y}{dx^2} + (\lambda+1-x)\dfrac{dy}{dx} + ny=0.$$ |
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| | =Properties= | | =Properties= |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$L_n^{(\lambda)}(x) = \displaystyle\sum_{k=0}^n (-1)^k {n+\lambda \choose n-k} \dfrac{x^k}{k!}.$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | =See also= |
| − | <strong>Theorem:</strong> The following formula holds:
| + | [[Laguerre L]]<br /> |
| − | $$L_n^{(\lambda+\beta+1)}(x+y) = \displaystyle\sum_{k=0}^n L_k^{(\lambda)}(x)L_{n-k}^{(\beta)}(x).$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div> | |
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| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | =References= |
| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$L_n^{(\lambda)}(x) = L_n^{(\lambda+1)}(x)-L_{n-1}^{(\lambda+1)}(x).$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | {{:Orthogonal polynomials footer}} |
| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$nL_n^{(\lambda+1)}(x) = (n-x)L_{n-1}^{(\lambda+1)}(x)-(n+\lambda)L_{n-1}^{(\lambda)}(x).$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$xL_n^{(\lambda+1)}(x)= (n+\lambda)L_{n-1}^{(\lambda)}(x)-(n-x)L_n^{(\lambda)}(x).$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$\dfrac{d^k}{dx^k} L_n^{(\lambda)}(x) = (-1)^kL_{n-k}^{(\lambda+k)}(x).$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | {{:Orthogonal polynomials footer}}
| + | [[Category:SpecialFunction]] |
Let $\lambda \in \mathbb{R}$. The associated Laguerre polynomials, $L_n^{(\lambda)}(x)$ are solutions of the differential equation
$$x\dfrac{d^2y}{dx^2} + (\lambda+1-x)\dfrac{dy}{dx} + ny=0.$$
The first few Laguerre polynomials are given by
$$\begin{array}{ll}
L_0^{(\lambda)}(x) &= 1 \\
L_1^{(\lambda)}(x) &= -x+\lambda+1 \\
L_2^{(\lambda)}(x) &= \dfrac{x^2}{2} -(\lambda+2)x+\dfrac{(\lambda+2)(\lambda+1)}{2} \\
L_3^{(\lambda)}(x) &= -\dfrac{x^3}{6} + \dfrac{(\lambda+3)x^2}{2} - \dfrac{(\lambda+2)(\lambda+3)x}{2} + \dfrac{(\lambda+1)(\lambda+2)(\lambda+3)}{6} \\
\vdots
\end{array}$$
