Difference between revisions of "Hermite (probabilist)"

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(Created page with "<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem:</strong> The Hermite polynomials $H_n$ satisfy the Rodrigues' formula $$H_n(t) = (-1)...")
 
 
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The (probabilist) Hermite polynomials $\{H_n\}_{n=0}^{\infty}$ (sometimes denoted as $He_n$) are a sequence of [[Orthogonal polynomial|orthogonal polynomials]] with weight function $e^{-\frac{x^2}{2}}$.
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$$\begin{array}{ll}
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H_0(x)=1 \\
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H_1(x)=x \\
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H_2(x)=x^2-1\\
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H_3(x)=x^3-3x\\
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H_4(x)=x^4-6x^2+3 \\
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\vdots
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\end{array}$$
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=Properties=
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
<strong>Theorem:</strong> The Hermite polynomials $H_n$ satisfy the Rodrigues' formula
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<strong>Theorem:</strong> The Hermite polynomials $H_n$ satisfy the [[Rodrigues' formula]]
 
$$H_n(t) = (-1)^ne^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$
 
$$H_n(t) = (-1)^ne^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
<strong>Proof:</strong> proof goes here █  
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<strong>Proof:</strong> █  
 
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</div>
 
</div>
 
</div>
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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<strong>Theorem:</strong> ([[Generating function]]) The Hermite polynomials obey
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$$e^{2tx-t^2} = \displaystyle\sum_{k=0}^{\infty} \dfrac{H_k(x)t^n}{n!}.$$
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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> ([[Orthogonal |Orthogonality]]) The Hermite polynomials obey
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$$\displaystyle\int_{-\infty}^{\infty} e^{-x^2}H_n(x)H_m(x)dx=\left\{ \begin{array}{ll}
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0 &; m \neq n \\
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2^nn!\sqrt{\pi} &; m=n
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\end{array} \right..$$
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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> $H_n(x)$ is an even function for even $n$ and an odd function for odd $n$.
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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}}
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[[Category:SpecialFunction]]

Latest revision as of 18:41, 24 May 2016

The (probabilist) Hermite polynomials $\{H_n\}_{n=0}^{\infty}$ (sometimes denoted as $He_n$) are a sequence of orthogonal polynomials with weight function $e^{-\frac{x^2}{2}}$.

$$\begin{array}{ll} H_0(x)=1 \\ H_1(x)=x \\ H_2(x)=x^2-1\\ H_3(x)=x^3-3x\\ H_4(x)=x^4-6x^2+3 \\ \vdots \end{array}$$

Properties

Theorem: The Hermite polynomials $H_n$ satisfy the Rodrigues' formula $$H_n(t) = (-1)^ne^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$

Proof:

Theorem: (Generating function) The Hermite polynomials obey $$e^{2tx-t^2} = \displaystyle\sum_{k=0}^{\infty} \dfrac{H_k(x)t^n}{n!}.$$

Proof:

Theorem: (Orthogonality) The Hermite polynomials obey $$\displaystyle\int_{-\infty}^{\infty} e^{-x^2}H_n(x)H_m(x)dx=\left\{ \begin{array}{ll} 0 &; m \neq n \\ 2^nn!\sqrt{\pi} &; m=n \end{array} \right..$$

Proof:

Theorem: $H_n(x)$ is an even function for even $n$ and an odd function for odd $n$.

Proof:

Orthogonal polynomials