Difference between revisions of "Hermite (probabilist)"
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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 $e^{-\frac{x^2}{2}}$. | ||
| + | |||
| + | $$\begin{array}{ll} | ||
| + | H_0(x)=1 \\ | ||
| + | H_1(x)=x \\ | ||
| + | H_2(x)=x^2-1\\ | ||
| + | H_3(x)=x^3-3x\\ | ||
| + | H_4(x)=x^4-6x^2+3 \\ | ||
| + | \vdots | ||
| + | \end{array}$$ | ||
| + | |||
| + | =Properties= | ||
<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]] | <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> | + | <strong>Proof:</strong> █ |
</div> | </div> | ||
</div> | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> ([[Generating function]]) The Hermite polynomials obey | ||
| + | $$e^{2tx-t^2} = \displaystyle\sum_{k=0}^{\infty} \dfrac{H_k(x)t^n}{n!}.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> ([[Orthogonal |Orthogonality]]) The Hermite polynomials obey | ||
| + | $$\displaystyle\int_{-\infty}^{\infty} e^{-x^2}H_n(x)H_m(x)dx=\left\{ \begin{array}{ll} | ||
| + | 0 &; m \neq n \\ | ||
| + | 2^nn!\sqrt{\pi} &; m=n | ||
| + | \end{array} \right..$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> $H_n(x)$ is an even function for even $n$ and an odd function for odd $n$. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | {{:Orthogonal polynomials footer}} | ||
| + | |||
| + | [[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: █
