Difference between revisions of "Hermite (probabilist)"
| Line 1: | Line 1: | ||
| + | The Hermite polynomials $\{H_n\}_{n=0}^{\infty}$ are a sequence of [[Orthogonal polynomial|orthogonal polynomials]] that satisfy the differential equation | ||
| + | $$\left\{ \begin{array}{ll} | ||
| + | |||
| + | \end{array} \right.$$ | ||
| + | |||
| + | =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]] | ||
Revision as of 03:49, 14 September 2014
The Hermite polynomials $\{H_n\}_{n=0}^{\infty}$ are a sequence of orthogonal polynomials that satisfy the differential equation $$\left\{ \begin{array}{ll}
\end{array} \right.$$
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: █
