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 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[edit]

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