M1: Hermite Polynomials
- Page ID
- 13498
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
The "physicists' Hermite polynomials" are given by
\[H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}=\left (2x-\frac{d}{dx} \right )^n \cdot 1 .\]
These are Hermite polynomial sequences of different variances; see the material on variances below. the first eleven physicists' Hermite polynomials are:
\(H_0(x)=1\,\)
\(H_1(x)=2x\,\)
\(H_2(x)=4x^2-2\,\)
\(H_3(x)=8x^3-12x\,\)
\(H_4(x)=16x^4-48x^2+12\,\)
\(H_5(x)=32x^5-160x^3+120x\,\)
\(H_6(x)=64x^6-480x^4+720x^2-120\,\)
\(H_7(x)=128x^7-1344x^5+3360x^3-1680x\,\)
\(H_8(x)=256x^8-3584x^6+13440x^4-13440x^2+1680\,\)
\(H_9(x)=512x^9-9216x^7+48384x^5-80640x^3+30240x\,\)
\(H_{10}(x)=1024x^{10}-23040x^8+161280x^6-403200x^4+302400x^2-30240\,\)