# 埃尔米特多项式

## 定义

${\displaystyle H_{n}^{\mathrm {prob} }(x)=(-1)^{n}e^{x^{2}/2}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}/2}\,\!}$

${\displaystyle H_{n}^{\mathrm {phys} }(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}\,\!}$

${\displaystyle H_{n}^{\mathrm {phys} }(x)=2^{n/2}H_{n}^{\mathrm {prob} }({\sqrt {2}}\,x).\,\!}$

${\displaystyle H_{0}(x)}$  ${\displaystyle 1\,}$  ${\displaystyle 1\,}$
${\displaystyle H_{1}(x)}$  ${\displaystyle x\,}$  ${\displaystyle 2x\,}$
${\displaystyle H_{2}(x)}$  ${\displaystyle x^{2}-1\,}$  ${\displaystyle 4x^{2}-2\,}$
${\displaystyle H_{3}(x)}$  ${\displaystyle x^{3}-3x\,}$  ${\displaystyle 8x^{3}-12x\,}$
${\displaystyle H_{4}(x)}$  ${\displaystyle x^{4}-6x^{2}+3\,}$  ${\displaystyle 16x^{4}-48x^{2}+12\,}$
${\displaystyle H_{5}(x)}$  ${\displaystyle x^{5}-10x^{3}+15x\,}$  ${\displaystyle 32x^{5}-160x^{3}+120x\,}$

## 性质

### 正交性

${\displaystyle w(x)=\mathrm {e} ^{-x^{2}/2}\,\!}$    （概率论）
${\displaystyle w(x)=\mathrm {e} ^{-x^{2}}\,\!}$    （物理学）

${\displaystyle \int _{-\infty }^{\infty }H_{m}(x)H_{n}(x)\,w(x)\,\mathrm {d} x=0}$

${\displaystyle \int _{-\infty }^{\infty }H_{m}^{\mathrm {prob} }(x)H_{n}^{\mathrm {prob} }(x)\,\mathrm {e} ^{-x^{2}/2}\,\mathrm {d} x=n!\,{\sqrt {2\pi }}\delta _{mn}}$    （概率论）
${\displaystyle \int _{-\infty }^{\infty }H_{m}^{\mathrm {phys} }(x)H_{n}^{\mathrm {phys} }(x)\,\mathrm {e} ^{-x^{2}}\,\mathrm {d} x=n!\,2^{n}{\sqrt {\pi }}\delta _{mn}}$    （物理学）

### 完备性

${\displaystyle \int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,w(x)\,\mathrm {d} x<\infty }$

${\displaystyle \langle f,g\rangle =\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,w(x)\,\mathrm {d} x}$

### 埃尔米特微分方程

${\displaystyle (e^{-x^{2}/2}u')'+\lambda e^{-x^{2}/2}u=0}$

${\displaystyle u''-2xu'+2\lambda u=0}$

## 參考文獻

• Arfken, Mathematical Methods for Physicists
• B Spain, M G Smith, Functions of mathematical physics, Van Nostrand Reinhold Company, London, 1970. Chapter 11 deals with Hermite polynomials.
• Bayin, S.S. (2006) Mathematical Methods in Science and Engineering, Wiley, Chapter 4.
• Courant, Richard; Hilbert, David, Methods of Mathematical Physics, Volume I, Wiley-Interscience, 1953.
• Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G., Higher transcendental functions. Vol. II, McGraw-Hill, 1955
• Fedoryuk, M.V., H/h046980, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4.
• Szegő, Gábor, Orthogonal Polynomials, American Mathematical Society, 1939, 1955
• Wiener, Norbert, The Fourier Integral and Certain of its Applications, New York: Dover Publications, 1958, ISBN 0-486-60272-9
• Whittaker, E. T.; Watson, G. N. A Course of Modern Analysis 4th Edition. London: Cambridge University Press. 1962.
• Temme, Nico, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, New York, 1996