# 埃尔米特多项式

## 定义

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

$(2)\ \ H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}\,\!$

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

$H_0(x)$ $1\,$ $1\,$
$H_1(x)$ $x\,$ $2x\,$
$H_2(x)$ $x^2-1\,$ $4x^2-2\,$
$H_3(x)$ $x^3-3x\,$ $8x^3-12x\,$
$H_4(x)$ $x^4-6x^2+3\,$ $16x^4-48x^2+12\,$
$H_5(x)$ $x^5-10x^3+15x\,$ $32x^5-160x^3+120x\,$

## 性质

### 正交性

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

$\int_{-\infty}^\infty H_m(x) H_n(x)\, w(x) \, \mathrm{d}x = 0$

$\int_{-\infty}^\infty H_m^\mathrm{prob}(x) H_n^\mathrm{prob}(x)\, \mathrm{e}^{-x^2/2} \, \mathrm{d}x = n! \, \sqrt{2 \pi}\delta_{mn}$   （概率论）
$\int_{-\infty}^\infty H_m^\mathrm{phys}(x) H_n^\mathrm{phys}(x)\, \mathrm{e}^{-x^2}\, \mathrm{d}x = n! \, 2^n \sqrt{\pi}\delta_{mn}$   （物理学）

### 完备性

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

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

### 埃尔米特微分方程

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

$u'' - 2xu'+2\lambda u=0$

## 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., 埃尔米特多项式//Hazewinkel, Michiel, 数学百科全书, 克鲁维尔学术出版社. 2001, ISBN 978-1556080104 .
• 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