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Hermite polynomials
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(Definition)
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The polynomial solutions of the Hermite differential equation, with $n$ a non-negative integer, are usually normed so that the highest degree term is $(2z)^n$ and called the Hermite polynomials $H_n(z)$ . The Hermite polynomials may be defined explicitly by
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(1) |
since this is a polynomial having the highest degree term $(2z)^n$ and satisfying the Hermite equation. The equation (1) is the Rodrigues's formula for Hermite polynomials. Using the Faà di Bruno's formula, one gets from (1) also $$H_n(x) \;=\; (-1)^n\!\sum_{m_1+2m_2=n}\frac{n!}{m_1!m_2!}(-1)^{m_1+m_2}(2x)^{m_1}.$$
The first six Hermite polynomials are
$H_0(z) \;\equiv\; 1,$
$H_1(z) \;\equiv\; 2z,$
$H_2(z) \;\equiv\; 4z^2\!-\!2,$
$H_3(z) \;\equiv\; 8z^3\!-\!12z,$
$H_4(z) \;\equiv\; 16z^4\!-\!48z^2\!+\!12,$
$H_5(z) \;\equiv\; 32z^5\!-\!160z^3\!+\!120z,$
and the general polynomial form is $$H_n(z) \;\equiv\; (2z)^n-\frac{n(n\!-\!1)}{1!}(2z)^{n-2} +\frac{n(n\!-\!1)(n\!-\!2)(n\!-\!3)}{2!}(2z)^{n-4}-+\ldots$$
Differentiating this termwise gives $$H'_n(z) \;=\; 2n\!\left[(2z)^{n-1}-\frac{(n\!-\!1)(n\!-\!2)}{1!}(2z)^{n-3}+ \frac{(n\!-\!1)(n\!-\!2)(n\!-\!3)(n\!-\!4)}{2!}(2z)^{n-5}-+\ldots\right]\!,$$ i.e.
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(2) |
The Hermite polynomials are sometimes scaled to such ones $\mathrm{He_n}$ which obey the differentiation rule
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(3) |
Such Hermite polynomials form an Appell sequence.
We shall now show that the Hermite polynomials form an orthogonal set on the interval $(-\infty,\,\infty)$ with the weight factor $e^{-x^2}$ . Let $m < n$ ; using (1) and integrating by parts we get
The substitution portion here equals to zero because $e^{-x^2}$ and its derivatives vanish at $\pm\infty$ . Using then (2) we obtain $$\int_{-\infty}^\infty H_m(x)H_n(x)e^{-x^2}\,dx \;=\; 2(-1)^{1+n}m\int_{-\infty}^\infty H_{m-1}(x)\frac{d^{n-1}e^{-x^2}}{dx^{n-1}}\,dx.$$ Repeating the integration by parts gives the result
whereas in the case $m = n$ the result $$\int_{-\infty}^\infty (H_n(x))^2e^{-x^2}\,dx \;=\; 2^n(-1)^{2n}n!\int_{-\infty}^\infty e^{-x^2}\,dx \;=\; 2^nn!\sqrt{\pi}$$ (see area under Gaussian curve). The results mean that the functions $x \mapsto\frac{H_n(x)}{\sqrt{2^nn!\sqrt{\pi}}}e^{-\frac{x^2}{2}}$ form an orthonormal set on $(-\infty,\,\infty)$ .
The Hermite polynomials are used in the quantum mechanical treatment of a harmonic oscillator, the wave functions of which have the form $$\xi \;\,\mapsto\,\; \Psi_n(\xi) \;=\; C_nH_n(\xi)e^{-\frac{\xi^2}{2}}.$$
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"Hermite polynomials" is owned by pahio.
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Cross-references: wave functions, harmonic oscillator, orthonormal set, functions, area under Gaussian curve, integration by parts, vanish, derivatives, substitution, interval, Appell sequence, differentiation, Faà di Bruno's formula, Rodrigues's Formula, equation, integer, Hermite differential equation, solutions, polynomial
There are 12 references to this entry.
This is version 25 of Hermite polynomials, born on 2005-05-17, modified 2010-01-03.
Object id is 7061, canonical name is HermitePolynomials.
Accessed 12381 times total.
Classification:
| AMS MSC: | 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous) | | | 26A09 (Real functions :: Functions of one variable :: Elementary functions) | | | 26C05 (Real functions :: Polynomials, rational functions :: Polynomials: analytic properties, etc.) | | | 33B99 (Special functions :: Elementary classical functions :: Miscellaneous) | | | 33E30 (Special functions :: Other special functions :: Other functions coming from differential, difference and integral equations) |
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Pending Errata and Addenda
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