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| In [[mathematics]], the '''Hermite polynomials''' are a classical [[orthogonal polynomials|orthogonal]] [[polynomial sequence]] that arise in [[probability]], such as the [[Edgeworth series]]; in [[combinatorics]], as an example of an [[Appell sequence]], obeying the [[umbral calculus]]; in [[numerical analysis]] as [[Gaussian quadrature]]; in [[finite element methods]] as shape functions for beams; and in [[physics]], where they give rise to the [[eigenstate]]s of the [[quantum harmonic oscillator]]. They are also used in [[systems theory]] in connection with nonlinear operations on [[Gaussian noise]]. They are named after [[Charles Hermite]] (1864)<ref>C. Hermite: ''Sur un nouveau développement en série de fonctions'' C. R Acad. Sci. Paris 58 1864 93-100; Oeuvres II 293-303</ref> although they were studied earlier by {{harvtxt|Laplace|1810}} and [[Chebyshev]] (1859).<ref>P.L.Chebyshev: ''Sur le développement des fonctions à une seule variable'' Bull. Acad. Sci. St. Petersb. I 1859 193-200; Oeuvres I 501-508</ref>
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| ==Definition==
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| There are two different ways of standardizing the Hermite polynomials:
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| :<math>(1)\ \ {\mathit{He}}_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n}e^{-x^2/2}=\bigg (x-\frac{d}{dx} \bigg )^n\cdot 1 ,</math>
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| (the '''"probabilists' Hermite polynomials"'''); and
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| :<math>(2)\ \ H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}=\bigg (2x-\frac{d}{dx} \bigg )^n\cdot 1 ,</math>
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| (the '''"physicists' Hermite polynomials"''').
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| These two definitions are ''not'' exactly identical; each one is a rescaling of the other,
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| :<math>H_n(x)=2^{n/2}{\mathit{He}}_n(\sqrt{2} \,x), \qquad {\mathit{He}}_n(x)=2^{-n/2}H_n\left(\frac x{\sqrt{2}} \right).</math>
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| These are Hermite polynomial sequences of different variances; see the material on variances below.
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| The notation ''He'' and ''H'' is that used in the standard references {{harvs|txt|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw|year=2010}} and [[Abramowitz & Stegun]].
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| The polynomials ''He''<sub>''n''</sub> are sometimes denoted by ''H''<sub>''n''</sub>, especially in probability theory, because
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| :<math>\frac{1}{\sqrt{2\pi}}e^{-x^2/2}</math> | |
| is the [[probability density function]] for the [[normal distribution]] with [[expected value]] 0 and [[standard deviation]] 1.
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| [[Image:Hermite poly.svg|thumb|right|390px|The first six (probabilists') Hermite polynomials ''He''<sub>''n''</sub>(''x'').]]
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| The first eleven probabilists' Hermite polynomials are:
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| :<math>{\mathit{He}}_0(x)=1\,</math>
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| :<math>{\mathit{He}}_1(x)=x\,</math>
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| :<math>{\mathit{He}}_2(x)=x^2-1\,</math>
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| :<math>{\mathit{He}}_3(x)=x^3-3x\,</math>
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| :<math>{\mathit{He}}_4(x)=x^4-6x^2+3\,</math>
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| :<math>{\mathit{He}}_5(x)=x^5-10x^3+15x\,</math>
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| :<math>{\mathit{He}}_6(x)=x^6-15x^4+45x^2-15\,</math>
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| :<math>{\mathit{He}}_7(x)=x^7-21x^5+105x^3-105x\,</math>
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| :<math>{\mathit{He}}_8(x)=x^8-28x^6+210x^4-420x^2+105\,</math>
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| :<math>{\mathit{He}}_9(x)=x^9-36x^7+378x^5-1260x^3+945x\,</math>
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| :<math>{\mathit{He}}_{10}(x)=x^{10}-45x^8+630x^6-3150x^4+4725x^2-945\,</math>
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| [[Image:Hermite poly phys.svg|thumb|right|390px|The first six (physicists') Hermite polynomials ''H''<sub>''n''</sub>(''x'').]]
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| and the first eleven physicists' Hermite polynomials are:
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| :<math>H_0(x)=1\,</math>
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| :<math>H_1(x)=2x\,</math>
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| :<math>H_2(x)=4x^2-2\,</math>
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| :<math>H_3(x)=8x^3-12x\,</math>
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| :<math>H_4(x)=16x^4-48x^2+12\,</math>
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| :<math>H_5(x)=32x^5-160x^3+120x\,</math>
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| :<math>H_6(x)=64x^6-480x^4+720x^2-120\,</math>
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| :<math>H_7(x)=128x^7-1344x^5+3360x^3-1680x\,</math>
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| :<math>H_8(x)=256x^8-3584x^6+13440x^4-13440x^2+1680\,</math>
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| :<math>H_9(x)=512x^9-9216x^7+48384x^5-80640x^3+30240x\,</math>
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| :<math>H_{10}(x)=1024x^{10}-23040x^8+161280x^6-403200x^4+302400x^2-30240\,</math>
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| <!--
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| As an alternative to calculating N<sup>th</sup> order derivatives of ''e''<sup>−''x''<sup>2</sup>/2</sup> and ''e''<sup>−''x''<sup>2</sup></sup>, an easier, less computationally intensive method of sequentially deriving individual terms of the ''N''th-order Hermite polynomials is to consider the combination of coefficients in the corresponding terms in the (''N'' − 1)th-order Hermite polynomial. For the probabilists' notation, follow the following rules:
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| 1) For the starting point in the sequence, the 0th-order polynomial (''H''<sub>0</sub>) is equal to 1.
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| 2) The first term has a power of ''x'' equal to the given ''N''th-order polynomial being derived, and the coefficient of this term is 1.
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| 3) The power of ''x'' of each successive term is 2 less than the preceding term.
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| 4) The coefficient of each term after the first term is calculated by taking the coefficient of the same-numbered term in the (''N'' − 1)th polynomial, and adding to it the product of the power of ''x'' and the corresponding coefficient of the immediately preceding term in the (''N'' − 1)th polynomial.
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| 5) All even-numbered terms in each polynomial are negative, and all odd-numbered terms are positive.
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| Thus, for the ''H''<sub>6</sub> polynomial, ''N'' = 6 and hence the first term is ''x''<sup>''N''</sup> = ''x''<sup>6</sup>, with a coefficient of 1. The second term has a power of ''x'' equal to 6 − 2 = 4. The coefficient is obtained by taking the coefficient of the second term in the ''H''<sub>5</sub> polynomial (which is 10) and adding to it the product of the power of ''x'' and its coefficient in the first term of the ''H''<sub>5</sub> polynomial (which are 5 and 1, respectively). Thus, 10 + 5·1 = 15. Make this coefficient negative, since this is an even-numbered term. The third term in the ''H''<sub>6</sub> polynomial has a power of ''x'' equal to 2 (which is 2 less than the power of ''x'' in the second term), and its coefficient is 15 + 3·10 = 45, where 15 = coefficient of the third term in the ''H''<sub>5</sub> polynomial; 3 = power of ''x'' in the second term of the ''H''<sub>5</sub> polynomial; and 10 = coefficient of the second term in the ''H''<sub>5</sub> polynomial. This coefficient (45) is positive, since this is an odd-numbered term. Finally, the fourth term in the ''H''<sub>6</sub> polynomial is 0th power in ''x'' (which is 2 less than the power of x in the third term), and its coefficient is 0 + 1·15 = 15, where 0 = coefficient of the (non-existent) fourth term in the ''H''<sub>5</sub> polynomial; 1 = power of ''x'' in the third term of the ''H''<sub>5</sub> polynomial, and 15 = coefficient of the third term in the ''H''<sub>5</sub> polynomial. This coefficient (15) is made negative, since this is an even-numbered term. Thus, ''H''<sub>6</sub>(''x'') = ''x''<sup>6</sup> − 15''x''<sup>4</sup> + 45''x''<sup>2</sup> − 15.
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| For the H<sub>7</sub> polynomial, the first term is x<sup>7</sup>; the second term is 15 + 6*1 = 21 (negative), with a 7-2 = 5th power of x (i.e., -21x<sup>5</sup>). The third term is 45 + 4*15 = 105 (positive), with a 5-2 = 3rd power of x (i.e., 105x<sup>3</sup>). The fourth term is 15 + 2*45 = 105 (negative), with a 3-2 = 1st power of x (i.e., -105x). Thus, H<sub>7</sub>(x) = x<sup>7</sup> - 21x<sup>5</sup> + 105x<sup>3</sup> - 105x.
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| For the physicists' notation, follow the following rules:
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| 1) For the starting point in the sequence, the 0th order polynomial (H<sub>0</sub>) is equal to 1.
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| 2) The first term has a power of x equal to the given N<sup>th</sup> order polynomial being derived, and the coefficient of this term is 2<sup>N</sup>.
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| 3) The power of x of each successive term is 2 less than the preceding term.
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| 4) The coefficient of each term after the first term is calculated by taking the coefficient of the same-numbered term in the (N-1)<sup>th</sup> polynomial, multiplying it by 2, and then adding to it the product of the power of x and the corresponding coefficient of the immediately preceding term in the (N-1)<sup>th</sup> polynomial.
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| 5) All even-numbered terms in each polynomial are negative, and all odd-numbered terms are positive.
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| Thus, for the H<sub>6</sub> polynomial, the first term is 2<sup>6</sup> = 64, with a 6th power of x (i.e., 64x<sup>6</sup>). The second term is 2*160 + 5*32 = 480 (negative), with a 6-2 = 4th power of x (i.e., -480x<sup>4</sup>). The third term is 2*120 + 3*160 = 720 (positive), with a 4-2 = 2nd power of x (i.e., 720x<sup>2</sup>). The fourth term is 2*0 + 1*120 = 120 (negative), with a 2-2 = 0th power of x (i.e., -120). Thus, H<sub>6</sub>(x) = 64x<sup>6</sup> - 480x<sup>4</sup> + 720x<sup>2</sup> - 120.
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| For the H<sub>7</sub> polynomial, the first term is 2<sup>7</sup> = 128, with a 7th power of x (i.e., 128x<sup>7</sup>). The second term is 2*480 + 6*64 = 1344 (negative), with a 7-2 = 5th power of x (i.e., -1344x<sup>5</sup>). The third term is 2*720 + 4*480 = 3360 (positive), with a 5-2 = 3rd power of x (i.e., 3360x<sup>3</sup>). The fourth term is 2*120 + 2*720 = 1680 (negative), with a 3-2 = 1st power of x (i.e., -1680x). Thus, H<sub>7</sub>(x) = 128x<sup>7</sup> - 1344x<sup>5</sup> + 3360x<sup>3</sup> - 1680x.
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| Recognizing that H<sub>0</sub> = 1, these rules can be followed to sequentially derive all N<sup>th</sup> order Hermite polynomials (from N = 1 toward infinity), and can be computer-coded relatively easily for practical applications. -->
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| ==Properties==
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| ''H<sub>n</sub>'' is a polynomial of degree ''n''. The probabilists' version ''He'' has leading coefficient 1, while the physicists' version ''H'' has leading coefficient 2<sup>''n''</sup>.
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| ===Orthogonality===
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| ''H''<sub>''n''</sub>(''x'') and ''He''<sub>''n''</sub>(''x'') are ''n''th-degree polynomials for ''n'' = 0, 1, 2, 3, .... These [[orthogonal polynomials|polynomials are orthogonal]] with respect to the ''weight function'' ([[measure (mathematics)|measure]])
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| :<math>w(x) = \mathrm{e}^{-x^2/2}\,\!</math> (''He'')
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| or
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| :<math>w(x) = \mathrm{e}^{-x^2}\,\!</math> (''H'')
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| i.e., we have
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| :<math>\int_{-\infty}^\infty H_m(x) H_n(x)\, w(x) \, \mathrm{d}x = 0</math>
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| when ''m'' ≠ ''n''. Furthermore,
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| :<math>\int_{-\infty}^\infty {\mathit{He}}_m(x) {\mathit{He}}_n(x)\, \mathrm{e}^{-x^2/2} \, \mathrm{d}x = \sqrt{2 \pi} n! \delta_{nm}</math> (probabilist)
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| or
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| :<math>\int_{-\infty}^\infty H_m(x) H_n(x)\, \mathrm{e}^{-x^2}\, \mathrm{d}x = \sqrt{ \pi} 2^n n! \delta_{nm}</math> (physicist).
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| The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.
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| === Completeness ===
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| The Hermite polynomials (probabilist or physicist) form an [[orthonormal basis|orthogonal basis]] of the [[Hilbert space]] of functions satisfying
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| :<math>\int_{-\infty}^\infty\left|f(x)\right|^2\, w(x) \, \mathrm{d}x <\infty,</math>
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| in which the inner product is given by the integral including the [[gaussian function|Gaussian]] weight function ''w''(''x'') defined in the preceding section,
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| :<math>\langle f,g\rangle=\int_{-\infty}^\infty f(x)\overline{g(x)}\, w(x) \, \mathrm{d}x.</math>
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| An orthogonal basis for [[Lp space|''L''<sup>2</sup>('''R''', ''w''(''x'') d''x'')]] is a [[Hilbert space#Orthonormal bases|''complete'' orthogonal system]]. For an orthogonal system, ''completeness'' is equivalent to the fact that the 0 function is the only function ''ƒ'' ∈ ''L''<sup>2</sup>('''R''', ''w''(''x'') d''x'') orthogonal to all functions in the system. Since the linear span of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that if ''ƒ'' satisfies
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| :<math>\int_{-\infty}^\infty f(x) x^n \mathrm{e}^{- x^2} \, \mathrm{d}x = 0</math>
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| for every ''n'' ≥ 0, then ''ƒ'' = 0. One possible way to do it is to see that the [[entire function]]
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| :<math>F(z) = \int_{-\infty}^\infty f(x) \, \mathrm{e}^{z x - x^2} \, \mathrm{d}x = \sum_{n=0}^\infty \frac{z^n}{n!}\int f(x) x^n \mathrm{e}^{- x^2} \, \mathrm{d}x = 0</math>
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| vanishes identically. The fact that ''F''(i''t'') = 0 for every ''t'' real means that the [[Fourier transform]] of ''ƒ''(''x'') exp(−''x''<sup>2</sup>) is 0, hence ''ƒ'' is 0 almost everywhere. Variants of the above completeness proof apply to other weights with exponential decay. In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see "''Completeness relation''" below).
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| An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for ''L''<sup>2</sup>('''R''', ''w''(''x'') d''x'') consists in introducing Hermite ''functions'' (see below), and in saying that the Hermite functions are an orthonormal basis for ''L''<sup>2</sup>('''R''').
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| ===Hermite's differential equation===
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| The probabilists' Hermite polynomials are solutions of the differential equation
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| :<math>(e^{-x^2/2}u')' + \lambda e^{-x^2/2}u = 0</math>
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| where λ is a constant, with the boundary conditions that ''u'' should be polynomially bounded at infinity. With these boundary conditions, the equation has solutions only if λ is a non-negative integer, and up to an overall scaling, the solution is uniquely given by ''u''(''x'') = ''He''<sub>λ</sub>(''x''). Rewriting the differential equation as an [[Eigenvalue|eigenvalue problem]]
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| :<math>L[u] = u'' - x u' = -\lambda u</math>
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| solutions are the [[eigenfunction]]s of the differential operator ''L''. This eigenvalue problem is called the '''Hermite equation''', although the term is also used for the closely related equation
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| :<math>u'' - 2xu'=-2\lambda u</math>
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| whose solutions are the physicists' Hermite polynomials.
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| With more general boundary conditions, the Hermite polynomials can be generalized to obtain more general [[analytic function]]s ''He''<sub>λ</sub>(''z'') for λ a complex index. An explicit formula can be given in terms of a [[contour integral]] {{harv|Courant|Hilbert|1953}}.
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| ===Recursion relation===
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| The sequence of Hermite polynomials also satisfies the [[recursion]]
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| :<math>{\mathit{He}}_{n+1}(x)=x{\mathit{He}}_n(x)-{\mathit{He}}_n'(x).\,\!</math> (probabilist)
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| Individual coefficients are related by the following recursion formula:
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| :<math>a_{n+1,k} = a_{n,k-1} - n a_{n-1,k} \ \ k>0</math>
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| :<math>a_{n+1,k} = - n a_{n-1,k} \ \ k =0</math>
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| and a[0,0]=1, a[1,0]=0, a[1,1]=1.
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| (Assuming :<math> H_n (x) = \sum\limits^{n}_{k=0} a_{n,k} \ x ^k </math>)
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| :<math>H_{n+1}(x)=2 xH_n(x)-H_n'(x).\,\!</math> (physicist)
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| Individual coefficients are related by the following recursion formula:
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| :<math>a_{n+1,k} =2 a_{n,k-1} - 2 n a_{n-1,k} \ \ k>0</math>
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| :<math>a_{n+1,k} = - 2 n a_{n-1,k} \ \ k =0</math>
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| and a[0,0]=1, a[1,0]=0, a[1,1]=2.
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| The Hermite polynomials constitute an [[Appell sequence]], i.e., they are a polynomial sequence satisfying the identity
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| :<math>{\mathit{He}}_n'(x)=n{\mathit{He}}_{n-1}(x),\,\!</math> (probabilist)
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| :<math>H_n'(x)=2nH_{n-1}(x),\,\!</math> (physicist)
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| or, equivalently, by Taylor expanding,
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| :<math>{\mathit{He}}_n(x+y)=\sum_{k=0}^n{n \choose k}x^{n-k} {\mathit{He}}_{k}(y)</math> (probabilist)
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| :<math>H_n(x+y)=\sum_{k=0}^n{n \choose k}H_{k}(x) (2y)^{(n-k)}= 2^{-\frac n 2}\cdot\sum_{k=0}^n {n \choose k} H_{n-k}\left(x\sqrt 2\right) H_k\left(y\sqrt 2\right).</math> (physicist)
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| In consequence, for the m-th derivatives the following relations hold:
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| :<math>{\mathit{He}}_n^{(m)}(x)=\frac{n!}{(n-m)!}\cdot{\mathit{He}}_{n-m}(x)=m!\cdot{n \choose m}\cdot{\mathit{He}}_{n-m}(x),\,\!</math> (probabilist)
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| :<math>{\mathit{H}}_n^{(m)}(x)=2^m\cdot\frac{n!}{(n-m)!}\cdot{\mathit{H}}_{n-m}(x)=2^m \cdot m!\cdot{n \choose m}\cdot{\mathit{H}}_{n-m}(x).\,\!</math> (physicist)
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| It follows that the Hermite polynomials also satisfy the [[recurrence relation]]
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| :<math>{\mathit{He}}_{n+1}(x)=x{\mathit{He}}_n(x)-n{\mathit{He}}_{n-1}(x),\,\!</math> (probabilist)
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| :<math>H_{n+1}(x)=2xH_n(x)-2nH_{n-1}(x).\,\!</math> (physicist)
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| These last relations, together with the initial polynomials ''H''<sub>0</sub>(''x'') and ''H''<sub>1</sub>(''x''), can be used in practice to compute the polynomials quickly.
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| [[Turán's inequalities]] are
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| :<math>{\mathit{He}}_n(x)^2 - {\mathit{He}}_{n-1}(x){\mathit{He}}_{n+1}(x)= (n-1)!\cdot \sum_{i=0}^{n-1}\frac{2^{n-i}}{i!}{\mathit{He}}_i(x)^2>0.</math>
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| Moreover, the following [[multiplication theorem]] holds:
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| :<math>{\mathit{H}}_n(\gamma x)=\sum_{i=0}^{\lfloor n/2 \rfloor} \gamma^{n-2i}(\gamma^2-1)^i {n \choose 2i} \frac{(2i)!}{i!}{\mathit{H}}_{n-2i}(x).</math>
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| ===Explicit expression===
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| The physicists' Hermite polynomials can be written explicitly as
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| :<math> H_n(x) = n! \sum_{\ell = 0}^{n/2} \frac{(-1)^{n/2 - \ell}}{(2\ell)! (n/2 - \ell)!} (2x)^{2\ell} </math>
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| for even values of ''n'' and
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| :<math> H_n(x) = n! \sum_{\ell = 0}^{(n-1)/2} \frac{(-1)^{(n-1)/2 - \ell}}{(2\ell + 1)! ((n-1)/2 - \ell)!} (2x)^{2\ell + 1} </math>
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| for odd values of ''n.'' These two equations may be combined into one using the [[Floor function|floor]] function:
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| :<math> H_n(x) = n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m!(n - 2m)!} (2x)^{n - 2m}. </math>
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| The probabilists' Hermite polynomials ''He'' have similar formulas, which may be obtained from these by replacing
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| the power of 2''x'' with the corresponding power of (√2)''x'', and multiplying the entire sum by 2<sup>−''n''/2</sup>.
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| :<math> He_n(x) = n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m!(n - 2m)!} \frac{x^{n - 2m}}{2^m}. </math>
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| ===Generating function===
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| The Hermite polynomials are given by the [[exponential generating function]]
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| :<math>\exp (xt-t^2/2) = \sum_{n=0}^\infty {\mathit{He}}_n(x) \frac {t^n}{n!}\,\!</math> (probabilist)
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| <!-- extra blank line between displayed [[TeX]] fractions, for legibility -->
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| :<math>\exp (2xt-t^2) = \sum_{n=0}^\infty H_n(x) \frac {t^n}{n!}\,\!</math> (physicist).
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| This equality is valid for all ''x'', ''t'' complex, and can be obtained by writing the Taylor expansion at ''x'' of the entire function ''z'' → exp(−''z''<sup>2</sup>) (in physicist's case).
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| One can also derive the (physicist's) generating function by using Cauchy's Integral Formula to write the Hermite polynomials as
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| :<math>H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}= (-1)^n e^{x^2}{n! \over 2\pi i} \oint_\gamma {e^{-z^2} \over (z-x)^{n+1}}\, dz.\,\!</math>
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| Using this in the sum <math>\sum_{n=0}^\infty H_n(x) \frac {t^n}{n!}\,\!</math>, one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.
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| ===Expected values===
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| If ''X'' is a [[random variable]] with a [[normal distribution]] with standard deviation 1 and expected value μ then
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| :<math>E({\mathit{He}}_n(X))=\mu^n.\,\!</math> (probabilist)
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| The moments of the standard normal may be read off directly from the relation for even indices
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| :<math>E(X^{2n})=(-1)^n {\mathit{He}}_{2n}(0)=(2n-1)!!\ </math>
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| where <math>(2n-1)!!</math> is the [[double factorial]]. Note that the above expression is a special case of the representation of the probabilists' Hermite polynomials as moments
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| :<math>{\mathit{He}}_n(x)=\int_{-\infty}^\infty (x+iy)^n\, \mathrm{e}^{- y^2/2} \, \mathrm{d}y/\sqrt{2\pi}\ .</math>
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| ===Asymptotic expansion===
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| Asymptotically, as <math>n</math> tends to infinity, the expansion
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| :<math>e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \frac{2^n}{\sqrt \pi}\Gamma\left(\frac{n+1}2\right) \cos \left(x \sqrt{2 n}- n\frac \pi 2 \right)</math> (physicist<ref>Abramowitz, [http://www.math.sfu.ca/~cbm/aands/page_508.htm p. 508-510], 13.6.38 and 13.5.16</ref>)
| |
| holds true.
| |
| For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude
| |
| :<math>e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \frac{2^n}{\sqrt \pi}\Gamma\left(\frac{n+1}2\right) \cos \left(x \sqrt{2 n}- n\frac \pi 2 \right)\left(1-\frac{x^2}{2n}\right)^{-\frac{1}{4}}=\frac{2 \Gamma\left(n\right)}{\Gamma\left(\frac{n}2\right)} \cos \left(x \sqrt{2 n}- n\frac \pi 2 \right)\left(1-\frac{x^2}{2n}\right)^{-\frac{1}{4}}</math>
| |
| Which, using [[Stirling's approximation]], can be further simplified, in the limit, to
| |
| :<math>e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \left(\frac{2 n}{e}\right)^{\frac{n}{2}} {\sqrt 2} \cos \left(x \sqrt{2 n}- n\frac \pi 2 \right)\left(1-\frac{x^2}{2n}\right)^{-\frac{1}{4}}</math>
| |
| This expansion is needed to resolve the wave-function of a [[quantum harmonic oscillator]] such that it agrees with the classical approximation in the limit of the [[correspondence principle]].
| |
| | |
| A finer approximation,<ref>{{harvnb|Szegő|1939, 1955|p=201}}</ref> which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution <math>x=\sqrt{ 2n+1}\cos(\phi)</math>, for <math>0<\epsilon\leq\phi\leq\pi-\epsilon</math>, with which one has the uniform approximation
| |
| :<math>e^{-\frac{x^2}{2}}\cdot H_n(x) = 2^{n/2+\frac{1}{4}}\sqrt{n!}(\pi n)^{-1/4}(\sin \phi)^{-1/2} \cdot \left[\sin\left(\left(\frac{n}{2}+\frac{1}{4}\right)\left(\sin(2\phi)-2\phi\right) +\frac{3\pi}{4}\right)+O(n^{-1}) \right].</math>
| |
| | |
| Similar approximations hold for the monotonic and transition regions. Specifically, if <math>x=\sqrt{2n+1} \cosh(\phi)</math> for <math>0<\epsilon\leq\phi\leq\omega<\infty</math> then
| |
| :<math>e^{-\frac{x^2}{2}}\cdot H_n(x) = 2^{n/2-\frac{3}{4}}\sqrt{n!}(\pi n)^{-1/4}(\sinh \phi)^{-1/2} \cdot \exp\left(\left(\frac{n}{2}+\frac{1}{4}\right)\left(2\phi-\sinh(2\phi)\right)\right)\left[1+O(n^{-1}) \right],</math>
| |
| while for <math>x=\sqrt{2n+1}-2^{-1/2}3^{-1/3}n^{-1/6}t</math> with <math>t</math> complex and bounded then
| |
| :<math>e^{-\frac{x^2}{2}}\cdot H_n(x) =\pi^{1/4}2^{n/2+\frac{1}{4}}\sqrt{n!} n^{-1/12}\left[ \mathrm{Ai}(-3^{-1/3}t)+ O(n^{-2/3}) \right]</math>
| |
| where Ai(·) is the [[Airy function]] of the first kind.
| |
| | |
| ===Special Values===
| |
| The Hermite polynomials evaluated at zero argument <math>H_n(0)</math> are called [[Hermite number]]s.
| |
| | |
| :<math>H_n(0) =
| |
| \begin{cases}
| |
| 0, & \mbox{if }n\mbox{ is odd} \\
| |
| (-1)^{n/2} 2^{n/2} (n-1)!! , & \mbox{if }n\mbox{ is even}
| |
| \end{cases}
| |
| </math>
| |
| | |
| In terms of the probabilist's polynomials this translates to
| |
| | |
| :<math>He_n(0) =
| |
| \begin{cases}
| |
| 0, & \mbox{if }n\mbox{ is odd} \\
| |
| (-1)^{n/2} (n-1)!! , & \mbox{if }n\mbox{ is even}
| |
| \end{cases}
| |
| </math>
| |
| | |
| ==Relations to other functions==
| |
| ===Laguerre polynomials===
| |
| The Hermite polynomials can be expressed as a special case of the [[Laguerre polynomials]].
| |
| | |
| :<math>H_{2n}(x) = (-4)^{n}\,n!\,L_{n}^{(-1/2)}(x^2)=4^n\, n! \sum_{i=0}^n (-1)^{n-i} {n-\frac{1}{2} \choose n-i} \frac{x^{2i}}{i!}\,\!</math> (physicist)
| |
| :<math>H_{2n+1}(x) = 2(-4)^{n}\,n!\,x\,L_{n}^{(1/2)}(x^2)=2\cdot 4^n\, n! \sum_{i=0}^n (-1)^{n-i} {n+\frac{1}{2} \choose n-i} \frac{x^{2i+1}}{i!}\,\!</math> (physicist)
| |
| | |
| ===Relation to confluent hypergeometric functions===
| |
| The Hermite polynomials can be expressed as a special case of the [[parabolic cylinder function]]s.
| |
| | |
| :<math>H_{n}(x) =
| |
| 2^n\,U\left(-\frac{n}{2},\frac{1}{2},x^2\right)</math> (physicist)
| |
| | |
| where <math>U(a,b,z)</math> is [[confluent hypergeometric function|Whittaker's confluent hypergeometric function]]. Similarly,
| |
| | |
| :<math>H_{2n}(x) = (-1)^{n}\,\frac{(2n)!}{n!}
| |
| \,_1F_1\left(-n,\frac{1}{2};x^2\right)</math> (physicist)
| |
| | |
| :<math>H_{2n+1}(x) = (-1)^{n}\,\frac{(2n+1)!}{n!}\,2x
| |
| \,_1F_1\left(-n,\frac{3}{2};x^2\right)</math> (physicist)
| |
| | |
| where <math>\,_1F_1(a,b;z)=M(a,b;z)</math> is [[confluent hypergeometric function|Kummer's confluent hypergeometric function]].
| |
| | |
| ==Differential operator representation==
| |
| The probabilists' Hermite polynomials satisfy the identity
| |
| | |
| :<math>{\mathit{He}}_n(x)=e^{-D^2/2}x^n\,\!</math>
| |
| | |
| where ''D'' represents differentiation with respect to ''x'', and the [[exponential function|exponential]] is interpreted by expanding it as a [[power series]]. There are no delicate questions of convergence of this series when it operates on polynomials, since all but finitely many terms vanish.
| |
| | |
| Since the power series coefficients of the exponential are well known, and higher order derivatives of the monomial ''x''<sup>''n''</sup> can be written down explicitly, this differential operator representation gives rise to a concrete formula for the coefficients of ''H<sub>n</sub>'' that can be used to quickly compute these polynomials.
| |
| | |
| Since the formal expression for the [[Weierstrass transform]] ''W'' is ''e''<sup>''D''<sup>2</sup></sup>, we see that the Weierstrass transform of (√2)<sup>''n''</sup>''He<sub>n</sub>''(''x''/√2) is ''x''<sup>''n''</sup>.
| |
| Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding [[Maclaurin series]].
| |
| | |
| The existence of some formal power series ''g''(''D'') with nonzero constant coefficient, such that ''He<sub>n</sub>''(''x'') = ''g''(''D'')''x<sup>n</sup>'', is another equivalent to the statement that these polynomials form an Appell sequence−−cf. [[Weierstrass_transform#The_inverse|''W'']]. Since they are an Appell sequence, they are ''a fortiori'' a [[Sheffer sequence]].
| |
| | |
| ==Contour integral representation==
| |
| The Hermite polynomials have a representation in terms of a [[contour integral]], as
| |
| | |
| :<math>{\mathit{He}}_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{tx-t^2/2}}{t^{n+1}}\,dt</math> (probabilist)
| |
| | |
| :<math>H_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{2tx-t^2}}{t^{n+1}}\,dt</math> (physicist)
| |
| | |
| with the contour encircling the origin.
| |
| | |
| ==Generalizations==
| |
| The (probabilists') Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is
| |
| | |
| :<math>\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\,\!</math>
| |
| | |
| which has expected value 0 and variance 1. One may speak of Hermite polynomials
| |
| | |
| :<math>{\mathit{He}}_n^{[\alpha]}(x)\,\!</math>
| |
| | |
| of variance α, where α is any positive number. These are orthogonal with respect to the normal probability distribution whose density function is
| |
| | |
| :<math>(2\pi\alpha)^{-1/2}e^{-x^2/(2\alpha)}.\,\!</math>
| |
| | |
| They are given by
| |
| | |
| :<math>{\mathit{He}}_n^{[\alpha]}(x) = \alpha^{-n/2}He_n^{[1]}\left(\frac{x}{\sqrt{\alpha}}\right)= (2 \alpha)^{-n/2} H_n\left( \frac{x}{\sqrt{2 \alpha}}\right) = e^{-\alpha D^2/2}x^n.\,\!</math>
| |
| | |
| In particular, the physicists' Hermite polynomials are
| |
| | |
| :<math>{\mathit{He}}^{[1/2]}(x).\,\!</math>
| |
| | |
| If
| |
| | |
| :<math>{\mathit{He}}_n^{[\alpha]}(x)=\sum_{k=0}^n h^{[\alpha]}_{n,k}x^k\,\!</math>
| |
| | |
| then the polynomial sequence whose ''n''th term is
| |
| | |
| :<math>\left({\mathit{He}}_n^{[\alpha]}\circ {\mathit{He}}^{[\beta]}\right)(x)=\sum_{k=0}^n h^{[\alpha]}_{n,k}\,{\mathit{He}}_k^{[\beta]}(x)\,\!</math>
| |
| | |
| is the '''umbral composition''' of the two polynomial sequences, and it can be
| |
| shown to satisfy the identities
| |
| | |
| :<math>\left({\mathit{He}}_n^{[\alpha]}\circ {\mathit{He}}^{[\beta]}\right)(x)={\mathit{He}}_n^{[\alpha+\beta]}(x)\,\!</math>
| |
| | |
| and | |
| | |
| :<math>{\mathit{He}}_n^{[\alpha+\beta]}(x+y)=\sum_{k=0}^n{n\choose k}{\mathit{He}}_k^{[\alpha]}(x) {\mathit{He}}_{n-k}^{[\beta]}(y).\,\!</math>
| |
| | |
| The last identity is expressed by saying that this [[parameterized family]] of polynomial sequences is a '''[[cross-sequence]]'''.
| |
| | |
| ==="Negative variance"===
| |
| Since polynomial sequences form a [[group (mathematics)|group]] under the operation of [[Binomial type#Umbral composition of polynomial sequences|umbral composition]], one may denote by
| |
| | |
| :<math>{\mathit{He}}_n^{[-\alpha]}(x)\,\!</math>
| |
| | |
| the sequence that is inverse to the one similarly denoted but without the minus sign, and thus speak of Hermite polynomials of negative variance. For α > 0, the coefficients of ''He''<sub>''n''</sub><sup>[−α]</sup>(''x'') are just the absolute values of the corresponding coefficients of ''He''<sub>''n''</sub><sup>[α]</sup>(''x'').
| |
| | |
| These arise as moments of normal probability distributions: The ''n''th moment of the normal distribution with expected value μ and variance σ<sup>2</sup> is
| |
| | |
| :<math>E(X^n)={\mathit{He}}_n^{[-\sigma^2]}(\mu)\,\!</math>
| |
| | |
| where ''X'' is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that
| |
| | |
| :<math>\sum_{k=0}^n {n\choose k}{\mathit{He}}_k^{[\alpha]}(x) {\mathit{He}}_{n-k}^{[-\alpha]}(y)={\mathit{He}}_n^{[0]}(x+y)=(x+y)^n.\,\!</math>
| |
| | |
| ==Applications==
| |
| ===Hermite functions===
| |
| One can define the '''Hermite functions''' from the physicists' polynomials:
| |
| | |
| :<math>\psi_n(x) = (2^n n! \sqrt{\pi})^{-1/2} \mathrm{e}^{-x^2/2} H_n(x) = (-1)^n(2^n n! \sqrt{\pi})^{-1/2} \mathrm{e}^{x^2/2} \frac{d^n}{dx^n} \mathrm{e}^{-x^2}</math>
| |
| | |
| Since these functions contain the square root of the weight function, and have been scaled
| |
| appropriately, they are orthonormal:
| |
| | |
| :<math>\int_{-\infty}^\infty \psi_n(x)\psi_m(x)\, \mathrm{d}x = \delta_{n\,m}\,</math>
| |
| | |
| and form an orthonormal basis of ''L''<sup>2</sup>('''R'''). This fact is equivalent to the corresponding statement for Hermite polynomials (see above).
| |
| | |
| The Hermite functions are closely related to the [[Whittaker function]] (Whittaker and Watson, 1962) <math>D_n(z)\,</math>:
| |
| | |
| :<math>D_n(z) = (n! \sqrt{\pi})^{1/2} \psi_n(z/\sqrt{2}) = \pi^{-1/4} \sqrt{2} \mathrm{e}^{z^2/4} \frac{d^n}{dz^n} \mathrm{e}^{-z^2}</math>
| |
| | |
| and thereby to other [[parabolic cylinder function]]s. The Hermite functions satisfy the differential equation:
| |
| | |
| :<math>\psi_n''(x) + (2n + 1 - x^2) \psi_n(x) = 0\,.</math>
| |
| | |
| This equation is equivalent to the [[Schrödinger equation]] for a harmonic oscillator in quantum mechanics, so these functions are the eigenfunctions.
| |
| | |
| [[Image:Herm5.svg|thumb|center|450px|Hermite functions 0 (black), 1 (red), 2 (blue), 3 (yellow), 4 (green), and 5 (magenta).]]
| |
| :<math>\psi_0(x) = \pi^{-1/4} \, \mathrm{e}^{-\frac{1}{2} x^2}</math>
| |
| :<math>\psi_1(x) = \sqrt{2} \, \pi^{-1/4} \, x \, \mathrm{e}^{-\frac{1}{2} x^2}</math>
| |
| :<math>\psi_2(x) = (\sqrt{2} \, \pi^{1/4})^{-1} \, (2x^2-1) \, \mathrm{e}^{-\frac{1}{2} x^2}</math>
| |
| :<math>\psi_3(x) = (\sqrt{3} \, \pi^{1/4})^{-1} \, (2x^3-3x) \, \mathrm{e}^{-\frac{1}{2} x^2}</math>
| |
| :<math>\psi_4(x) = (2 \sqrt{6} \, \pi^{1/4})^{-1} \, (4x^4-12x^2+3) \, \mathrm{e}^{-\frac{1}{2} x^2}</math>
| |
| :<math>\psi_5(x) = (2 \sqrt{15} \, \pi^{1/4})^{-1} \, (4x^5-20x^3+15x) \, \mathrm{e}^{-\frac{1}{2} x^2}</math>
| |
| [[Image:Herm50.svg|thumb|center|680px|Hermite functions 0 (black), 2 (blue), 4 (green), and 50 (magenta).]]
| |
| | |
| === Recursion relation ===
| |
| Following recursion relations of Hermite polynomials, the Hermite functions obey
| |
| | |
| :<math>\psi_n'(x) = \sqrt{\frac{n}{2}}\psi_{n-1}(x) - \sqrt{\frac{n+1}{2}}\psi_{n+1}(x)</math>
| |
| | |
| as well as
| |
| | |
| :<math>x\;\psi_n(x) = \sqrt{\frac{n}{2}}\psi_{n-1}(x) + \sqrt{\frac{n+1}{2}}\psi_{n+1}(x).\,\!</math>
| |
| | |
| Extending the first relation to the arbitrary m-th derivatives for any positive integer ''m'' leads to
| |
| | |
| :<math>\psi_n^{(m)}(x) = \sum_{k=0}^m{m \choose k} (-1)^k 2^{(m-k)/2}\sqrt{\frac{n!}{(n-m+k)!}} \cdot \psi_{n-m+k}(x) \cdot {\mathit{He}}_{k}(x).\,\!</math>
| |
| | |
| This formula can be used in connection with the recurrence relations for ''He''<sub>n</sub> and <math>\psi_n</math> to calculate any derivative of the Hermite functions efficiently.
| |
| | |
| ===Cramér's inequality===
| |
| The Hermite functions satisfy the following bound due to [[Harald Cramér]]<ref>{{harvnb|Erdélyi|Magnus|Oberhettinger|Tricomi|1955|p=207}}</ref><ref>{{harvnb|Szegő|1939, 1955}}</ref>
| |
| | |
| :<math> |\psi_n(x)| \le K \pi^{-1/4}</math>
| |
| | |
| for ''x'' real, where the constant ''K'' is less than 1.086435.
| |
| | |
| ===Hermite functions as eigenfunctions of the Fourier transform===
| |
| The Hermite functions ψ<sub>''n''</sub>(''x'') are a set of eigenfunctions of the [[continuous Fourier transform]]. To see this, take the physicist's version of the generating function and multiply by exp(−''x''<sup> 2</sup>/2). This gives
| |
| | |
| : <math>\exp (-x^2/2 + 2xt-t^2) = \sum_{n=0}^\infty \exp (-x^2/2) H_n(x) \frac {t^n}{n!}.\,\!</math>
| |
| | |
| Choosing the unitary representation of the Fourier transform, the Fourier transform of the left hand side is given by
| |
| | |
| : <math>
| |
| \begin{align}
| |
| \mathcal{F} \{ \exp (-x^2/2 + 2xt-t^2)\}(k) & {} =
| |
| \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^\infty \exp (-ixk)\exp (-x^2/2 + 2xt-t^2)\, \mathrm{d}x \\
| |
| & {} = \exp (-k^2/2 - 2kit+t^2) \\
| |
| & {} = \sum_{n=0}^\infty \exp (-k^2/2) H_n(k) \frac {(-it)^n}{n!}.
| |
| \end{align}
| |
| </math>
| |
| | |
| The Fourier transform of the right hand side is given by
| |
| : <math>\mathcal{F} \left\{ \sum_{n=0}^\infty \exp (-x^2/2) H_n(x) \frac {t^n}{n!} \right\} = \sum_{n=0}^\infty \mathcal{F} \left \{ \exp(-x^2/2) H_n(x) \right\} \frac{t^n}{n!}. \,</math>
| |
| | |
| Equating like powers of ''t'' in the transformed versions of the left- and right-hand sides finally yields
| |
| | |
| :<math> \mathcal{F} \left\{ \exp (-x^2/2) H_n(x) \right\} = (-i)^n \exp (-k^2/2) H_n(k). \,\!</math>
| |
| | |
| The Hermite functions ψ<sub>''n''</sub>(''x'') are thus an orthonormal basis of ''L''<sup>2</sup>('''R''') which diagonalizes the Fourier transform operator. In this case, we chose the unitary version of the Fourier transform, so the [[eigenvalue]]s are (−''i'')<sup> ''n''</sup>.
| |
| The ensuing resolution of the identity then serves to define powers, including fractional ones, of the Fourier transform, to wit a [[Fractional Fourier transform]] generalization.
| |
| | |
| ===Combinatorial interpretation of coefficients===
| |
| In the Hermite polynomial ''He''<sub>''n''</sub>(''x'') of variance 1, the absolute value of the coefficient of ''x''<sup>''k''</sup> is the number of (unordered) partitions of an ''n''-member set into ''k'' singletons and (''n'' − ''k'')/2 (unordered) pairs. The sum of the absolute values of the coefficients gives the total number of partitions into singletons and pairs, the so-called [[Telephone number (mathematics)|telephone numbers]]
| |
| :1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... {{OEIS|A000085}}.
| |
| These numbers may also be expressed as a special value of the Hermite polynomials
| |
| :<math>T(n)=\frac{\mathop{He}_n(i)}{i^n}.</math><ref name="gfgt">{{citation
| |
| | last1 = Banderier | first1 = Cyril
| |
| | last2 = Bousquet-Mélou | first2 = Mireille
| |
| | last3 = Denise | first3 = Alain
| |
| | last4 = Flajolet | first4 = Philippe | author4-link = Philippe Flajolet
| |
| | last5 = Gardy | first5 = Danièle
| |
| | last6 = Gouyou-Beauchamps | first6 = Dominique
| |
| | arxiv = math/0411250
| |
| | doi = 10.1016/S0012-365X(01)00250-3
| |
| | issue = 1-3
| |
| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
| |
| | mr = 1884885
| |
| | pages = 29–55
| |
| | title = Generating functions for generating trees
| |
| | volume = 246
| |
| | year = 2002}}.</ref>
| |
| | |
| === Completeness relation ===
| |
| The [[Christoffel–Darboux formula]] for Hermite polynomials reads
| |
| :<math>\sum_{i=0}^n \frac{H_i(x) H_i(y)}{i!2^i}= \frac{1}{n!2^{n+1}}\frac{H_n(y)H_{n+1}(x)- H_n(x)H_{n+1}(y)}{x-y}.</math>
| |
| | |
| Moreover, the following identity holds in the sense of [[distribution (mathematics)|distributions]]
| |
| | |
| : <math>\sum_{n=0}^\infty \psi_n (x) \psi_n (y)= \delta(x-y),</math>
| |
| | |
| where ''δ'' is the [[Dirac delta function]], (''ψ''<sub>''n''</sub>) the Hermite functions, and ''δ''(''x'' − ''y'') represents the [[Lebesgue measure]] on the line ''y'' = ''x'' in '''R'''<sup>2</sup>, normalized so that its projection on the horizontal axis is the usual Lebesgue measure. This distributional identity follows by letting ''u'' → 1 in [[Mehler's formula]], valid when −1 < ''u'' < 1:
| |
| | |
| :<math>E(x, y; u) := \sum_{n=0}^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac 1 {\sqrt{\pi (1 - u^2)}} \, \mathrm{exp} \left( - \frac{1 - u}{1 + u} \, \frac{(x + y)^2}{4} \,-\, \frac{1 + u}{1 - u} \, \frac{(x - y)^2}{4}\right),</math>
| |
| which is often stated equivalently as
| |
| :<math>\sum_{n=0}^\infty \frac{H_n(x)H_n(y)}{n!}\left(\frac u 2\right)^n= \frac 1 {\sqrt{1-u^2}} \mathrm{e}^{\frac{2u}{1+u}x y-\frac{u^2}{1-u^2}(x-y)^2}.</math> | |
| | |
| The function (''x'', ''y'') → ''E''(''x'', ''y''; ''u'') is the density for a Gaussian measure on '''R'''<sup>2</sup> which is, when ''u'' is close to 1, very concentrated around the line ''y'' = ''x'', and very spread out on that line. It follows that
| |
| | |
| :<math> \left\langle \left( \sum_{n=0}^\infty u^n \langle f, \psi_n \rangle \psi_n\right), g \right\rangle = \int \int E(x, y; u) f(x) \overline{g(y)} \, \mathrm{d}x \, \mathrm{d}y \rightarrow \int f(x) \overline{g(x)} \, \mathrm{d} x = \langle f, g \rangle,</math>
| |
| | |
| when ''ƒ'', ''g'' are continuous and compactly supported. This yields that ''ƒ'' can be expressed from the Hermite functions, as sum of a series of vectors in ''L''<sup>2</sup>('''R'''), namely
| |
| | |
| :<math> f = \sum_{n=0}^\infty \langle f, \psi_n \rangle \psi_n.</math>
| |
| | |
| In order to prove the equality above for ''E''(''x'', ''y''; ''u''), the [[Fourier transform]] of [[Gaussian function]]s will be used several times,
| |
| | |
| : <math> \rho \sqrt{\pi} \, \mathrm{e}^{-\rho^2 x^2 / 4} = \int \mathrm{e}^{isx- s^2/\rho^2}\, \mathrm{d}s, \quad \rho > 0. </math>
| |
| | |
| The Hermite polynomial is then represented as
| |
| | |
| :<math> H_n(x) = (-1)^{n} \mathrm{e}^{x^2} \frac {\mathrm{d}^n}{\mathrm{d}x^n} \Bigl( \frac {1}{2\sqrt{\pi}} \int \mathrm{e}^{isx - s^2/4}\, \mathrm{d}s \Bigr) = (-1)^n \mathrm{e}^{x^2}\frac {1}{2\sqrt{\pi}}\int (is)^n \, \mathrm{e}^{isx- s^2/4}\, \mathrm{d}s. </math>
| |
| | |
| With this representation for ''H<sub>n</sub>''(''x'') and ''H<sub>n</sub>''(''y''), one sees that
| |
| | |
| : <math>
| |
| \begin{align}E(x, y; u) &= \sum_{n=0}^\infty \frac{u^n}{2^n n! \sqrt{\pi}} \, H_n(x) H_n(y) \, \mathrm{e}^{ - (x^2+y^2)/2} \\
| |
| & =\frac{\mathrm{e}^{(x^2+y^2)/2}}{4\pi\sqrt{\pi}}\int \!\! \int \Bigl( \sum_{n=0}^\infty \frac{1}{2^n n! } (-ust)^n \Bigr) \, \mathrm{e}^{isx+ity - s^2/4 - t^2/4}\, \mathrm{d}s\,\mathrm{d}t \\
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| & =\frac{\mathrm{e}^{(x^2+y^2)/2} }{4\pi\sqrt{\pi}}\int \!\! \int \mathrm{e}^{-ust/2} \, \mathrm{e}^{isx+ity - s^2/4 - t^2/4}\, \mathrm{d}s\,\mathrm{d}t,\end{align}
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| </math>
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| and this implies the desired result, using again the Fourier transform of Gaussian kernels after performing the substitution
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| : <math>s = \frac{\sigma + \tau}{\sqrt 2},\qquad\qquad t = \frac{\sigma - \tau}{\sqrt 2}.</math>
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| The proof of completeness by Mehler's formula is due to N.Wiener ''The Fourier integral and certain of its applications'' Cambridge Univ. Press 1933 reprinted Dover 1958
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| ==See also==
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| *[[Mehler kernel]]
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| *[[Kibble–Slepian formula]]
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| *[[Turán's inequalities]]
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| *[[Parabolic cylinder function]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Abramowitz Stegun ref|22|773}}
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| * {{citation|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|title=Methods of Mathematical Physics, Volume I|publisher=Wiley-Interscience|year=1953}}.
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| *{{citation|first=Arthur|last= Erdélyi|authorlink1=Arthur Erdélyi|first2= Wilhelm |last2=Magnus|authorlink2=Wilhelm Magnus|first3= Fritz|last3= Oberhettinger|first4=Francesco G.|last4= Tricomi|authorlink4=Francesco Tricomi|title= Higher transcendental functions. Vol. II|publisher= McGraw-Hill|year=1955}} ([http://www.nr.com/legacybooks scan])
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| *{{eom|id=H/h046980|first=M.V.|last=Fedoryuk|title=Hermite functions}}.
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| *{{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|authorlink1 = Tom H. Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
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| *{{citation|first=P.S. |last=Laplace|journal= Mém. Cl. Sci. Math. Phys. Inst. France |volume= 58 |year=1810|pages= 279–347}}
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| *{{eom|id=H/h047010|first=P. K. |last=Suetin}}.
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| *{{citation|first=Gábor|last=Szegő|authorlink=Gábor Szegő|title=Orthogonal Polynomials|publisher=American Mathematical Society|year=1939, 1955}}
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| *{{citation|title=The Fourier Integral and Certain of its Applications|last=Wiener |first=Norbert |authorlink=Norbert Wiener |year=1958 |publisher=Dover Publications |location=New York|isbn=0-486-60272-9}}
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| *{{Citation |title=A Course of Modern Analysis|last=Whittaker |first=E. T. |authorlink=E. T. Whittaker |coauthors=Watson, G. N. |year=1962 |publisher=Cambridge University Press |location=London |editor=4th Edition}}
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| *Temme, Nico, ''Special Functions: An Introduction to the Classical Functions of Mathematical Physics'', Wiley, New York, 1996
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| ==External links==
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| *{{MathWorld|urlname=HermitePolynomial|title=Hermite Polynomial}}
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| *[http://math.fullerton.edu/mathews/n2003/HermitePolyMod.html Module for Hermite Polynomial Interpolation by John H. Mathews]
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| {{DEFAULTSORT:Hermite Polynomials}}
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| [[Category:Special hypergeometric functions]]
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| [[Category:Polynomials]]
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| [[Category:Orthogonal polynomials]]
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| {{Link GA|uk}}
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