Английская Википедия:Hermite polynomials

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In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.

The polynomials arise in:

• signal processing as Hermitian wavelets for wavelet transform analysis
• probability, such as the Edgeworth series, as well as in connection with Brownian motion;
• combinatorics, as an example of an Appell sequence, obeying the umbral calculus;
• numerical analysis as Gaussian quadrature;
• physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term <math>\begin{align}xu_{x}\end{align}</math> is present);
• systems theory in connection with nonlinear operations on Gaussian noise.
• random matrix theory in Gaussian ensembles.

Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new. They were consequently not new, although Hermite was the first to define the multidimensional polynomials.

Definition

Like the other classical orthogonal polynomials, the Hermite polynomials can be defined from several different starting points. Noting from the outset that there are two different standardizations in common use, one convenient method is as follows:

• The "probabilist's Hermite polynomials" are given by <math display="block">\mathit{He}_n(x) = (-1)^n e^{\frac{x^2}{2}}\frac{d^n}{dx^n}e^{-\frac{x^2}{2}},</math>
• while the "physicist's Hermite polynomials" are given by <math display="block">H_n(x) = (-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}.</math>

These equations have the form of a Rodrigues' formula and can also be written as, <math display="block">\mathit{He}_n(x) = \left(x - \frac{d}{dx} \right)^n \cdot 1, \quad H_n(x) = \left(2x - \frac{d}{dx} \right)^n \cdot 1.</math>

The two definitions are not exactly identical; each is a rescaling of the other: <math display="block">H_n(x)=2^\frac{n}{2} \mathit{He}_n\left(\sqrt{2} \,x\right), \quad \mathit{He}_n(x)=2^{-\frac{n}{2}} H_n\left(\frac {x}{\sqrt 2} \right).</math>

These are Hermite polynomial sequences of different variances; see the material on variances below.

The notation Шаблон:Mvar and Шаблон:Mvar is that used in the standard references. The polynomials Шаблон:Mvar are sometimes denoted by Шаблон:Mvar, especially in probability theory, because <math display="block">\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}</math> is the probability density function for the normal distribution with expected value 0 and standard deviation 1.

Файл:Hermite poly.svg

The first six probabilist's Hermite polynomials Шаблон:Math

Файл:Hermite poly phys.svg

The first six (physicist's) Hermite polynomials Шаблон:Math

• The first eleven probabilist's Hermite polynomials are: <math display="block">\begin{align}

\mathit{He}_0(x) &= 1, \\ \mathit{He}_1(x) &= x, \\ \mathit{He}_2(x) &= x^2 - 1, \\ \mathit{He}_3(x) &= x^3 - 3x, \\ \mathit{He}_4(x) &= x^4 - 6x^2 + 3, \\ \mathit{He}_5(x) &= x^5 - 10x^3 + 15x, \\ \mathit{He}_6(x) &= x^6 - 15x^4 + 45x^2 - 15, \\ \mathit{He}_7(x) &= x^7 - 21x^5 + 105x^3 - 105x, \\ \mathit{He}_8(x) &= x^8 - 28x^6 + 210x^4 - 420x^2 + 105, \\ \mathit{He}_9(x) &= x^9 - 36x^7 + 378x^5 - 1260x^3 + 945x, \\ \mathit{He}_{10}(x) &= x^{10} - 45x^8 + 630x^6 - 3150x^4 + 4725x^2 - 945. \end{align}</math>

• The first eleven physicist's Hermite polynomials are: <math display="block">\begin{align}

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. \end{align}</math>

Properties

The Шаблон:Mvarth-order Hermite polynomial is a polynomial of degree Шаблон:Mvar. The probabilist's version Шаблон:Mvar has leading coefficient 1, while the physicist's version Шаблон:Mvar has leading coefficient Шаблон:Math.

Symmetry

From the Rodrigues formulae given above, we can see that Шаблон:Math and Шаблон:Math are even or odd functions depending on Шаблон:Mvar: <math display="block">H_n(-x)=(-1)^nH_n(x),\quad \mathit{He}_n(-x)=(-1)^n\mathit{He}_n(x).</math>

Orthogonality

Шаблон:Math and Шаблон:Math are Шаблон:Mvarth-degree polynomials for Шаблон:Math. These polynomials are orthogonal with respect to the weight function (measure) <math display="block">w(x) = e^{-\frac{x^2}{2}} \quad (\text{for }\mathit{He})</math> or <math display="block">w(x) = e^{-x^2} \quad (\text{for } H),</math> i.e., we have <math display="block">\int_{-\infty}^\infty H_m(x) H_n(x)\, w(x) \,dx = 0 \quad \text{for all }m \neq n.</math>

Furthermore, <math display="block">\int_{-\infty}^\infty H_m(x) H_n(x)\, e^{-x^2} \,dx = \sqrt{\pi}\, 2^n n!\, \delta_{nm},</math> and <math display="block">\int_{-\infty}^\infty \mathit{He}_m(x) \mathit{He}_n(x)\, e^{-\frac{x^2}{2}} \,dx = \sqrt{2 \pi}\, n!\, \delta_{nm},</math> where <math>\delta_{nm}</math> is the Kronecker delta.

The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.

Completeness

The Hermite polynomials (probabilist's or physicist's) form an orthogonal basis of the Hilbert space of functions satisfying <math display="block">\int_{-\infty}^\infty \bigl|f(x)\bigr|^2\, w(x) \,dx < \infty,</math> in which the inner product is given by the integral <math display="block">\langle f,g\rangle = \int_{-\infty}^\infty f(x) \overline{g(x)}\, w(x) \,dx</math> including the Gaussian weight function Шаблон:Math defined in the preceding section

An orthogonal basis for Шаблон:Math is a complete orthogonal system. For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function Шаблон:Math 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 Шаблон:Mvar satisfies <math display="block">\int_{-\infty}^\infty f(x) x^n e^{- x^2} \,dx = 0</math> for every Шаблон:Math, then Шаблон:Math.

One possible way to do this is to appreciate that the entire function <math display="block">F(z) = \int_{-\infty}^\infty f(x) e^{z x - x^2} \,dx = \sum_{n=0}^\infty \frac{z^n}{n!} \int f(x) x^n e^{- x^2} \,dx = 0</math> vanishes identically. The fact then that Шаблон:Math for every real Шаблон:Mvar means that the Fourier transform of Шаблон:Math is 0, hence Шаблон:Mvar 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 section on the Completeness relation below).

An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for Шаблон:Math consists in introducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis for Шаблон:Math.

Hermite's differential equation

The probabilist's Hermite polynomials are solutions of the differential equation <math display="block">\left(e^{-\frac12 x^2}u'\right)' + \lambda e^{-\frac 1 2 x^2}u = 0,</math> where Шаблон:Mvar is a constant. Imposing the boundary condition that Шаблон:Mvar should be polynomially bounded at infinity, the equation has solutions only if Шаблон:Mvar is a non-negative integer, and the solution is uniquely given by <math>u(x) = C_1 He_\lambda(x) </math>, where <math>C_{1}</math> denotes a constant.

Rewriting the differential equation as an eigenvalue problem <math display="block">L[u] = u - x u' = -\lambda u,</math> the Hermite polynomials <math>He_\lambda(x) </math> may be understood as eigenfunctions of the differential operator <math>L[u]</math> . This eigenvalue problem is called the Hermite equation, although the term is also used for the closely related equation <math display="block">u - 2xu' = -2\lambda u.</math> whose solution is uniquely given in terms of physicist's Hermite polynomials in the form <math>u(x) = C_1 H_\lambda(x) </math>, where <math>C_{1}</math> denotes a constant, after imposing the boundary condition that Шаблон:Mvar should be polynomially bounded at infinity.

The general solutions to the above second-order differential equations are in fact linear combinations of both Hermite polynomials and confluent hypergeometric functions of the first kind. For example, for the physicist's Hermite equation <math display="block">u - 2xu' + 2\lambda u = 0,</math> the general solution takes the form <math display="block">u(x) = C_1 H_\lambda(x) + C_2 h_\lambda(x),</math> where <math>C_{1}</math> and <math>C_{2}</math> are constants, <math>H_\lambda(x)</math> are physicist's Hermite polynomials (of the first kind), and <math>h_\lambda(x)</math> are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as <math> h_\lambda(x) = {}_1F_1(-\tfrac{\lambda}{2};\tfrac{1}{2};x^2)</math> where <math>{}_1F_1(a;b;z)</math> are Confluent hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions, see below.

With more general boundary conditions, the Hermite polynomials can be generalized to obtain more general analytic functions for complex-valued Шаблон:Mvar. An explicit formula of Hermite polynomials in terms of contour integrals Шаблон:Harv is also possible.

Recurrence relation

The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation <math display="block">\mathit{He}_{n+1}(x) = x \mathit{He}_n(x) - \mathit{He}_n'(x).</math> Individual coefficients are related by the following recursion formula: <math display="block">a_{n+1,k} = \begin{cases}

- (k+1) a_{n,k+1} &  k = 0, \\
a_{n,k-1} - (k+1) a_{n,k+1} & k > 0,

\end{cases}</math> and Шаблон:Math, Шаблон:Math, Шаблон:Math.

For the physicist's polynomials, assuming <math display="block">H_n(x) = \sum^n_{k=0} a_{n,k} x^k,</math> we have <math display="block">H_{n+1}(x) = 2xH_n(x) - H_n'(x).</math> Individual coefficients are related by the following recursion formula: <math display="block">a_{n+1,k} = \begin{cases}

- a_{n,k+1} & k = 0, \\ 
2 a_{n,k-1} - (k+1)a_{n,k+1} & k > 0,

\end{cases}</math> and Шаблон:Math, Шаблон:Math, Шаблон:Math.

The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity <math display="block">\begin{align}

\mathit{He}_n'(x) &= n\mathit{He}_{n-1}(x), \\
H_n'(x) &= 2nH_{n-1}(x).

\end{align}</math>

An integral recurrence that is deduced and demonstrated in is as follows: <math display="block">He_{n+1}(x) = (n+1)\int_0^x He_n(t)dt - He'_n(0),</math>

<math display="block">H_{n+1}(x) = 2(n+1)\int_0^x H_n(t)dt - H'_n(0).</math>

Equivalently, by Taylor-expanding, <math display="block">\begin{align}

\mathit{He}_n(x+y) &= \sum_{k=0}^n \binom{n}{k}x^{n-k} \mathit{He}_{k}(y)
 &&= 2^{-\frac n 2} \sum_{k=0}^n \binom{n}{k} \mathit{He}_{n-k}\left(x\sqrt 2\right) \mathit{He}_k\left(y\sqrt 2\right), \\
H_n(x+y) &= \sum_{k=0}^n \binom{n}{k}H_{k}(x) (2y)^{(n-k)}
 &&= 2^{-\frac n 2}\cdot\sum_{k=0}^n \binom{n}{k} H_{n-k}\left(x\sqrt 2\right) H_k\left(y\sqrt 2\right).

\end{align}</math> These umbral identities are self-evident and included in the differential operator representation detailed below, <math display="block">\begin{align}

\mathit{He}_n(x) &= e^{-\frac{D^2}{2}} x^n, \\
H_n(x) &= 2^n e^{-\frac{D^2}{4}} x^n.

\end{align}</math>

In consequence, for the Шаблон:Mvarth derivatives the following relations hold: <math display="block">\begin{align}

\mathit{He}_n^{(m)}(x) &= \frac{n!}{(n-m)!} \mathit{He}_{n-m}(x)
 &&= m! \binom{n}{m} \mathit{He}_{n-m}(x), \\
H_n^{(m)}(x) &= 2^m \frac{n!}{(n-m)!} H_{n-m}(x)
 &&= 2^m m! \binom{n}{m} H_{n-m}(x).

\end{align}</math>

It follows that the Hermite polynomials also satisfy the recurrence relation <math display="block">\begin{align}

\mathit{He}_{n+1}(x) &= x\mathit{He}_n(x) - n\mathit{He}_{n-1}(x), \\
H_{n+1}(x) &= 2xH_n(x) - 2nH_{n-1}(x).

\end{align}</math>

These last relations, together with the initial polynomials Шаблон:Math and Шаблон:Math, can be used in practice to compute the polynomials quickly.

Turán's inequalities are <math display="block">\mathit{H}_n(x)^2 - \mathit{H}_{n-1}(x) \mathit{H}_{n+1}(x) = (n-1)! \sum_{i=0}^{n-1} \frac{2^{n-i}}{i!}\mathit{H}_i(x)^2 > 0.</math>

Moreover, the following multiplication theorem holds: <math display="block">\begin{align}

H_n(\gamma x) &= \sum_{i=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \gamma^{n-2i}(\gamma^2 - 1)^i \binom{n}{2i} \frac{(2i)!}{i!} H_{n-2i}(x), \\
\mathit{He}_n(\gamma x) &= \sum_{i=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \gamma^{n-2i}(\gamma^2 - 1)^i \binom{n}{2i} \frac{(2i)!}{i!}2^{-i} \mathit{He}_{n-2i}(x).

\end{align}</math>

Explicit expression

The physicist's Hermite polynomials can be written explicitly as <math display="block">H_n(x) = \begin{cases} \displaystyle n! \sum_{l = 0}^{\frac{n}{2}} \frac{(-1)^{\tfrac{n}{2} - l}}{(2l)! \left(\tfrac{n}{2} - l \right)!} (2x)^{2l} & \text{for even } n, \\ \displaystyle n! \sum_{l = 0}^{\frac{n-1}{2}} \frac{(-1)^{\frac{n-1}{2} - l}}{(2l + 1)! \left (\frac{n-1}{2} - l \right )!} (2x)^{2l + 1} & \text{for odd } n. \end{cases}</math>

These two equations may be combined into one using the floor function: <math display="block">H_n(x) = n! \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{(-1)^m}{m!(n - 2m)!} (2x)^{n - 2m}.</math>

The probabilist's Hermite polynomials Шаблон:Mvar have similar formulas, which may be obtained from these by replacing the power of Шаблон:Math with the corresponding power of Шаблон:Math and multiplying the entire sum by Шаблон:Math: <math display="block">He_n(x) = n! \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{(-1)^m}{m!(n - 2m)!} \frac{x^{n - 2m}}{2^m}.</math>

Inverse explicit expression

The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials Шаблон:Mvar are <math display="block">x^n = n! \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{1}{2^m m!(n-2m)!} He_{n-2m}(x).</math>

The corresponding expressions for the physicist's Hermite polynomials Шаблон:Mvar follow directly by properly scaling this: <math display="block">x^n = \frac{n!}{2^n} \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{1}{m!(n-2m)! } H_{n-2m}(x).</math>

Generating function

The Hermite polynomials are given by the exponential generating function <math display="block">\begin{align}

e^{xt - \frac12 t^2} &= \sum_{n=0}^\infty \mathit{He}_n(x) \frac{t^n}{n!}, \\
e^{2xt - t^2} &= \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}.

\end{align}</math>

This equality is valid for all complex values of Шаблон:Mvar and Шаблон:Mvar, and can be obtained by writing the Taylor expansion at Шаблон:Mvar of the entire function Шаблон:Math (in the physicist's case). One can also derive the (physicist's) generating function by using Cauchy's integral formula to write the Hermite polynomials as <math display="block">H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} = (-1)^n e^{x^2} \frac{n!}{2\pi i} \oint_\gamma \frac{e^{-z^2}}{(z-x)^{n+1}} \,dz.</math>

Using this in the sum <math display="block">\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.

Expected values

If Шаблон:Mvar is a random variable with a normal distribution with standard deviation 1 and expected value Шаблон:Mvar, then <math display="block">\operatorname{\mathbb E}\left[\mathit{He}_n(X)\right] = \mu^n.</math>

The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices: <math display="block">\operatorname{\mathbb E}\left[X^{2n}\right] = (-1)^n \mathit{He}_{2n}(0) = (2n-1)!!,</math> where Шаблон:Math is the double factorial. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments: <math display="block">\mathit{He}_n(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty (x + iy)^n e^{-\frac{y^2}{2}} \,dy.</math>

Asymptotic expansion

Asymptotically, as Шаблон:Math, the expansion <math display="block">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}- \frac{n\pi}{2} \right)</math> holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude: <math display="block">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}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}=\frac{2 \Gamma(n)}{\Gamma\left(\frac{n}2\right)} \cos \left(x \sqrt{2 n}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14},</math> which, using Stirling's approximation, can be further simplified, in the limit, to <math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \left(\frac{2n}{e}\right)^{\frac{n}{2}} \sqrt{2} \cos \left(x \sqrt{2n}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}.</math>

This expansion is needed to resolve the wavefunction of a quantum harmonic oscillator such that it agrees with the classical approximation in the limit of the correspondence principle.

A better approximation, which accounts for the variation in frequency, is given by <math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \left(\frac{2n}{e}\right)^{\frac{n}{2}} \sqrt{2} \cos \left(x \sqrt{2n+1-\frac{x^2}{3}}- \frac {n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}.</math>

A finer approximation, which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution <math display="block">x = \sqrt{2n + 1}\cos(\varphi), \quad 0 < \varepsilon \leq \varphi \leq \pi - \varepsilon,</math> with which one has the uniform approximation <math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) = 2^{\frac{n}{2}+\frac14}\sqrt{n!}(\pi n)^{-\frac14}(\sin \varphi)^{-\frac12} \cdot \left(\sin\left(\frac{3\pi}{4} + \left(\frac{n}{2} + \frac{1}{4}\right)\left(\sin 2\varphi-2\varphi\right) \right)+O\left(n^{-1}\right) \right).</math>

Similar approximations hold for the monotonic and transition regions. Specifically, if <math display="block">x = \sqrt{2n+1} \cosh(\varphi), \quad 0 < \varepsilon \leq \varphi \leq \omega < \infty,</math> then <math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) = 2^{\frac{n}{2}-\frac34}\sqrt{n!}(\pi n)^{-\frac14}(\sinh \varphi)^{-\frac12} \cdot e^{\left(\frac{n}{2}+\frac{1}{4}\right)\left(2\varphi-\sinh 2\varphi\right)}\left(1+O\left(n^{-1}\right) \right),</math> while for <math display="block">x = \sqrt{2n + 1} + t</math> with Шаблон:Mvar complex and bounded, the approximation is <math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) =\pi^{\frac14}2^{\frac{n}{2}+\frac14}\sqrt{n!}\, n^{-\frac{1}{12}}\left( \operatorname{Ai}\left(2^{\frac12}n^{\frac16}t\right)+ O\left(n^{-\frac23}\right) \right),</math> where Шаблон:Math is the Airy function of the first kind.

Special values

The physicist's Hermite polynomials evaluated at zero argument Шаблон:Math are called Hermite numbers.

<math display="block">H_n(0) = \begin{cases}

0 & \text{for odd }n, \\
(-2)^\frac{n}{2} (n-1)!! & \text{for even }n,

\end{cases}</math> which satisfy the recursion relation Шаблон:Math.

In terms of the probabilist's polynomials this translates to <math display="block">He_n(0) = \begin{cases}

0 & \text{for odd }n, \\
(-1)^\frac{n}{2} (n-1)!! & \text{for even }n.

\end{cases}</math>

Relations to other functions

Laguerre polynomials

The Hermite polynomials can be expressed as a special case of the Laguerre polynomials: <math display="block">\begin{align}

H_{2n}(x) &= (-4)^n n! L_n^{\left(-\frac12\right)}(x^2)
 &&= 4^n n! \sum_{k=0}^n (-1)^{n-k} \binom{n-\frac12}{n-k} \frac{x^{2k}}{k!}, \\
H_{2n+1}(x) &= 2(-4)^n n! x L_n^{\left(\frac12\right)}(x^2)
 &&= 2\cdot 4^n n!\sum_{k=0}^n (-1)^{n-k} \binom{n+\frac12}{n-k} \frac{x^{2k+1}}{k!}.

\end{align}</math>

Relation to confluent hypergeometric functions

The physicist's Hermite polynomials can be expressed as a special case of the parabolic cylinder functions: <math display="block">H_n(x) = 2^n U\left(-\tfrac12 n, \tfrac12, x^2\right)</math> in the right half-plane, where Шаблон:Math is Tricomi's confluent hypergeometric function. Similarly, <math display="block">\begin{align}

H_{2n}(x) &= (-1)^n \frac{(2n)!}{n!} \,_1F_1\big(-n, \tfrac12; x^2\big), \\
H_{2n+1}(x) &= (-1)^n \frac{(2n+1)!}{n!}\,2x \,_1F_1\big(-n, \tfrac32; x^2\big),

\end{align}</math> where Шаблон:Math is Kummer's confluent hypergeometric function.

Hermite polynomial expansion

Similar to Taylor expansion, some functions are expressible as an infinite sum of Hermite polynomials. Specifically, if <math>\int e^{-x^2}f(x)^2 dx < \infty</math>, then it has an expansion in the physicist's Hermite polynomials.

Given such <math>f</math>, the partial sums of the Hermite expansion of <math>f</math> converges to in the <math>L^p</math> norm if and only if <math>4 / 3<p<4</math>.<math display="block">x^n = \frac{n!}{2^n} \,\sum_{k= 0}^{\left\lfloor n/2 \right\rfloor} \frac{1}{k! \,(n-2k)!} \, H_{n-2k} (x) = n! \sum_{k= 0}^{\left\lfloor n/2 \right\rfloor} \frac{1}{k! \,2^k \,(n-2k)!} \, He_{n-2k} (x) , \qquad n \in \mathbb{Z}_{+} . </math><math display="block">e^{ax} = e^{a^2 /4} \sum_{n\ge 0} \frac{a^n}{n! \,2^n} \, H_n (x) , \qquad a\in \mathbb{C}, \quad x\in \mathbb{R} .</math><math display="block">e^{-a^2 x^2} = \sum_{n\ge 0} \frac{(-1)^n a^{2n}}{n! \left( 1 + a^2 \right)^{n + 1/2} 2^{2n}}\, H_{2n} (x) .</math><math display="block">\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \mathrm{~d} t=\frac{1}{\sqrt{2 \pi}} \sum_{k \geq 0} \frac{(-1)^k}{k !(2 k+1) 2^{3 k}} H_{2 k}(x) .</math><math display="block">\cosh (2x) = e \sum_{k\ge 0} \frac{1}{(2k)!}\, H_{2k} (x) , \qquad \sinh (2x) = e \sum_{k\ge 0} \frac{1}{(2k+1)!} \, H_{2k+1} (x) .</math><math display="block">\cos (x) = e^{-1/4} \,\sum_{k\ge 0} \frac{(-1)^k}{2^{2k} \, (2k)!} \, H_{2k} (x) \quad \sin (x) = e^{-1/4} \,\sum_{k\ge 0} \frac{(-1)^k}{2^{2k+1} \, (2k+1)!} \, H_{2k+1} (x) </math>

Differential-operator representation

The probabilist's Hermite polynomials satisfy the identity <math display="block">\mathit{He}_n(x) = e^{-\frac{D^2}{2}}x^n,</math> where Шаблон:Mvar represents differentiation with respect to Шаблон:Mvar, and the 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 Шаблон:Math can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients of Шаблон:Math that can be used to quickly compute these polynomials.

Since the formal expression for the Weierstrass transform Шаблон:Mvar is Шаблон:Math, we see that the Weierstrass transform of Шаблон:Math is Шаблон:Math. Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series.

The existence of some formal power series Шаблон:Math with nonzero constant coefficient, such that Шаблон:Math, is another equivalent to the statement that these polynomials form an Appell sequence. Since they are an Appell sequence, they are a fortiori a Sheffer sequence.

Contour-integral representation

From the generating-function representation above, we see that the Hermite polynomials have a representation in terms of a contour integral, as <math display="block">\begin{align}

\mathit{He}_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{tx-\frac{t^2}{2}}}{t^{n+1}}\,dt, \\
H_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{2tx-t^2}}{t^{n+1}}\,dt,

\end{align}</math> with the contour encircling the origin.

Generalizations

The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is <math display="block">\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}},</math> which has expected value 0 and variance 1.

Scaling, one may analogously speak of generalized Hermite polynomials <math display="block">\mathit{He}_n^{[\alpha]}(x)</math> of variance Шаблон:Mvar, where Шаблон:Mvar is any positive number. These are then orthogonal with respect to the normal probability distribution whose density function is <math display="block">(2\pi\alpha)^{-\frac12} e^{-\frac{x^2}{2\alpha}}.</math> They are given by <math display="block">\mathit{He}_n^{[\alpha]}(x) = \alpha^{\frac{n}{2}}\mathit{He}_n\left(\frac{x}{\sqrt{\alpha}}\right) = \left(\frac{\alpha}{2}\right)^{\frac{n}{2}} H_n\left( \frac{x}{\sqrt{2 \alpha}}\right) = e^{-\frac{\alpha D^2}{2}} \left(x^n\right).</math>

Now, if <math display="block">\mathit{He}_n^{[\alpha]}(x) = \sum_{k=0}^n h^{[\alpha]}_{n,k} x^k,</math> then the polynomial sequence whose Шаблон:Mvarth term is <math display="block">\left(\mathit{He}_n^{[\alpha]} \circ \mathit{He}^{[\beta]}\right)(x) \equiv \sum_{k=0}^n h^{[\alpha]}_{n,k}\,\mathit{He}_k^{[\beta]}(x)</math> is called the umbral composition of the two polynomial sequences. It can be shown to satisfy the identities <math display="block">\left(\mathit{He}_n^{[\alpha]} \circ \mathit{He}^{[\beta]}\right)(x) = \mathit{He}_n^{[\alpha+\beta]}(x)</math> and <math display="block">\mathit{He}_n^{[\alpha+\beta]}(x + y) = \sum_{k=0}^n \binom{n}{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 known as a cross-sequence. (See the above section on Appell sequences and on the differential-operator representation, which leads to a ready derivation of it. This binomial type identity, for Шаблон:Math, has already been encountered in the above section on #Recursion relations.)

"Negative variance"

Since polynomial sequences form a group under the operation of umbral composition, one may denote by <math display="block">\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 Шаблон:Math, the coefficients of <math>\mathit{He}_n^{[-\alpha]}(x)</math> are just the absolute values of the corresponding coefficients of <math>\mathit{He}_n^{[\alpha]}(x)</math>.

These arise as moments of normal probability distributions: The Шаблон:Mvarth moment of the normal distribution with expected value Шаблон:Mvar and variance Шаблон:Math is <math display="block">E[X^n] = \mathit{He}_n^{[-\sigma^2]}(\mu),</math> where Шаблон:Mvar is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that <math display="block">\sum_{k=0}^n \binom{n}{k} \mathit{He}_k^{[\alpha]}(x) \mathit{He}_{n-k}^{[-\alpha]}(y) = \mathit{He}_n^{[0]}(x + y) = (x + y)^n.</math>

Hermite functions

Definition

One can define the Hermite functions (often called Hermite-Gaussian functions) from the physicist's polynomials: <math display="block">\psi_n(x) = \left (2^n n! \sqrt{\pi} \right )^{-\frac12} e^{-\frac{x^2}{2}} H_n(x) = (-1)^n \left (2^n n! \sqrt{\pi} \right)^{-\frac12} e^{\frac{x^2}{2}}\frac{d^n}{dx^n} e^{-x^2}.</math> Thus, <math display="block">\sqrt{2(n+1)}~~\psi_{n+1}(x)= \left ( x- {d\over dx}\right ) \psi_n(x).</math>

Since these functions contain the square root of the weight function and have been scaled appropriately, they are orthonormal: <math display="block">\int_{-\infty}^\infty \psi_n(x) \psi_m(x) \,dx = \delta_{nm},</math> and they form an orthonormal basis of Шаблон:Math. This fact is equivalent to the corresponding statement for Hermite polynomials (see above).

The Hermite functions are closely related to the Whittaker function Шаблон:Harv Шаблон:Math: <math display="block">D_n(z) = \left(n! \sqrt{\pi}\right)^{\frac12} \psi_n\left(\frac{z}{\sqrt 2}\right) = (-1)^n e^\frac{z^2}{4} \frac{d^n}{dz^n} e^\frac{-z^2}{2}</math> and thereby to other parabolic cylinder functions.

The Hermite functions satisfy the differential equation <math display="block">\psi_n(x) + \left(2n + 1 - x^2\right) \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.

Файл:Herm5.svg

Hermite functions: 0 (blue, solid), 1 (orange, dashed), 2 (green, dot-dashed), 3 (red, dotted), 4 (purple, solid), and 5 (brown, dashed)

<math display="block">\begin{align}

\psi_0(x) &= \pi^{-\frac14} \, e^{-\frac12 x^2}, \\
\psi_1(x) &= \sqrt{2} \, \pi^{-\frac14} \, x \, e^{-\frac12 x^2}, \\
\psi_2(x) &= \left(\sqrt{2} \, \pi^{\frac14}\right)^{-1} \, \left(2x^2-1\right) \, e^{-\frac12 x^2}, \\
\psi_3(x) &= \left(\sqrt{3} \, \pi^{\frac14}\right)^{-1} \, \left(2x^3-3x\right) \, e^{-\frac12 x^2}, \\
\psi_4(x) &= \left(2 \sqrt{6} \, \pi^{\frac14}\right)^{-1} \, \left(4x^4-12x^2+3\right) \, e^{-\frac12 x^2}, \\
\psi_5(x) &= \left(2 \sqrt{15} \, \pi^{\frac14}\right)^{-1} \, \left(4x^5-20x^3+15x\right) \, e^{-\frac12 x^2}.

\end{align}</math>

Файл:Herm50.svg

Hermite functions: 0 (blue, solid), 2 (orange, dashed), 4 (green, dot-dashed), and 50 (red, solid)

Recursion relation

Following recursion relations of Hermite polynomials, the Hermite functions obey <math display="block">\psi_n'(x) = \sqrt{\frac{n}{2}}\,\psi_{n-1}(x) - \sqrt{\frac{n+1}{2}}\psi_{n+1}(x)</math> and <math display="block">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 Шаблон:Mvarth derivatives for any positive integer Шаблон:Mvar leads to <math display="block">\psi_n^{(m)}(x) = \sum_{k=0}^m \binom{m}{k} (-1)^k 2^\frac{m-k}{2} \sqrt{\frac{n!}{(n-m+k)!}} \psi_{n-m+k}(x) \mathit{He}_k(x).</math>

This formula can be used in connection with the recurrence relations for Шаблон:Math and Шаблон:Math to calculate any derivative of the Hermite functions efficiently.

Cramér's inequality

For real Шаблон:Mvar, the Hermite functions satisfy the following bound due to Harald Cramér and Jack Indritz: <math display="block"> \bigl|\psi_n(x)\bigr| \le \pi^{-\frac14}.</math>

Hermite functions as eigenfunctions of the Fourier transform

The Hermite functions Шаблон:Math are a set of eigenfunctions of the continuous Fourier transform Шаблон:Mathcal. To see this, take the physicist's version of the generating function and multiply by Шаблон:Math. This gives <math display="block">e^{-\frac12 x^2 + 2xt - t^2} = \sum_{n=0}^\infty e^{-\frac12 x^2} H_n(x) \frac{t^n}{n!}.</math>

The Fourier transform of the left side is given by <math display="block">\begin{align}

\mathcal{F} \left\{ e^{ -\frac12 x^2 + 2xt - t^2 } \right\}(k)
 &= \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^\infty e^{-ixk}e^{-\frac12 x^2 + 2xt - t^2}\, dx \\
 &= e^{-\frac12 k^2 - 2kit + t^2 } \\
 &= \sum_{n=0}^\infty e^{ -\frac12 k^2 } H_n(k) \frac{(-it)^n}{n!}.

\end{align}</math>

The Fourier transform of the right side is given by <math display="block">\mathcal{F} \left\{ \sum_{n=0}^\infty e^{-\frac12 x^2} H_n(x) \frac {t^n}{n!} \right\} = \sum_{n=0}^\infty \mathcal{F} \left \{ e^{-\frac12 x^2} H_n(x) \right\} \frac{t^n}{n!}.</math>

Equating like powers of Шаблон:Mvar in the transformed versions of the left and right sides finally yields <math display="block">\mathcal{F} \left\{ e^{-\frac12 x^2} H_n(x) \right\} = (-i)^n e^{-\frac12 k^2} H_n(k).</math>

The Hermite functions Шаблон:Math are thus an orthonormal basis of Шаблон:Math, which diagonalizes the Fourier transform operator.

Wigner distributions of Hermite functions

The Wigner distribution function of the Шаблон:Mvarth-order Hermite function is related to the Шаблон:Mvarth-order Laguerre polynomial. The Laguerre polynomials are <math display="block">L_n(x) := \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{k!}x^k,</math> leading to the oscillator Laguerre functions <math display="block">l_n (x) := e^{-\frac{x}{2}} L_n(x).</math> For all natural integers Шаблон:Mvar, it is straightforward to see that <math display="block">W_{\psi_n}(t,f) = (-1)^n l_n \big(4\pi (t^2 + f^2) \big),</math> where the Wigner distribution of a function Шаблон:Math is defined as <math display="block"> W_x(t,f) = \int_{-\infty}^\infty x\left(t + \frac{\tau}{2}\right) \, x\left(t - \frac{\tau}{2}\right)^* \, e^{-2\pi i\tau f} \,d\tau.</math> This is a fundamental result for the quantum harmonic oscillator discovered by Hip Groenewold in 1946 in his PhD thesis. It is the standard paradigm of quantum mechanics in phase space.

There are further relations between the two families of polynomials.

Combinatorial interpretation of coefficients

In the Hermite polynomial Шаблон:Math of variance 1, the absolute value of the coefficient of Шаблон:Math is the number of (unordered) partitions of an Шаблон:Mvar-element set into Шаблон:Mvar singletons and Шаблон:Math (unordered) pairs. Equivalently, it is the number of involutions of an Шаблон:Mvar-element set with precisely Шаблон:Mvar fixed points, or in other words, the number of matchings in the complete graph on Шаблон:Mvar vertices that leave Шаблон:Mvar vertices uncovered (indeed, the Hermite polynomials are the matching polynomials of these graphs). The sum of the absolute values of the coefficients gives the total number of partitions into singletons and pairs, the so-called telephone numbers

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496,... Шаблон:OEIS.

This combinatorial interpretation can be related to complete exponential Bell polynomials as <math display="block">\mathit{He}_n(x) = B_n(x, -1, 0, \ldots, 0),</math> where Шаблон:Math for all Шаблон:Math.

These numbers may also be expressed as a special value of the Hermite polynomials: <math display="block">T(n) = \frac{\mathit{He}_n(i)}{i^n}.</math>

Completeness relation

The Christoffel–Darboux formula for Hermite polynomials reads <math display="block">\sum_{k=0}^n \frac{H_k(x) H_k(y)}{k!2^k} = \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 completeness identity for the above Hermite functions holds in the sense of distributions: <math display="block">\sum_{n=0}^\infty \psi_n(x) \psi_n(y) = \delta(x - y),</math> where Шаблон:Mvar is the Dirac delta function, Шаблон:Math the Hermite functions, and Шаблон:Math represents the Lebesgue measure on the line Шаблон:Math in Шаблон:Math, normalized so that its projection on the horizontal axis is the usual Lebesgue measure.

This distributional identity follows Шаблон:Harvtxt by taking Шаблон:Math in Mehler's formula, valid when Шаблон:Math: <math display="block">E(x, y; u) := \sum_{n=0}^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac{1}{\sqrt{\pi (1 - u^2)}} \, \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 a separable kernel, <math display="block">\sum_{n=0}^\infty \frac{H_n(x) H_n(y)}{n!} \left(\frac u 2\right)^n = \frac{1}{\sqrt{1 - u^2}} e^{\frac{2u}{1 + u}xy - \frac{u^2}{1 - u^2}(x - y)^2}.</math>

The function Шаблон:Math is the bivariate Gaussian probability density on Шаблон:Math, which is, when Шаблон:Mvar is close to 1, very concentrated around the line Шаблон:Math, and very spread out on that line. It follows that <math display="block">\sum_{n=0}^\infty u^n \langle f, \psi_n \rangle \langle \psi_n, g \rangle = \iint E(x, y; u) f(x) \overline{g(y)} \,dx \,dy \to \int f(x) \overline{g(x)} \,dx = \langle f, g \rangle</math> when Шаблон:Math and Шаблон:Math are continuous and compactly supported.

This yields that Шаблон:Mvar can be expressed in Hermite functions as the sum of a series of vectors in Шаблон:Math, namely, <math display="block">f = \sum_{n=0}^\infty \langle f, \psi_n \rangle \psi_n.</math>

In order to prove the above equality for Шаблон:Math, the Fourier transform of Gaussian functions is used repeatedly: <math display="block">\rho \sqrt{\pi} e^{-\frac{\rho^2 x^2}{4}} = \int e^{isx - \frac{s^2}{\rho^2}} \,ds \quad \text{for }\rho > 0.</math>

The Hermite polynomial is then represented as <math display="block"> H_n(x) = (-1)^n e^{x^2} \frac {d^n}{dx^n} \left( \frac {1}{2\sqrt{\pi}} \int e^{isx - \frac{s^2}{4}} \,ds \right) = (-1)^n e^{x^2}\frac{1}{2\sqrt{\pi}} \int (is)^n e^{isx - \frac{s^2}{4}} \,ds.</math>

With this representation for Шаблон:Math and Шаблон:Math, it is evident that <math display="block">\begin{align}

E(x, y; u) &= \sum_{n=0}^\infty \frac{u^n}{2^n n! \sqrt{\pi}} \, H_n(x) H_n(y) e^{-\frac{x^2+y^2}{2}} \\
 &= \frac{e^{\frac{x^2+y^2}{2}}}{4\pi\sqrt{\pi}}\iint\left( \sum_{n=0}^\infty \frac{1}{2^n n!} (-ust)^n \right ) e^{isx+ity - \frac{s^2}{4} - \frac{t^2}{4}}\, ds\,dt \\
 & =\frac{e^{\frac{x^2+y^2}{2}}}{4\pi\sqrt{\pi}}\iint e^{-\frac{ust}{2}} \, e^{isx+ity - \frac{s^2}{4} - \frac{t^2}{4}}\, ds\,dt,

\end{align}</math> and this yields the desired resolution of the identity result, using again the Fourier transform of Gaussian kernels under the substitution <math display="block">s = \frac{\sigma + \tau}{\sqrt 2}, \quad t = \frac{\sigma - \tau}{\sqrt 2}.</math>

See also

• Hermite transform
• Legendre polynomials
• Mehler kernel
• Parabolic cylinder function
• Romanovski polynomials
• Turán's inequalities

References

• Шаблон:Abramowitz Stegun ref
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Eom
• Шаблон:Dlmf
• Шаблон:Citation Oeuvres complètes 12, pp.357-412, English translation Шаблон:Webarchive.
• Шаблон:Citation - 2000 references of Bibliography on Hermite polynomials.
• Шаблон:Eom
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation

External links

• Шаблон:Commons category-inline
• Шаблон:MathWorld
• GNU Scientific Library — includes C version of Hermite polynomials, functions, their derivatives and zeros (see also GNU Scientific Library)