Template:Short description Template:About Template:Use American English
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,<ref>Template:Cite journal</ref><ref>Template:Citation Collected in Œuvres complètes VII.</ref> though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859.<ref>Template:Cite journal Collected in Œuvres I, 501–508.</ref> Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new.<ref>Template:Cite journal Collected in Œuvres II, 293–308.</ref> They were consequently not new, although Hermite was the first to define the multidimensional polynomials.
DefinitionEdit
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">\operatorname{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">\operatorname{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} \operatorname{He}_n\left(\sqrt{2} \,x\right), \quad \operatorname{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 Template:Mvar and Template:Mvar is that used in the standard references.<ref>Template:Harvs and Abramowitz & Stegun.</ref> The polynomials Template:Mvar are sometimes denoted by Template: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.
- The first eleven probabilist's Hermite polynomials are: <math display="block">\begin{align}
\operatorname{He}_0(x) &= 1, \\ \operatorname{He}_1(x) &= x, \\ \operatorname{He}_2(x) &= x^2 - 1, \\ \operatorname{He}_3(x) &= x^3 - 3x, \\ \operatorname{He}_4(x) &= x^4 - 6x^2 + 3, \\ \operatorname{He}_5(x) &= x^5 - 10x^3 + 15x, \\ \operatorname{He}_6(x) &= x^6 - 15x^4 + 45x^2 - 15, \\ \operatorname{He}_7(x) &= x^7 - 21x^5 + 105x^3 - 105x, \\ \operatorname{He}_8(x) &= x^8 - 28x^6 + 210x^4 - 420x^2 + 105, \\ \operatorname{He}_9(x) &= x^9 - 36x^7 + 378x^5 - 1260x^3 + 945x, \\ \operatorname{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>
| physicist's | probabilist's | |
|---|---|---|
| symbol | <math>H_n</math> | <math>\operatorname{He}_n</math> |
| head coefficient | <math>2^n</math> | <math>1</math> |
| differential operator | <math>(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}</math> | <math>(-1)^n e^{\frac{x^2}{2}}\frac{d^n}{dx^n}e^{-\frac{x^2}{2}}</math> |
| orthogonal to | <math>e^{-x^2}</math> | <math>e^{-\frac 12 x^2}</math> |
| inner product | <math>\int H_m(x) H_n(x) \frac{e^{-x^2}}{\sqrt{\pi}}dx = 2^n n! \delta_{mn}</math> | <math>\int \operatorname{He}_m(x) \operatorname{He}_n(x)\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}} \,dx = n!\, \delta_{nm}, </math> |
| generating function | <math>e^{2xt - t^2} = \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}</math> | <math>e^{xt - \frac12 t^2} = \sum_{n=0}^\infty \operatorname{He}_n(x) \frac{t^n}{n!} </math> |
| Rodrigues' formula | <math>\left(2x - \frac{d}{dx} \right)^n \cdot 1 </math> | <math>\left(x - \frac{d}{dx} \right)^n \cdot 1 </math> |
| recurrence relation | <math>H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)</math> | <math>\operatorname{He}_{n+1}(x) = x\operatorname{He}_n(x) - n\operatorname{He}_{n-1}(x)</math> |
- Hermite poly.svg
The first six probabilist's Hermite polynomials <math>\operatorname{He}_n(x)</math>
- Hermite poly phys.svg
The first six physicist's Hermite polynomials <math>H_n(x)</math>
PropertiesEdit
The Template:Mvarth-order Hermite polynomial is a polynomial of degree Template:Mvar. The probabilist's version Template:Mvar has leading coefficient 1, while the physicist's version Template:Mvar has leading coefficient Template:Math.
SymmetryEdit
From the Rodrigues formulae given above, we can see that Template:Math and Template:Math are even or odd functions depending on Template:Mvar: <math display="block">H_n(-x)=(-1)^nH_n(x),\quad \operatorname{He}_n(-x)=(-1)^n\operatorname{He}_n(x).</math>
OrthogonalityEdit
Template:Math and Template:Math are Template:Mvarth-degree polynomials for Template: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 }\operatorname{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 \operatorname{He}_m(x) \operatorname{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.
CompletenessEdit
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 Template:Math defined in the preceding section.
An orthogonal basis for Template:Math is a complete orthogonal system. For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function Template: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 Template:Mvar satisfies <math display="block">\int_{-\infty}^\infty f(x) x^n e^{- x^2} \,dx = 0</math> for every Template:Math, then Template: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 Template:Math for every real Template:Mvar means that the Fourier transform of Template:Math is 0, hence Template: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 Template:Math consists in introducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis for Template:Math.
Hermite's differential equationEdit
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 Template:Mvar is a constant. Imposing the boundary condition that Template:Mvar should be polynomially bounded at infinity, the equation has solutions only if Template:Mvar is a non-negative integer, and the solution is uniquely given by <math>u(x) = C_1 \operatorname{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>\operatorname{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 Template: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 Template:Mvar. An explicit formula of Hermite polynomials in terms of contour integrals Template:Harv is also possible.
Recurrence relationEdit
The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation <math display="block">\operatorname{He}_{n+1}(x) = x \operatorname{He}_n(x) - \operatorname{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 Template:Math, Template:Math, Template: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 Template:Math, Template:Math, Template:Math.
The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity <math display="block">\begin{align}
\operatorname{He}_n'(x) &= n\operatorname{He}_{n-1}(x), \\
H_n'(x) &= 2nH_{n-1}(x).
\end{align}</math>
An integral recurrence that is deduced and demonstrated in <ref>Hurtado Benavides, Miguel Ángel. (2020). De las sumas de potencias a las sucesiones de Appell y su caracterización a través de funcionales. [Tesis de maestría]. Universidad Sergio Arboleda.</ref> is as follows: <math display="block">\operatorname{He}_{n+1}(x) = (n+1)\int_0^x \operatorname{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}
\operatorname{He}_n(x+y) &= \sum_{k=0}^n \binom{n}{k}x^{n-k} \operatorname{He}_{k}(y)
&&= 2^{-\frac n 2} \sum_{k=0}^n \binom{n}{k} \operatorname{He}_{n-k}\left(x\sqrt 2\right) \operatorname{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}
\operatorname{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 Template:Mvarth derivatives the following relations hold: <math display="block">\begin{align}
\operatorname{He}_n^{(m)}(x) &= \frac{n!}{(n-m)!} \operatorname{He}_{n-m}(x)
&&= m! \binom{n}{m} \operatorname{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}
\operatorname{He}_{n+1}(x) &= x\operatorname{He}_n(x) - n\operatorname{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 Template:Math and Template: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), \\
\operatorname{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} \operatorname{He}_{n-2i}(x).
\end{align}</math>
Explicit expressionEdit
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 Template:Mvar have similar formulas, which may be obtained from these by replacing the power of Template:Math with the corresponding power of Template:Math and multiplying the entire sum by Template:Math: <math display="block">\operatorname{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 expressionEdit
The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials Template: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)!} \operatorname{He}_{n-2m}(x).</math>
The corresponding expressions for the physicist's Hermite polynomials Template:Mvar follow directly by properly scaling this:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <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 functionEdit
The Hermite polynomials are given by the exponential generating function <math display="block">\begin{align}
e^{xt - \frac12 t^2} &= \sum_{n=0}^\infty \operatorname{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 Template:Mvar and Template:Mvar, and can be obtained by writing the Taylor expansion at Template:Mvar of the entire function Template: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.
A slight generalization states<ref>(Rainville 1971), p. 198</ref><math display="block">e^{2 x t-t^2} H_k(x-t) = \sum_{n=0}^{\infty} \frac{H_{n+k}(x) t^n}{n!}</math>
Expected valuesEdit
If Template:Mvar is a random variable with a normal distribution with standard deviation 1 and expected value Template:Mvar, then <math display="block">\operatorname{\mathbb E}\left[\operatorname{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 \operatorname{He}_{2n}(0) = (2n-1)!!,</math> where Template: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">\operatorname{He}_n(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty (x + iy)^n e^{-\frac{y^2}{2}} \,dy.</math>
Integral representationsEdit
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}
\operatorname{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.
Using the Fourier transform of the gaussian <math>e^{-x^2}=\frac{1}{\sqrt{ \pi}} \int e^{-t^2+2 i x t} dt </math>, we have<math display="block">\begin{align} H_n(x) &= (-1)^n e^{x^2} \frac {d^n}{dx^n} e^{-x^2} = \frac{(-2 i)^n e^{x^2}}{\sqrt{\pi}} \int t^n e^{-t^2+2 i x t} d t \\
\operatorname{He}_n(x) &= \frac{(-i)^n e^{x^2/2}}{\sqrt{2\pi}} \int t^n \, e^{-t^2/2 + i x t}\, dt.
\end{align}</math>
Other propertiesEdit
The addition theorem, or the summation theorem, states that<ref name=":1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite book</ref>Template:Pg<math display="block">\frac{\left(\sum_{k=1}^r a_k^2\right)^{\frac{n}{2}}}{n!} H_n\left(\frac{\sum_{k=1}^r a_k x_k}{\sqrt{\sum_{k=1}^r a_k^2}}\right)=\sum_{m_1+m_2+\ldots+m_r=n, m_i \geq 0} \prod_{k=1}^r\left\{\frac{a_k^{m_k}}{m_{k}!} H_{m_k}\left(x_k\right)\right\} </math>for any nonzero vector <math>a_{1:r}</math>.
The multiplication theorem states that<ref name=":1" /><math display="block">H_{n}\left(\lambda x\right)=\lambda^{n}\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{{\left(-n\right)_{2\ell}}}{\ell!}(1-\lambda^{-2})^{\ell}H_{n-2\ell}\left(x\right)</math>for any nonzero <math>\lambda</math>.
Feldheim formula<ref name=":2">Feldheim, Ervin. "Développements en série de polynômes d’Hermite et de Laguerrea l’aide des transformations de Gauss et de Hankel." Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 435 (1940). Part I, II, III</ref>Template:Pg<math display="block">\begin{aligned} \frac{1}{\sqrt{a \pi}} & \int_{-\infty}^{+\infty} e^{-\frac{x^2}{a}} H_m\left(\frac{x+y}{\lambda}\right) H_n\left(\frac{x+z}{\mu}\right) d x \\ & = \left(1-\frac{a}{\lambda^2}\right)^{\frac{m}{2}}\left(1-\frac{a}{\mu^2}\right)^{\frac{n}{2}} \sum_{r=0}^{\min (m, n)} r!\binom{m}{r}\binom{n}{r} \left(\frac{2 a}{\sqrt{\left(\lambda^2-a\right)\left(\mu^2-a\right)}}\right)^r H_{m-r}\left(\frac{y}{\sqrt{\lambda^2-a}}\right) H_{n-r}\left(\frac{z}{\sqrt{\mu^2-a}}\right) \end{aligned}</math>where <math>a \in \mathbb C</math> has a positive real part. As a special case,<ref name=":2" />Template:Pg<math display="block">\frac{1}{\sqrt{\pi}} \int_{-\infty}^{+\infty} e^{-t^2} H_m(t \sin \theta+v \cos \theta) H_n(t \cos \theta-v \sin \theta) d t =(-1)^n \cos ^m \theta \sin ^n \theta H_{m+n}(v)</math>
Asymptotic expansionEdit
Asymptotically, as Template:Math, the expansion<ref>Template:Harvnb, 13.6.38 and 13.5.16.</ref> <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{\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,<ref>Template:Harvnb</ref> 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 Template: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 Template:Math is the Airy function of the first kind.
Special valuesEdit
The physicist's Hermite polynomials evaluated at zero argument Template: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 Template:Math. Equivalently, <math>H_{2n}(0) = (-2)^n (2n-1)!!</math>.
In terms of the probabilist's polynomials this translates to <math display="block">\operatorname{He}_n(0) = \begin{cases}
0 & \text{for odd }n, \\
(-1)^\frac{n}{2} (n-1)!! & \text{for even }n.
\end{cases}</math>
Kibble–Slepian formulaEdit
Let <math display="inline">M</math> be a real <math display="inline">n\times n</math> symmetric matrix, then the Kibble–Slepian formula states that<math display="block">\det(I+M)^{-\frac 12 } e^{x^T M (I+M)^{-1}x} = \sum_K \left[\prod_{1\leq i \leq j \leq n} \frac{(M_{ij}/2)^{k_{ij}}}{k_{ij}!}\right] 2^{-tr(K)} H_{k_1}(x_1) \cdots H_{k_n}(x_n)</math> where <math display="inline">\sum_K</math> is the <math>\frac{n(n+1)}{2}</math>-fold summation over all <math display="inline">n \times n</math> symmetric matrices with non-negative integer entries, <math>tr(K)</math> is the trace of <math>K</math>, and <math display="inline">k_i</math> is defined as <math display="inline">k_{ii} + \sum_{j=1}^n k_{ij}</math>. This gives Mehler's formula when <math>M = \begin{bmatrix} 0 & u \\ u & 0\end{bmatrix}</math>.
Equivalently stated, if <math display="inline">T</math> is a positive semidefinite matrix, then set <math display="inline">M = -T(I+T)^{-1}</math>, we have <math display="inline">M(I+M)^{-1} = -T</math>, so <math display="block"> e^{-x^T T x} = \det(I+T)^{-\frac 12} \sum_K \left[\prod_{1\leq i \leq j \leq n} \frac{(M_{ij}/2)^{k_{ij}}}{k_{ij}!}\right] 2^{-tr(K)} H_{k_1}(x_1) \dots H_{k_n}(x_n) </math>Equivalently stated in a form closer to the boson quantum mechanics of the harmonic oscillator:<ref name=":0">Template:Cite journal</ref><math display="block"> \pi^{-n/4}\det(I+M)^{-\frac 12 }e^{- \frac 12 x^T(I-M)(I+M)^{-1} x}= \sum_K\left[\prod_{1 \leq i \leq j \leq n} M_{i j}^{k_{i j}} / k_{i j}!\right]\left[\prod_{1 \leq i \leq n} k_{i}!\right]^{1 / 2} 2^{-\operatorname{tr} K} \psi_{k_1}\left(x_1\right) \cdots \psi_{k_n}\left(x_n\right) . </math> where each <math display="inline">\psi_n(x)</math> is the <math display="inline">n</math>-th eigenfunction of the harmonic oscillator, defined as <math display="block">\psi_n(x) := \frac{1}{\sqrt{2^n n!}}\left(\frac{1}{\pi}\right)^{\frac{1}{4}} e^{-\frac{1}{2} x^2} H_n(x) </math>The Kibble–Slepian formula was proposed by Kibble in 1945<ref>Template:Cite journal</ref> and proven by Slepian in 1972 using Fourier analysis.<ref>Template:Cite journal</ref> Foata gave a combinatorial proof<ref>Template:Cite journal</ref> while Louck gave a proof via boson quantum mechanics.<ref name=":0" /> It has a generalization for complex-argument Hermite polynomials.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
Relations to other functionsEdit
Laguerre polynomialsEdit
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>
Hypergeometric functionsEdit
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 Template: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 Template:Math is Kummer's confluent hypergeometric function.
There is also<ref>DLMF Equation 18.5.13</ref><math display="block">H_{n}\left(x\right)=(2x)^{n}{{}_{2}F_{0}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right).</math>
Limit relationsEdit
The Hermite polynomials can be obtained as the limit of various other polynomials.<ref>DLMF §18.7(iii) Limit Relations</ref>
As a limit of Jacobi polynomials:<math display="block">\lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}P^{(\alpha,\alpha)}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{2^{n}n!}.</math>As a limit of ultraspherical polynomials:<math display="block">\lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}C^{(\lambda)}_{n}\left(\lambda^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{n!}.</math>As a limit of associated Laguerre polynomials:<math display="block">\lim_{\alpha\to\infty}\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}L^{(\alpha)}_{n}\left((2\alpha)^{\frac{1}{2}}x+\alpha\right)=\frac{(-1)^{n}}{n!}H_{n}\left(x\right).</math>
Hermite polynomial expansionEdit
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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
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>.<ref>Template:Cite journal</ref><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)!} \, \operatorname{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} ~dt=\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 representationEdit
The probabilist's Hermite polynomials satisfy the identity<ref>Template:Cite book</ref> <math display="block">\operatorname{He}_n(x) = e^{-\frac{D^2}{2}}x^n,</math> where Template:Mvar represents differentiation with respect to Template: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 Template:Math can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients of Template:Math that can be used to quickly compute these polynomials.
Since the formal expression for the Weierstrass transform Template:Mvar is Template:Math, we see that the Weierstrass transform of Template:Math is Template:Math. Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series.
The existence of some formal power series Template:Math with nonzero constant coefficient, such that Template: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.
GeneralizationsEdit
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<ref>Template:Citation</ref> <math display="block">\operatorname{He}_n^{[\alpha]}(x)</math> of variance Template:Mvar, where Template:Mvar is any positive number. These are then orthogonal with respect to the normal probability distribution whose density function is <math display="block">\frac{1}{\sqrt{2 \pi \alpha}} e^{-\frac{x^2}{2\alpha}}.</math> They are given by <math display="block">\operatorname{He}_n^{[\alpha]}(x) = \alpha^{\frac{n}{2}}\operatorname{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">\operatorname{He}_n^{[\alpha]}(x) = \sum_{k=0}^n h^{[\alpha]}_{n,k} x^k,</math> then the polynomial sequence whose Template:Mvarth term is <math display="block">\left(\operatorname{He}_n^{[\alpha]} \circ \operatorname{He}^{[\beta]}\right)(x) \equiv \sum_{k=0}^n h^{[\alpha]}_{n,k}\,\operatorname{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(\operatorname{He}_n^{[\alpha]} \circ \operatorname{He}^{[\beta]}\right)(x) = \operatorname{He}_n^{[\alpha+\beta]}(x)</math> and <math display="block">\operatorname{He}_n^{[\alpha+\beta]}(x + y) = \sum_{k=0}^n \binom{n}{k} \operatorname{He}_k^{[\alpha]}(x) \operatorname{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 Template:Math, has already been encountered in the above section on #Recursion relations.)
"Negative variance"Edit
Since polynomial sequences form a group under the operation of umbral composition, one may denote by <math display="block">\operatorname{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 Template:Math, the coefficients of <math>\operatorname{He}_n^{[-\alpha]}(x)</math> are just the absolute values of the corresponding coefficients of <math>\operatorname{He}_n^{[\alpha]}(x)</math>.
These arise as moments of normal probability distributions: The Template:Mvarth moment of the normal distribution with expected value Template:Mvar and variance Template:Math is <math display="block">E[X^n] = \operatorname{He}_n^{[-\sigma^2]}(\mu),</math> where Template: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} \operatorname{He}_k^{[\alpha]}(x) \operatorname{He}_{n-k}^{[-\alpha]}(y) = \operatorname{He}_n^{[0]}(x + y) = (x + y)^n.</math>
Hermite functionsEdit
DefinitionEdit
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 Template:Math. This fact is equivalent to the corresponding statement for Hermite polynomials (see above).
The Hermite functions are closely related to the Whittaker function Template:Harv Template: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.
<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>
Recursion relationEdit
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 Template:Mvarth derivatives for any positive integer Template: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) \operatorname{He}_k(x).</math>
This formula can be used in connection with the recurrence relations for Template:Math and Template:Math to calculate any derivative of the Hermite functions efficiently.
Cramér's inequalityEdit
For real Template:Mvar, the Hermite functions satisfy the following bound due to Harald Cramér<ref>Template:Harvnb.</ref><ref>Template:Harvnb.</ref> and Jack Indritz:<ref name="indritz">Template:Citation</ref> <math display="block"> \bigl|\psi_n(x)\bigr| \le \pi^{-\frac14}.</math>
Hermite functions as eigenfunctions of the Fourier transformEdit
The Hermite functions Template:Math are a set of eigenfunctions of the continuous Fourier transform Template:Mathcal. To see this, take the physicist's version of the generating function and multiply by Template: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 Template: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 Template:Math are thus an orthonormal basis of Template:Math, which diagonalizes the Fourier transform operator.<ref>In this case, we used the unitary version of the Fourier transform, so the eigenvalues are Template:Math. 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, in effect a Mehler kernel.</ref> In short, we have:<math display="block">\frac{1}{\sqrt{2\pi}} \int e^{-ikx} \psi_n(x) dx = (-i)^n \psi_n(k), \quad \frac{1}{\sqrt{2\pi}} \int e^{+ikx} \psi_n(k) dk = i^n \psi_n(x)</math>
Wigner distributions of Hermite functionsEdit
The Wigner distribution function of the Template:Mvarth-order Hermite function is related to the Template: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 Template:Mvar, it is straightforward to see<ref>Template:Citation</ref> 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 Template: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.<ref name="Groenewold1946">Template:Cite journal</ref> It is the standard paradigm of quantum mechanics in phase space.
There are further relations between the two families of polynomials.
Partial Overlap IntegralsEdit
It can be shown<ref>Template:Cite arXiv</ref><ref>Template:Cite arXiv</ref> that the overlap between two different Hermite functions (<math> k\neq \ell </math>) over a given interval has the exact result: <math display="block">\int_{x_1}^{x_2}\psi_{k}(x) \psi_{\ell}(x)\,dx =\frac{1}{2(\ell - k)}\left(\psi_k'(x_2)\psi_\ell(x_2)-\psi_\ell'(x_2)\psi_k(x_2)-\psi_k'(x_1)\psi_\ell(x_1)+\psi_\ell'(x_1)\psi_k(x_1)\right). </math>
Combinatorial interpretation of coefficientsEdit
In the Hermite polynomial Template:Math of variance 1, the absolute value of the coefficient of Template:Math is the number of (unordered) partitions of an Template:Mvar-element set into Template:Mvar singletons and Template:Math (unordered) pairs. Equivalently, it is the number of involutions of an Template:Mvar-element set with precisely Template:Mvar fixed points, or in other words, the number of matchings in the complete graph on Template:Mvar vertices that leave Template: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
This combinatorial interpretation can be related to complete exponential Bell polynomials as <math display="block">\operatorname{He}_n(x) = B_n(x, -1, 0, \ldots, 0),</math> where Template:Math for all Template:Math.
These numbers may also be expressed as a special value of the Hermite polynomials:<ref name="gfgt">Template:Citation</ref> <math display="block">T(n) = \frac{\operatorname{He}_n(i)}{i^n}.</math>
Completeness relationEdit
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 Template:Mvar is the Dirac delta function, Template:Math the Hermite functions, and Template:Math represents the Lebesgue measure on the line Template:Math in Template:Math, normalized so that its projection on the horizontal axis is the usual Lebesgue measure.
This distributional identity follows Template:Harvtxt by taking Template:Math in Mehler's formula, valid when Template: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,<ref>Template:Citation. See p. 174, eq. (18) and p. 173, eq. (13).</ref><ref>Template:Harvnb, 10.13 (22).</ref> <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 Template:Math is the bivariate Gaussian probability density on Template:Math, which is, when Template:Mvar is close to 1, very concentrated around the line Template: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 Template:Math and Template:Math are continuous and compactly supported.
This yields that Template:Mvar can be expressed in Hermite functions as the sum of a series of vectors in Template: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 Template: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 Template:Math and Template: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 alsoEdit
- Hermite transform
- Legendre polynomials
- Mehler kernel
- Parabolic cylinder function
- Romanovski polynomials
- Turán's inequalities
NotesEdit
ReferencesEdit
- Template:Abramowitz Stegun ref
- Template:Citation
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- Template:Dlmf
- Template:Citation Oeuvres complètes 12, pp.357-412, English translation Template:Webarchive.
- Template:Cite book
- Template:Citation - 2000 references of Bibliography on Hermite polynomials.
- Template:Eom
- Template:Citation
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External linksEdit
- Template:Commons category-inline
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:HermitePolynomial%7CHermitePolynomial.html}} |title = Hermite Polynomial |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}
- GNU Scientific Library — includes C version of Hermite polynomials, functions, their derivatives and zeros (see also GNU Scientific Library)