Hermite polynomials

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 $\begin{array}{c}x{u}_x\end{array}$ 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 $${\mathrm{He}}_n(x)=(-1{)}^n{e}^{\frac{x^2}{2}}\frac{d^n}{d{x}^n}{e}^{-\frac{x^2}{2}},$$
• while the "physicist's Hermite polynomials" are given by $$H_n(x)=(-1{)}^n{e}^{x^2}\frac{d^n}{d{x}^n}{e}^{-x^2}.$$

These equations have the form of a Rodrigues' formula and can also be written as, $${\mathrm{He}}_n(x)={(x-\frac{d}{dx})}^n\cdot 1,H_n(x)={(2x-\frac{d}{dx})}^n\cdot 1.$$

The two definitions are not exactly identical; each is a rescaling of the other: $$H_n(x)=2^{\frac{n}{2}}{\mathrm{He}}_n\left(\sqrt{2}x\right),{\mathrm{He}}_n(x)=2^{-\frac{n}{2}}{H}_n\left(\frac{x}{\sqrt{2}}\right).$$

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. The polynomials Template:Mvar are sometimes denoted by Template:Mvar, especially in probability theory, because $$\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}$$ is the probability density function for the normal distribution with expected value 0 and standard deviation 1.

The first six probabilist's Hermite polynomials Template:Math

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

• The first eleven probabilist's Hermite polynomials are: $$\begin{array}{cc}{\mathrm{He}}_0(x)& =1,\\ {\mathrm{He}}_1(x)& =x,\\ {\mathrm{He}}_2(x)& =x^2-1,\\ {\mathrm{He}}_3(x)& =x^3-3x,\\ {\mathrm{He}}_4(x)& =x^4-6x^2+3,\\ {\mathrm{He}}_5(x)& =x^5-10x^3+15x,\\ {\mathrm{He}}_6(x)& =x^6-15x^4+45x^2-15,\\ {\mathrm{He}}_7(x)& =x^7-21x^5+105x^3-105x,\\ {\mathrm{He}}_8(x)& =x^8-28x^6+210x^4-420x^2+105,\\ {\mathrm{He}}_9(x)& =x^9-36x^7+378x^5-1260x^3+945x,\\ {\mathrm{He}}_{10}(x)& =x^{10}-45x^8+630x^6-3150x^4+4725x^2-945.\end{array}$$
• The first eleven physicist's Hermite polynomials are: $$\begin{array}{cc}{H}_0(x)& =1,\\ H_1(x)& =2x,\\ H_2(x)& =4x^2-2,\\ H_3(x)& =8x^3-12x,\\ H_4(x)& =16x^4-48x^2+12,\\ H_5(x)& =32x^5-160x^3+120x,\\ H_6(x)& =64x^6-480x^4+720x^2-120,\\ H_7(x)& =128x^7-1344x^5+3360x^3-1680x,\\ H_8(x)& =256x^8-3584x^6+13440x^4-13440x^2+1680,\\ H_9(x)& =512x^9-9216x^7+48384x^5-80640x^3+30240x,\\ H_{10}(x)& =1024x^{10}-23040x^8+161280x^6-403200x^4+302400x^2-30240.\end{array}$$

Properties

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.

Symmetry

From the Rodrigues formulae given above, we can see that Template:Math and Template:Math are even or odd functions depending on Template:Mvar: $$H_n(-x)=(-1{)}^n{H}_n(x),{\mathrm{He}}_n(-x)=(-1{)}^n{\mathrm{He}}_n(x).$$

Orthogonality

Template:Math and Template:Math are Template:Mvarth-degree polynomials for Template:Math. These polynomials are orthogonal with respect to the weight function (measure) $$w(x)=e^{-\frac{x^2}{2}}(\text{for }\mathrm{He})$$ or $$w(x)=e^{-x^2}(\text{for }H),$$ i.e., we have $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{H}_m(x)H_n(x)w(x)dx=0\text{for all }m\ne n.$$

Furthermore, $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{H}_m(x)H_n(x)e^{-x^2}dx=\sqrt{\pi }{2}^{n}n!{\delta }_{nm},$$ and $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\mathrm{He}}_m(x){\mathrm{He}}_n(x)e^{-\frac{x^2}{2}}dx=\sqrt{2\pi }n!{\delta }_{nm},$$ where ${\delta }_{nm}$ 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 $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|f(x){|}^{2}w(x)dx<\mathrm{\infty },$$ in which the inner product is given by the integral $$⟨f,g⟩=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)\overline{g(x)}w(x)dx$$ 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 $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)x^n{e}^{-x^2}dx=0$$ for every Template:Math, then Template:Math.

One possible way to do this is to appreciate that the entire function $$F(z)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)e^{zx-x^2}dx=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}\int f(x)x^n{e}^{-x^2}dx=0$$ 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 equation

The probabilist's Hermite polynomials are solutions of the differential equation $$\left(e^{-\frac{1}{2}{x}^2}{u}^{\prime }\right)+\lambda e^{-\frac{1}{2}{x}^2}u=0,$$ 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 $u(x)=C_1{\mathrm{He}}_{\lambda }(x)$, where $C_1$ denotes a constant.

Rewriting the differential equation as an eigenvalue problem $$L[u]=u^{″}-x{u}^{\prime }=-\lambda u,$$ the Hermite polynomials ${\mathrm{He}}_{\lambda }(x)$ may be understood as eigenfunctions of the differential operator $L[u]$ . This eigenvalue problem is called the Hermite equation, although the term is also used for the closely related equation $$u^{″}-2x{u}^{\prime }=-2\lambda u.$$ whose solution is uniquely given in terms of physicist's Hermite polynomials in the form $u(x)=C_1{H}_{\lambda }(x)$, where $C_1$ 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 $$u^{″}-2x{u}^{\prime }+2\lambda u=0,$$ the general solution takes the form $$u(x)=C_1{H}_{\lambda }(x)+C_2{h}_{\lambda }(x),$$ where $C_1$ and $C_2$ are constants, $H_{\lambda }(x)$ are physicist's Hermite polynomials (of the first kind), and $h_{\lambda }(x)$ are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as $h_{\lambda }(x)={}_1{F}_1(-\frac{\lambda }{2};\frac{1}{2};x^2)$ where ${}_1{F}_1(a;b;z)$ 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 relation

The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation $${\mathrm{He}}_{n+1}(x)=x{\mathrm{He}}_n(x)-{{\mathrm{He}}_n}^{\prime }(x).$$ Individual coefficients are related by the following recursion formula: $$a_{n+1,k}=\{\begin{array}{cc}-(k+1)a_{n,k+1}& k=0,\\ a_{n,k-1}-(k+1)a_{n,k+1}& k>0,\end{array}$$ and Template:Math, Template:Math, Template:Math.

For the physicist's polynomials, assuming $$H_n(x)=\sum _{k=0}^n{a}_{n,k}{x}^k,$$ we have $$H_{n+1}(x)=2x{H}_n(x)-{H_n}^{\prime }(x).$$ Individual coefficients are related by the following recursion formula: $$a_{n+1,k}=\{\begin{array}{cc}-a_{n,k+1}& k=0,\\ 2a_{n,k-1}-(k+1)a_{n,k+1}& k>0,\end{array}$$ and Template:Math, Template:Math, Template:Math.

The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity $$\begin{array}{cc}{{\mathrm{He}}_n}^{\prime }(x)& =n{\mathrm{He}}_{n-1}(x),\\ {H_n}^{\prime }(x)& =2n{H}_{n-1}(x).\end{array}$$

An integral recurrence that is deduced and demonstrated in is as follows: $${\mathrm{He}}_{n+1}(x)=(n+1)\underset{0}{\overset{x}{\int }}{\mathrm{He}}_n(t)dt-He{\text{'}}_n(0),$$

$$H_{n+1}(x)=2(n+1)\underset{0}{\overset{x}{\int }}{H}_n(t)dt-H{\text{'}}_n(0).$$

Equivalently, by Taylor-expanding, $$\begin{array}{cccc}{\mathrm{He}}_n(x+y)& =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{n-k}{\mathrm{He}}_k(y)& & =2^{-\frac{n}{2}}\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k}){\mathrm{He}}_{n-k}\left(x\sqrt{2}\right){\mathrm{He}}_k\left(y\sqrt{2}\right),\\ H_n(x+y)& =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})H_k(x)(2y{)}^{n-k}& & =2^{-\frac{n}{2}}\cdot \sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})H_{n-k}\left(x\sqrt{2}\right)H_k\left(y\sqrt{2}\right).\end{array}$$ These umbral identities are self-evident and included in the differential operator representation detailed below, $$\begin{array}{cc}{\mathrm{He}}_n(x)& =e^{-\frac{D^2}{2}}{x}^n,\\ H_n(x)& =2^n{e}^{-\frac{D^2}{4}}{x}^n.\end{array}$$

In consequence, for the Template:Mvarth derivatives the following relations hold: $$\begin{array}{cccc}{\mathrm{He}}_n^{(m)}(x)& =\frac{n!}{(n-m)!}{\mathrm{He}}_{n-m}(x)& & =m!(\genfrac{}{}{0ex}{}{n}{m}){\mathrm{He}}_{n-m}(x),\\ H_n^{(m)}(x)& =2^m\frac{n!}{(n-m)!}{H}_{n-m}(x)& & =2^{m}m!(\genfrac{}{}{0ex}{}{n}{m})H_{n-m}(x).\end{array}$$

It follows that the Hermite polynomials also satisfy the recurrence relation $$\begin{array}{cc}{\mathrm{He}}_{n+1}(x)& =x{\mathrm{He}}_n(x)-n{\mathrm{He}}_{n-1}(x),\\ H_{n+1}(x)& =2x{H}_n(x)-2n{H}_{n-1}(x).\end{array}$$

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 $${𝐻}_n(x{)}^2-{𝐻}_{n-1}(x){𝐻}_{n+1}(x)=(n-1)!\sum _{i=0}^{n-1}\frac{2^{n-i}}{i!}{𝐻}_i(x{)}^2>0.$$

Moreover, the following multiplication theorem holds: $$\begin{array}{cc}{H}_n(\gamma x)& =\sum _{i=0}^{\lfloor \frac{n}{2}\rfloor }{\gamma }^{n-2i}({\gamma }^2-1{)}^i(\genfrac{}{}{0ex}{}{n}{2i})\frac{(2i)!}{i!}{H}_{n-2i}(x),\\ {\mathrm{He}}_n(\gamma x)& =\sum _{i=0}^{\lfloor \frac{n}{2}\rfloor }{\gamma }^{n-2i}({\gamma }^2-1{)}^i(\genfrac{}{}{0ex}{}{n}{2i})\frac{(2i)!}{i!}{2}^{-i}{\mathrm{He}}_{n-2i}(x).\end{array}$$

Explicit expression

The physicist's Hermite polynomials can be written explicitly as $$H_n(x)=\{\begin{array}{cc}n!\sum _{l=0}^{\frac{n}{2}}\frac{(-1{)}^{\frac{n}{2}-l}}{(2l)!(\frac{n}{2}-l)!}(2x{)}^{2l}& \text{for even }n,\\ n!\sum _{l=0}^{\frac{n-1}{2}}\frac{(-1{)}^{\frac{n-1}{2}-l}}{(2l+1)!(\frac{n-1}{2}-l)!}(2x{)}^{2l+1}& \text{for odd }n.\end{array}$$

These two equations may be combined into one using the floor function: $$H_n(x)=n!\sum _{m=0}^{\lfloor \frac{n}{2}\rfloor }\frac{(-1{)}^m}{m!(n-2m)!}(2x{)}^{n-2m}.$$

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: $${\mathrm{He}}_n(x)=n!\sum _{m=0}^{\lfloor \frac{n}{2}\rfloor }\frac{(-1{)}^m}{m!(n-2m)!}\frac{x^{n-2m}}{2^m}.$$

Inverse explicit expression

The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials Template:Mvar are $$x^n=n!\sum _{m=0}^{\lfloor \frac{n}{2}\rfloor }\frac{1}{2^{m}m!(n-2m)!}{\mathrm{He}}_{n-2m}(x).$$

The corresponding expressions for the physicist's Hermite polynomials Template:Mvar follow directly by properly scaling this: $$x^n=\frac{n!}{2^n}\sum _{m=0}^{\lfloor \frac{n}{2}\rfloor }\frac{1}{m!(n-2m)!}{H}_{n-2m}(x).$$

Generating function

The Hermite polynomials are given by the exponential generating function $$\begin{array}{cc}{e}^{xt-\frac{1}{2}{t}^2}& =\sum _{n=0}^{\mathrm{\infty }}{\mathrm{He}}_n(x)\frac{t^n}{n!},\\ e^{2xt-t^2}& =\sum _{n=0}^{\mathrm{\infty }}{H}_n(x)\frac{t^n}{n!}.\end{array}$$

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 $$H_n(x)=(-1{)}^n{e}^{x^2}\frac{d^n}{d{x}^n}{e}^{-x^2}=(-1{)}^n{e}^{x^2}\frac{n!}{2\pi i}{\oint }_{\gamma }\frac{e^{-z^2}}{(z-x{)}^{n+1}}dz.$$

Using this in the sum $$\sum _{n=0}^{\mathrm{\infty }}{H}_n(x)\frac{t^n}{n!},$$ one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.

Expected values

If Template:Mvar is a random variable with a normal distribution with standard deviation 1 and expected value Template:Mvar, then $$𝔼[{\mathrm{He}}_n(X)]={\mu }^n.$$

The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices: $$𝔼\left[X^{2n}\right]=(-1{)}^n{\mathrm{He}}_{2n}(0)=(2n-1)!!,$$ 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: $${\mathrm{He}}_n(x)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(x+iy{)}^n{e}^{-\frac{y^2}{2}}dy.$$

Asymptotic expansion

Asymptotically, as Template:Math, the expansion $$e^{-\frac{x^2}{2}}\cdot H_n(x)\sim \frac{2^n}{\sqrt{\pi }}\mathrm{\Gamma }\left(\frac{n+1}{2}\right)\cos (x\sqrt{2n}-\frac{n\pi }{2})$$ holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude: $$e^{-\frac{x^2}{2}}\cdot H_n(x)\sim \frac{2^n}{\sqrt{\pi }}\mathrm{\Gamma }\left(\frac{n+1}{2}\right)\cos (x\sqrt{2n}-\frac{n\pi }{2}){(1-\frac{x^2}{2n+1})}^{-\frac{1}{4}}=\frac{\mathrm{\Gamma }(n)}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}\cos (x\sqrt{2n}-\frac{n\pi }{2}){(1-\frac{x^2}{2n+1})}^{-\frac{1}{4}},$$ which, using Stirling's approximation, can be further simplified, in the limit, to $$e^{-\frac{x^2}{2}}\cdot H_n(x)\sim {\left(\frac{2n}{e}\right)}^{\frac{n}{2}}\sqrt{2}\cos (x\sqrt{2n}-\frac{n\pi }{2}){(1-\frac{x^2}{2n+1})}^{-\frac{1}{4}}.$$

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 $$e^{-\frac{x^2}{2}}\cdot H_n(x)\sim {\left(\frac{2n}{e}\right)}^{\frac{n}{2}}\sqrt{2}\cos (x\sqrt{2n+1-\frac{x^2}{3}}-\frac{n\pi }{2}){(1-\frac{x^2}{2n+1})}^{-\frac{1}{4}}.$$

A finer approximation, which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution $$x=\sqrt{2n+1}\cos (\phi ),0<\epsilon \le \phi \le \pi -\epsilon ,$$ with which one has the uniform approximation $$e^{-\frac{x^2}{2}}\cdot H_n(x)=2^{\frac{n}{2}+\frac{1}{4}}\sqrt{n!}(\pi n{)}^{-\frac{1}{4}}(\sin \phi {)}^{-\frac{1}{2}}\cdot (\sin (\frac{3\pi }{4}+(\frac{n}{2}+\frac{1}{4})(\sin 2\phi -2\phi ))+O\left(n^{-1}\right)).$$

Similar approximations hold for the monotonic and transition regions. Specifically, if $$x=\sqrt{2n+1}\cosh (\phi ),0<\epsilon \le \phi \le \omega <\mathrm{\infty },$$ then $$e^{-\frac{x^2}{2}}\cdot H_n(x)=2^{\frac{n}{2}-\frac{3}{4}}\sqrt{n!}(\pi n{)}^{-\frac{1}{4}}(\sinh \phi {)}^{-\frac{1}{2}}\cdot e^{(\frac{n}{2}+\frac{1}{4})(2\phi -\sinh 2\phi )}(1+O\left(n^{-1}\right)),$$ while for $$x=\sqrt{2n+1}+t$$ with Template:Mvar complex and bounded, the approximation is $$e^{-\frac{x^2}{2}}\cdot H_n(x)={\pi }^{\frac{1}{4}}{2}^{\frac{n}{2}+\frac{1}{4}}\sqrt{n!}{n}^{-\frac{1}{12}}(\mathrm{Ai}\left(2^{\frac{1}{2}}{n}^{\frac{1}{6}}t\right)+O\left(n^{-\frac{2}{3}}\right)),$$ where Template:Math is the Airy function of the first kind.

Special values

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

$$H_n(0)=\{\begin{array}{cc}0& \text{for odd }n,\\ (-2{)}^{\frac{n}{2}}(n-1)!!& \text{for even }n,\end{array}$$ which satisfy the recursion relation Template:Math.

In terms of the probabilist's polynomials this translates to $${\mathrm{He}}_n(0)=\{\begin{array}{cc}0& \text{for odd }n,\\ (-1{)}^{\frac{n}{2}}(n-1)!!& \text{for even }n.\end{array}$$

Relations to other functions

Laguerre polynomials

The Hermite polynomials can be expressed as a special case of the Laguerre polynomials: $$\begin{array}{cccc}{H}_{2n}(x)& =(-4{)}^{n}n!L_n^{(-\frac{1}{2})}(x^2)& & =4^{n}n!\sum _{k=0}^n(-1{)}^{n-k}(\genfrac{}{}{0ex}{}{n-\frac{1}{2}}{n-k})\frac{x^{2k}}{k!},\\ H_{2n+1}(x)& =2(-4{)}^{n}n!x{L}_n^{\left(\frac{1}{2}\right)}(x^2)& & =2\cdot 4^{n}n!\sum _{k=0}^n(-1{)}^{n-k}(\genfrac{}{}{0ex}{}{n+\frac{1}{2}}{n-k})\frac{x^{2k+1}}{k!}.\end{array}$$

Relation to confluent hypergeometric functions

The physicist's Hermite polynomials can be expressed as a special case of the parabolic cylinder functions: $$H_n(x)=2^{n}U(-\frac{1}{2}n,\frac{1}{2},x^2)$$ in the right half-plane, where Template:Math is Tricomi's confluent hypergeometric function. Similarly, $$\begin{array}{cc}{H}_{2n}(x)& =(-1{)}^n\frac{(2n)!}{n!}{}_1{F}_1(-n,\frac{1}{2};x^2),\\ H_{2n+1}(x)& =(-1{)}^n\frac{(2n+1)!}{n!}2x{}_1{F}_1(-n,\frac{3}{2};x^2),\end{array}$$ where Template: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 $\int e^{-x^2}f(x{)}^{2}dx<\mathrm{\infty }$, then it has an expansion in the physicist's Hermite polynomials.

Given such $f$, the partial sums of the Hermite expansion of $f$ converges to in the $L^p$ norm if and only if $4/3<p<4$.$$x^n=\frac{n!}{2^n}\sum _{k=0}^{\lfloor n/2\rfloor }\frac{1}{k!(n-2k)!}{H}_{n-2k}(x)=n!\sum _{k=0}^{\lfloor n/2\rfloor }\frac{1}{k!2^k(n-2k)!}{\mathrm{He}}_{n-2k}(x),n\in {\mathbb{Z}}_{+}.$$$$e^{ax}=e^{a^2/4}\sum _{n\ge 0}\frac{a^n}{n!2^n}{H}_n(x),a\in ℂ,x\in ℝ.$$$$e^{-a^2{x}^2}=\sum _{n\ge 0}\frac{(-1{)}^n{a}^{2n}}{n!{(1+a^2)}^{n+1/2}{2}^{2n}}{H}_{2n}(x).$$$$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}\underset{0}{\overset{x}{\int }}{e}^{-t^2}dt=\frac{1}{\sqrt{2\pi }}\sum _{k\ge 0}\frac{(-1{)}^k}{k!(2k+1)2^{3k}}{H}_{2k}(x).$$$$\cosh (2x)=e\sum _{k\ge 0}\frac{1}{(2k)!}{H}_{2k}(x),\sinh (2x)=e\sum _{k\ge 0}\frac{1}{(2k+1)!}{H}_{2k+1}(x).$$$$\cos (x)=e^{-1/4}\sum _{k\ge 0}\frac{(-1{)}^k}{2^{2k}(2k)!}{H}_{2k}(x)\sin (x)=e^{-1/4}\sum _{k\ge 0}\frac{(-1{)}^k}{2^{2k+1}(2k+1)!}{H}_{2k+1}(x)$$

Differential-operator representation

The probabilist's Hermite polynomials satisfy the identity $${\mathrm{He}}_n(x)=e^{-\frac{D^2}{2}}{x}^n,$$ 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.

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 $$\begin{array}{cc}{\mathrm{He}}_n(x)& =\frac{n!}{2\pi i}{\oint }_C\frac{e^{tx-\frac{t^2}{2}}}{t^{n+1}}dt,\\ H_n(x)& =\frac{n!}{2\pi i}{\oint }_C\frac{e^{2tx-t^2}}{t^{n+1}}dt,\end{array}$$ 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 $$\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}},$$ which has expected value 0 and variance 1.

Scaling, one may analogously speak of generalized Hermite polynomials $${\mathrm{He}}_n^{[\alpha ]}(x)$$ 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 $$\frac{1}{\sqrt{2\pi \alpha }}{e}^{-\frac{x^2}{2\alpha }}.$$ They are given by $${\mathrm{He}}_n^{[\alpha ]}(x)={\alpha }^{\frac{n}{2}}{\mathrm{He}}_n\left(\frac{x}{\sqrt{\alpha }}\right)={\left(\frac{\alpha }{2}\right)}^{\frac{n}{2}}{H}_n\left(\frac{x}{\sqrt{2\alpha }}\right)=e^{-\frac{\alpha D^2}{2}}\left(x^n\right).$$

Now, if $${\mathrm{He}}_n^{[\alpha ]}(x)=\sum _{k=0}^n{h}_{n,k}^{[\alpha ]}{x}^k,$$ then the polynomial sequence whose Template:Mvarth term is $$({\mathrm{He}}_n^{[\alpha ]}\circ {\mathrm{He}}^{[\beta ]})(x)\equiv \sum _{k=0}^n{h}_{n,k}^{[\alpha ]}{\mathrm{He}}_k^{[\beta ]}(x)$$ is called the umbral composition of the two polynomial sequences. It can be shown to satisfy the identities $$({\mathrm{He}}_n^{[\alpha ]}\circ {\mathrm{He}}^{[\beta ]})(x)={\mathrm{He}}_n^{[\alpha +\beta ]}(x)$$ and $${\mathrm{He}}_n^{[\alpha +\beta ]}(x+y)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k}){\mathrm{He}}_k^{[\alpha ]}(x){\mathrm{He}}_{n-k}^{[\beta ]}(y).$$ 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"

Since polynomial sequences form a group under the operation of umbral composition, one may denote by $${\mathrm{He}}_n^{[-\alpha ]}(x)$$ 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 ${\mathrm{He}}_n^{[-\alpha ]}(x)$ are just the absolute values of the corresponding coefficients of ${\mathrm{He}}_n^{[\alpha ]}(x)$.

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 $$E[X^n]={\mathrm{He}}_n^{[-{\sigma }^2]}(\mu ),$$ where Template:Mvar is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that $$\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k}){\mathrm{He}}_k^{[\alpha ]}(x){\mathrm{He}}_{n-k}^{[-\alpha ]}(y)={\mathrm{He}}_n^{[0]}(x+y)=(x+y{)}^n.$$

Hermite functions

Definition

One can define the Hermite functions (often called Hermite-Gaussian functions) from the physicist's polynomials: $${\psi }_n(x)={(2^{n}n!\sqrt{\pi })}^{-\frac{1}{2}}{e}^{-\frac{x^2}{2}}{H}_n(x)=(-1{)}^n{(2^{n}n!\sqrt{\pi })}^{-\frac{1}{2}}{e}^{\frac{x^2}{2}}\frac{d^n}{d{x}^n}{e}^{-x^2}.$$ Thus, $$\sqrt{2(n+1)}{\psi }_{n+1}(x)=(x-\frac{d}{dx}){\psi }_n(x).$$

Since these functions contain the square root of the weight function and have been scaled appropriately, they are orthonormal: $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\psi }_n(x){\psi }_m(x)dx={\delta }_{nm},$$ 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: $$D_n(z)={(n!\sqrt{\pi })}^{\frac{1}{2}}{\psi }_n\left(\frac{z}{\sqrt{2}}\right)=(-1{)}^n{e}^{\frac{z^2}{4}}\frac{d^n}{d{z}^n}{e}^{\frac{-z^2}{2}}$$ and thereby to other parabolic cylinder functions.

The Hermite functions satisfy the differential equation $${{\psi }_n}^{″}(x)+(2n+1-x^2){\psi }_n(x)=0.$$ This equation is equivalent to the Schrödinger equation for a harmonic oscillator in quantum mechanics, so these functions are the eigenfunctions.

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

$$\begin{array}{cc}{\psi }_0(x)& ={\pi }^{-\frac{1}{4}}{e}^{-\frac{1}{2}{x}^2},\\ {\psi }_1(x)& =\sqrt{2}{\pi }^{-\frac{1}{4}}x{e}^{-\frac{1}{2}{x}^2},\\ {\psi }_2(x)& ={\left(\sqrt{2}{\pi }^{\frac{1}{4}}\right)}^{-1}(2x^2-1)e^{-\frac{1}{2}{x}^2},\\ {\psi }_3(x)& ={\left(\sqrt{3}{\pi }^{\frac{1}{4}}\right)}^{-1}(2x^3-3x)e^{-\frac{1}{2}{x}^2},\\ {\psi }_4(x)& ={\left(2\sqrt{6}{\pi }^{\frac{1}{4}}\right)}^{-1}(4x^4-12x^2+3)e^{-\frac{1}{2}{x}^2},\\ {\psi }_5(x)& ={\left(2\sqrt{15}{\pi }^{\frac{1}{4}}\right)}^{-1}(4x^5-20x^3+15x)e^{-\frac{1}{2}{x}^2}.\end{array}$$

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 $${{\psi }_n}^{\prime }(x)=\sqrt{\frac{n}{2}}{\psi }_{n-1}(x)-\sqrt{\frac{n+1}{2}}{\psi }_{n+1}(x)$$ and $$x{\psi }_n(x)=\sqrt{\frac{n}{2}}{\psi }_{n-1}(x)+\sqrt{\frac{n+1}{2}}{\psi }_{n+1}(x).$$

Extending the first relation to the arbitrary Template:Mvarth derivatives for any positive integer Template:Mvar leads to $${\psi }_n^{(m)}(x)=\sum _{k=0}^m(\genfrac{}{}{0ex}{}{m}{k})(-1{)}^k{2}^{\frac{m-k}{2}}\sqrt{\frac{n!}{(n-m+k)!}}{\psi }_{n-m+k}(x){\mathrm{He}}_k(x).$$

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 inequality

For real Template:Mvar, the Hermite functions satisfy the following bound due to Harald Cramér and Jack Indritz: $$|{\psi }_n(x)|\le {\pi }^{-\frac{1}{4}}.$$

Hermite functions as eigenfunctions of the Fourier transform

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 $$e^{-\frac{1}{2}{x}^2+2xt-t^2}=\sum _{n=0}^{\mathrm{\infty }}{e}^{-\frac{1}{2}{x}^2}{H}_n(x)\frac{t^n}{n!}.$$

The Fourier transform of the left side is given by $$\begin{array}{cc}ℱ\left\{e^{-\frac{1}{2}{x}^2+2xt-t^2}\right\}(k)& =\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-ixk}{e}^{-\frac{1}{2}{x}^2+2xt-t^2}dx\\ & =e^{-\frac{1}{2}{k}^2-2kit+t^2}\\ & =\sum _{n=0}^{\mathrm{\infty }}{e}^{-\frac{1}{2}{k}^2}{H}_n(k)\frac{(-it{)}^n}{n!}.\end{array}$$

The Fourier transform of the right side is given by $$ℱ\{\sum _{n=0}^{\mathrm{\infty }}{e}^{-\frac{1}{2}{x}^2}{H}_n(x)\frac{t^n}{n!}\}=\sum _{n=0}^{\mathrm{\infty }}ℱ\{e^{-\frac{1}{2}{x}^2}{H}_n(x)\}\frac{t^n}{n!}.$$

Equating like powers of Template:Mvar in the transformed versions of the left and right sides finally yields $$ℱ\{e^{-\frac{1}{2}{x}^2}{H}_n(x)\}=(-i{)}^n{e}^{-\frac{1}{2}{k}^2}{H}_n(k).$$

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

Wigner distributions of Hermite functions

The Wigner distribution function of the Template:Mvarth-order Hermite function is related to the Template:Mvarth-order Laguerre polynomial. The Laguerre polynomials are $$L_n(x):=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\frac{(-1{)}^k}{k!}{x}^k,$$ leading to the oscillator Laguerre functions $$l_n(x):=e^{-\frac{x}{2}}{L}_n(x).$$ For all natural integers Template:Mvar, it is straightforward to see that $$W_{{\psi }_n}(t,f)=(-1{)}^n{l}_n(4\pi (t^2+f^2)),$$ where the Wigner distribution of a function Template:Math is defined as $$W_x(t,f)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t+\frac{\tau }{2})x{(t-\frac{\tau }{2})}^{*}{e}^{-2\pi i\tau f}d\tau .$$ 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.

Partial Overlap Integrals

It can be shown that the overlap between two different Hermite functions ($k\ne \ell$) over a given interval has the exact result: $$\underset{x_1}{\overset{x_2}{\int }}{\psi }_k(x){\psi }_{\ell }(x)dx=\frac{1}{2(\ell -k)}({{\psi }_k}^{\prime }(x_2){\psi }_{\ell }(x_2)-{{\psi }_{\ell }}^{\prime }(x_2){\psi }_k(x_2)-{{\psi }_k}^{\prime }(x_1){\psi }_{\ell }(x_1)+{{\psi }_{\ell }}^{\prime }(x_1){\psi }_k(x_1)).$$

Combinatorial interpretation of coefficients

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

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

This combinatorial interpretation can be related to complete exponential Bell polynomials as $${\mathrm{He}}_n(x)=B_n(x,-1,0,\dots ,0),$$ where Template:Math for all Template:Math.

These numbers may also be expressed as a special value of the Hermite polynomials: $$T(n)=\frac{{\mathrm{He}}_n(i)}{i^n}.$$

Completeness relation

The Christoffel–Darboux formula for Hermite polynomials reads $$\sum _{k=0}^n\frac{H_k(x)H_k(y)}{k!2^k}=\frac{1}{n!2^{n+1}}\frac{H_n(y)H_{n+1}(x)-H_n(x)H_{n+1}(y)}{x-y}.$$

Moreover, the following completeness identity for the above Hermite functions holds in the sense of distributions: $$\sum _{n=0}^{\mathrm{\infty }}{\psi }_n(x){\psi }_n(y)=\delta (x-y),$$ 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: $$E(x,y;u):=\sum _{n=0}^{\mathrm{\infty }}{u}^n{\psi }_n(x){\psi }_n(y)=\frac{1}{\sqrt{\pi (1-u^2)}}\exp (-\frac{1-u}{1+u}\frac{(x+y{)}^2}{4}-\frac{1+u}{1-u}\frac{(x-y{)}^2}{4}),$$ which is often stated equivalently as a separable kernel, $$\sum _{n=0}^{\mathrm{\infty }}\frac{H_n(x)H_n(y)}{n!}{\left(\frac{u}{2}\right)}^n=\frac{1}{\sqrt{1-u^2}}{e}^{\frac{2u}{1+u}xy-\frac{u^2}{1-u^2}(x-y{)}^2}.$$

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 $$\sum _{n=0}^{\mathrm{\infty }}{u}^n⟨f,{\psi }_n⟩⟨{\psi }_n,g⟩=\iint E(x,y;u)f(x)\overline{g(y)}dxdy\to \int f(x)\overline{g(x)}dx=⟨f,g⟩$$ 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, $$f=\sum _{n=0}^{\mathrm{\infty }}⟨f,{\psi }_n⟩{\psi }_n.$$

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

The Hermite polynomial is then represented as $$H_n(x)=(-1{)}^n{e}^{x^2}\frac{d^n}{d{x}^n}(\frac{1}{2\sqrt{\pi }}\int e^{isx-\frac{s^2}{4}}ds)=(-1{)}^n{e}^{x^2}\frac{1}{2\sqrt{\pi }}\int (is{)}^n{e}^{isx-\frac{s^2}{4}}ds.$$

With this representation for Template:Math and Template:Math, it is evident that $$\begin{array}{cc}E(x,y;u)& =\sum _{n=0}^{\mathrm{\infty }}\frac{u^n}{2^{n}n!\sqrt{\pi }}{H}_n(x)H_n(y)e^{-\frac{x^2+y^2}{2}}\\ & =\frac{e^{\frac{x^2+y^2}{2}}}{4\pi \sqrt{\pi }}\iint (\sum _{n=0}^{\mathrm{\infty }}\frac{1}{2^{n}n!}(-ust{)}^n)e^{isx+ity-\frac{s^2}{4}-\frac{t^2}{4}}dsdt\\ & =\frac{e^{\frac{x^2+y^2}{2}}}{4\pi \sqrt{\pi }}\iint e^{-\frac{ust}{2}}{e}^{isx+ity-\frac{s^2}{4}-\frac{t^2}{4}}dsdt,\end{array}$$ and this yields the desired resolution of the identity result, using again the Fourier transform of Gaussian kernels under the substitution $$s=\frac{\sigma +\tau }{\sqrt{2}},t=\frac{\sigma -\tau }{\sqrt{2}}.$$

See also

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

References

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

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