Hermite polynomials explained

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{align}xu _{x} \end{align} 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,^{[1] [2]} though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859.^[3] Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new.^[4] 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 $$\backslash operatorname\_n(x)\; =\; (-1)^n\; e^\backslash frace^,$$
• while the "physicist's Hermite polynomials" are given by $$H\_n(x)\; =\; (-1)^n\; e^\backslash frace^.$$

These equations have the form of a Rodrigues' formula and can also be written as,$$\backslash operatorname\_n(x)\; =\; \backslash left(x\; -\; \backslash frac\; \backslash right)^n\; \backslash cdot\; 1,\; \backslash quad\; H\_n(x)\; =\; \backslash left(2x\; -\; \backslash frac\; \backslash right)^n\; \backslash cdot\; 1.$$

The two definitions are not exactly identical; each is a rescaling of the other:$$H\_n(x)=2^\backslash frac\; \backslash operatorname\_n\backslash left(\backslash sqrt\; \backslash ,x\backslash right),\; \backslash quad\; \backslash operatorname\_n(x)=2^\; H\_n\backslash left(\backslash frac\; \backslash right).$$

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

The notation and is that used in the standard references.^[5] The polynomials are sometimes denoted by, especially in probability theory, because$$\backslash frace^$$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: $$\backslash begin$$

\operatorname_0(x) &= 1, \\\operatorname_1(x) &= x, \\\operatorname_2(x) &= x^2 - 1, \\\operatorname_3(x) &= x^3 - 3x, \\\operatorname_4(x) &= x^4 - 6x^2 + 3, \\\operatorname_5(x) &= x^5 - 10x^3 + 15x, \\\operatorname_6(x) &= x^6 - 15x^4 + 45x^2 - 15, \\\operatorname_7(x) &= x^7 - 21x^5 + 105x^3 - 105x, \\\operatorname_8(x) &= x^8 - 28x^6 + 210x^4 - 420x^2 + 105, \\\operatorname_9(x) &= x^9 - 36x^7 + 378x^5 - 1260x^3 + 945x, \\\operatorname_(x) &= x^ - 45x^8 + 630x^6 - 3150x^4 + 4725x^2 - 945.\end

• The first eleven physicist's Hermite polynomials are: $$\backslash begin$$

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_(x) &= 1024x^ - 23040x^8 + 161280x^6 - 403200x^4 + 302400x^2 - 30240.\end

Properties

The th-order Hermite polynomial is a polynomial of degree . The probabilist's version has leading coefficient 1, while the physicist's version has leading coefficient .

Symmetry

From the Rodrigues formulae given above, we can see that and are even or odd functions depending on :$$H\_n(-x)=(-1)^nH\_n(x),\backslash quad\; \backslash operatorname\_n(-x)=(-1)^n\backslash operatorname\_n(x).$$

Orthogonality

and are th-degree polynomials for . These polynomials are orthogonal with respect to the weight function (measure)$$w(x)\; =\; e^\; \backslash quad\; (\backslash text\backslash operatorname)$$ or $$w(x)\; =\; e^\; \backslash quad\; (\backslash text\; H),$$i.e., we have$$\backslash int\_^\backslash infty\; H\_m(x)\; H\_n(x)\backslash ,\; w(x)\; \backslash ,dx\; =\; 0\; \backslash quad\; \backslash textm\; \backslash neq\; n.$$

Furthermore,$$\backslash int\_^\backslash infty\; H\_m(x)\; H\_n(x)\backslash ,\; e^\; \backslash ,dx\; =\; \backslash sqrt\backslash ,\; 2^n\; n!\backslash ,\; \backslash delta\_,$$and$$\backslash int\_^\backslash infty\; \backslash operatorname\_m(x)\; \backslash operatorname\_n(x)\backslash ,\; e^\; \backslash ,dx\; =\; \backslash sqrt\backslash ,\; n!\backslash ,\; \backslash delta\_,$$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 satisfyingin which the inner product is given by the integral$$\backslash langle\; f,g\backslash rangle\; =\; \backslash int\_^\backslash infty\; f(x)\; \backslash overline\backslash ,\; w(x)\; \backslash ,dx$$including the Gaussian weight function defined in the preceding section

An orthogonal basis for is a complete orthogonal system. For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function 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 satisfies$$\backslash int\_^\backslash infty\; f(x)\; x^n\; e^\; \backslash ,dx\; =\; 0$$for every, then .

One possible way to do this is to appreciate that the entire function$$F(z)\; =\; \backslash int\_^\backslash infty\; f(x)\; e^\; \backslash ,dx\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash int\; f(x)\; x^n\; e^\; \backslash ,dx\; =\; 0$$vanishes identically. The fact then that for every real means that the Fourier transform of is 0, hence 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 consists in introducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis for .

Hermite's differential equation

The probabilist's Hermite polynomials are solutions of the differential equation$$\backslash left(e^u\text{'}\backslash right)\text{'}\; +\; \backslash lambda\; e^u\; =\; 0,$$where is a constant. Imposing the boundary condition that should be polynomially bounded at infinity, the equation has solutions only if is a non-negative integer, and the solution is uniquely given by

u(x) = C _{1} \operatorname{He} _{λ(x)} , where C _{1} denotes a constant.

Rewriting the differential equation as an eigenvalue problem$$L[u]\; =\; u$$ - x u' = -\lambda u,the Hermite polynomials

\operatorname{He} _{λ(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$$u$$ - 2xu' = -2\lambda u. whose solution is uniquely given in terms of physicist's Hermite polynomials in the form u(x) = C _{1} H _{λ(x)} , where C _{1} denotes a constant, after imposing the boundary condition that 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$$u$$ - 2xu' + 2\lambda u = 0,the general solution takes the form$$u(x)\; =\; C\_1\; H\_\backslash lambda(x)\; +\; C\_2\; h\_\backslash lambda(x),$$where

C _{1} and C _{2} are constants, H _{λ(x)} are physicist's Hermite polynomials (of the first kind), and h _{λ(x)} are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as h _{λ(x)} = {} _{1F} ^{2 )} ^{1 ( - \tfrac{ λ }{ 2 };\tfrac{ 1 }{ 2 };x} where {} _{1F} _{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 . An explicit formula of Hermite polynomials in terms of contour integrals is also possible.

Recurrence relation

The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation$$\backslash operatorname\_(x)\; =\; x\; \backslash operatorname\_n(x)\; -\; \backslash operatorname\_n\text{'}(x).$$Individual coefficients are related by the following recursion formula: and,, .

For the physicist's polynomials, assuming $$H\_n(x)\; =\; \backslash sum^n\_\; a\_\; x^k,$$we have$$H\_(x)\; =\; 2xH\_n(x)\; -\; H\_n\text{'}(x).$$Individual coefficients are related by the following recursion formula:and,, .

The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity

An integral recurrence that is deduced and demonstrated in ^[6] is as follows:$$\backslash operatorname\_(x)\; =\; (n+1)\backslash int\_0^x\; \backslash operatorname\_n(t)dt\; -\; He\text{'}\_n(0),$$

$$H\_(x)\; =\; 2(n+1)\backslash int\_0^x\; H\_n(t)dt\; -\; H\text{'}\_n(0).$$

Equivalently, by Taylor-expanding,These umbral identities are self-evident and included in the differential operator representation detailed below,

In consequence, for the th derivatives the following relations hold:

It follows that the Hermite polynomials also satisfy the recurrence relation

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

Turán's inequalities are$$\backslash mathit\_n(x)^2\; -\; \backslash mathit\_(x)\; \backslash mathit\_(x)\; =\; (n-1)!\; \backslash sum\_^\; \backslash frac\backslash mathit\_i(x)^2\; >\; 0.$$

Moreover, the following multiplication theorem holds:

Explicit expression

The physicist's Hermite polynomials can be written explicitly as

These two equations may be combined into one using the floor function:$$H\_n(x)\; =\; n!\; \backslash sum\_^\; \backslash frac\; (2x)^.$$

The probabilist's Hermite polynomials have similar formulas, which may be obtained from these by replacing the power of with the corresponding power of and multiplying the entire sum by :$$\backslash operatorname\_n(x)\; =\; n!\; \backslash sum\_^\; \backslash frac\; \backslash frac.$$

Inverse explicit expression

The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials are$$x^n\; =\; n!\; \backslash sum\_^\; \backslash frac\; \backslash operatorname\_(x).$$

The corresponding expressions for the physicist's Hermite polynomials follow directly by properly scaling this:^[7] $$x^n\; =\; \backslash frac\; \backslash sum\_^\; \backslash frac\; H\_(x).$$

Generating function

The Hermite polynomials are given by the exponential generating function

This equality is valid for all complex values of and, and can be obtained by writing the Taylor expansion at of the entire function (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$$H\_n(x)\; =\; (-1)^n\; e^\; \backslash frac\; e^\; =\; (-1)^n\; e^\; \backslash frac\; \backslash oint\_\backslash gamma\; \backslash frac\; \backslash ,dz.$$

Using this in the sum $$\backslash sum\_^\backslash infty\; H\_n(x)\; \backslash frac,$$ one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.

Expected values

If is a random variable with a normal distribution with standard deviation 1 and expected value, then$$\backslash operatorname\backslash left[\backslash operatorname\{He\}\_n(X)\backslash right]\; =\; \backslash mu^n.$$

The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices:$$\backslash operatorname\backslash left[X^\{2n\}\backslash right]\; =\; (-1)^n\; \backslash operatorname\_(0)\; =\; (2n-1)!!,$$where is the double factorial. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments:$$\backslash operatorname\_n(x)\; =\; \backslash frac\; \backslash int\_^\backslash infty\; (x\; +\; iy)^n\; e^\; \backslash ,dy.$$

Asymptotic expansion

Asymptotically, as, the expansion^[8] $$e^\backslash cdot\; H\_n(x)\; \backslash sim\; \backslash frac\backslash Gamma\backslash left(\backslash frac2\backslash right)\; \backslash cos\; \backslash left(x\; \backslash sqrt-\; \backslash frac\; \backslash right)$$ holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude:$$e^\backslash cdot\; H\_n(x)\; \backslash sim\; \backslash frac\backslash Gamma\backslash left(\backslash frac2\backslash right)\; \backslash cos\; \backslash left(x\; \backslash sqrt-\; \backslash frac\; \backslash right)\backslash left(1-\backslash frac\backslash right)^=\backslash frac\; \backslash cos\; \backslash left(x\; \backslash sqrt-\; \backslash frac\; \backslash right)\backslash left(1-\backslash frac\backslash right)^,$$which, using Stirling's approximation, can be further simplified, in the limit, to$$e^\backslash cdot\; H\_n(x)\; \backslash sim\; \backslash left(\backslash frac\backslash right)^\; \backslash sqrt\; \backslash cos\; \backslash left(x\; \backslash sqrt-\; \backslash frac\; \backslash right)\backslash left(1-\backslash frac\backslash right)^.$$

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$$e^\backslash cdot\; H\_n(x)\; \backslash sim\; \backslash left(\backslash frac\backslash right)^\; \backslash sqrt\; \backslash cos\; \backslash left(x\; \backslash sqrt-\; \backslash frac\; \backslash right)\backslash left(1-\backslash frac\backslash right)^.$$

A finer approximation, which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution with which one has the uniform approximation$$e^\backslash cdot\; H\_n(x)\; =\; 2^\backslash sqrt(\backslash pi\; n)^(\backslash sin\; \backslash varphi)^\; \backslash cdot\; \backslash left(\backslash sin\backslash left(\backslash frac\; +\; \backslash left(\backslash frac\; +\; \backslash frac\backslash right)\backslash left(\backslash sin\; 2\backslash varphi-2\backslash varphi\backslash right)\; \backslash right)+O\backslash left(n^\backslash right)\; \backslash right).$$

Similar approximations hold for the monotonic and transition regions. Specifically, if then$$e^\backslash cdot\; H\_n(x)\; =\; 2^\backslash sqrt(\backslash pi\; n)^(\backslash sinh\; \backslash varphi)^\; \backslash cdot\; e^\backslash left(1+O\backslash left(n^\backslash right)\; \backslash right),$$while for $$x\; =\; \backslash sqrt\; +\; t$$ with complex and bounded, the approximation is$$e^\backslash cdot\; H\_n(x)\; =\backslash pi^2^\backslash sqrt\backslash ,\; n^\backslash left(\backslash operatorname\backslash left(2^n^t\backslash right)+\; O\backslash left(n^\backslash right)\; \backslash right),$$where is the Airy function of the first kind.

Special values

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

which satisfy the recursion relation .

In terms of the probabilist's polynomials this translates to

Relations to other functions

Laguerre polynomials

The Hermite polynomials can be expressed as a special case of the Laguerre polynomials:

Relation to confluent hypergeometric functions

The physicist's Hermite polynomials can be expressed as a special case of the parabolic cylinder functions:$$H\_n(x)\; =\; 2^n\; U\backslash left(-\backslash tfrac12\; n,\; \backslash tfrac12,\; x^2\backslash right)$$in the right half-plane, where is Tricomi's confluent hypergeometric function. Similarly,where 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 < inf ty , then it has an expansion in the physicist's Hermite polynomials.^[9]

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 .^[10] $$x^n\; =\; \backslash frac\; \backslash ,\backslash sum\_^\; \backslash frac\; \backslash ,\; H\_\; (x)\; =\; n!\; \backslash sum\_^\; \backslash frac\; \backslash ,\; \backslash operatorname\_\; (x),\; \backslash qquad\; n\; \backslash in\; \backslash mathbb\_\; .$$$$e^\; =\; e^\; \backslash sum\_\; \backslash frac\; \backslash ,\; H\_n\; (x),\; \backslash qquad\; a\backslash in\; \backslash mathbb,\; \backslash quad\; x\backslash in\; \backslash mathbb\; .$$$$e^\; =\; \backslash sum\_\; \backslash frac\backslash ,\; H\_\; (x)\; .$$$$\backslash operatorname(x)=\backslash frac\; \backslash int\_0^x\; e^\; ~dt=\backslash frac\; \backslash sum\_\; \backslash frac\; H\_(x)\; .$$$$\backslash cosh\; (2x)\; =\; e\; \backslash sum\_\; \backslash frac\backslash ,\; H\_\; (x),\; \backslash qquad\; \backslash sinh\; (2x)\; =\; e\; \backslash sum\_\; \backslash frac\; \backslash ,\; H\_\; (x)\; .$$$$\backslash cos\; (x)\; =\; e^\; \backslash ,\backslash sum\_\; \backslash frac\; \backslash ,\; H\_\; (x)\; \backslash quad\; \backslash sin\; (x)\; =\; e^\; \backslash ,\backslash sum\_\; \backslash frac\; \backslash ,\; H\_\; (x)$$

Differential-operator representation

The probabilist's Hermite polynomials satisfy the identity $$\backslash operatorname\_n(x)\; =\; e^x^n,$$ where represents differentiation with respect to, 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 can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients of that can be used to quickly compute these polynomials.

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

The existence of some formal power series with nonzero constant coefficient, such that, 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, aswith 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$$\backslash frac\; e^,$$which has expected value 0 and variance 1.

Scaling, one may analogously speak of generalized Hermite polynomials$$\backslash operatorname\_n^(x)$$of variance, where is any positive number. These are then orthogonal with respect to the normal probability distribution whose density function is$$(2\backslash pi\backslash alpha)^\; e^.$$They are given by$$\backslash operatorname\_n^(x)\; =\; \backslash alpha^\backslash operatorname\_n\backslash left(\backslash frac\backslash right)\; =\; \backslash left(\backslash frac\backslash right)^\; H\_n\backslash left(\backslash frac\backslash right)\; =\; e^\; \backslash left(x^n\backslash right).$$

Now, if $$\backslash operatorname\_n^(x)\; =\; \backslash sum\_^n\; h^\_\; x^k,$$then the polynomial sequence whose th term is$$\backslash left(\backslash operatorname\_n^\; \backslash circ\; \backslash operatorname^\backslash right)(x)\; \backslash equiv\; \backslash sum\_^n\; h^\_\backslash ,\backslash operatorname\_k^(x)$$is called the umbral composition of the two polynomial sequences. It can be shown to satisfy the identities$$\backslash left(\backslash operatorname\_n^\; \backslash circ\; \backslash operatorname^\backslash right)(x)\; =\; \backslash operatorname\_n^(x)$$and$$\backslash operatorname\_n^(x\; +\; y)\; =\; \backslash sum\_^n\; \backslash binom\; \backslash operatorname\_k^(x)\; \backslash operatorname\_^(y).$$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, has already been encountered in the above section on

1. Recursion relation

s.)

"Negative variance"

Since polynomial sequences form a group under the operation of umbral composition, one may denote by$$\backslash operatorname\_n^(x)$$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, the coefficients of

\operatorname{He} ^{[ - \alpha]} _{n} (x) are just the absolute values of the corresponding coefficients of \operatorname{He} ^{[\alpha]} _{n} (x) .

These arise as moments of normal probability distributions: The th moment of the normal distribution with expected value and variance is$$E[X^n]\; =\; \backslash operatorname\_n^(\backslash mu),$$where is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that$$\backslash sum\_^n\; \backslash binom\; \backslash operatorname\_k^(x)\; \backslash operatorname\_^(y)\; =\; \backslash operatorname\_n^(x\; +\; y)\; =\; (x\; +\; y)^n.$$

Hermite functions

Definition

One can define the Hermite functions (often called Hermite-Gaussian functions) from the physicist's polynomials:$$\backslash psi\_n(x)\; =\; \backslash left\; (2^n\; n!\; \backslash sqrt\; \backslash right)^\; e^\; H\_n(x)\; =\; (-1)^n\; \backslash left\; (2^n\; n!\; \backslash sqrt\; \backslash right)^\; e^\backslash frac\; e^.$$Thus, $$\backslash sqrt~~\backslash psi\_(x)=\; \backslash left\; (x-\; \backslash right)\; \backslash psi\_n(x).$$

Since these functions contain the square root of the weight function and have been scaled appropriately, they are orthonormal:$$\backslash int\_^\backslash infty\; \backslash psi\_n(x)\; \backslash psi\_m(x)\; \backslash ,dx\; =\; \backslash delta\_,$$and they form an orthonormal basis of . This fact is equivalent to the corresponding statement for Hermite polynomials (see above).

The Hermite functions are closely related to the Whittaker function :$$D\_n(z)\; =\; \backslash left(n!\; \backslash sqrt\backslash right)^\; \backslash psi\_n\backslash left(\backslash frac\backslash right)\; =\; (-1)^n\; e^\backslash frac\; \backslash frac\; e^\backslash frac$$and thereby to other parabolic cylinder functions.

The Hermite functions satisfy the differential equation$$\backslash psi\_n$$(x) + \left(2n + 1 - x^2\right) \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.

Recursion relation

Following recursion relations of Hermite polynomials, the Hermite functions obey$$\backslash psi\_n\text{'}(x)\; =\; \backslash sqrt\backslash ,\backslash psi\_(x)\; -\; \backslash sqrt\backslash psi\_(x)$$and$$x\backslash psi\_n(x)\; =\; \backslash sqrt\backslash ,\backslash psi\_(x)\; +\; \backslash sqrt\backslash psi\_(x).$$

Extending the first relation to the arbitrary th derivatives for any positive integer leads to$$\backslash psi\_n^(x)\; =\; \backslash sum\_^m\; \backslash binom\; (-1)^k\; 2^\backslash frac\; \backslash sqrt\; \backslash psi\_(x)\; \backslash operatorname\_k(x).$$

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

Cramér's inequality

For real, the Hermite functions satisfy the following bound due to Harald Cramér^{[11] [12]} and Jack Indritz:$$\backslash bigl|\backslash psi\_n(x)\backslash bigr|\; \backslash le\; \backslash pi^.$$

Hermite functions as eigenfunctions of the Fourier transform

The Hermite functions are a set of eigenfunctions of the continuous Fourier transform . To see this, take the physicist's version of the generating function and multiply by . This gives$$e^\; =\; \backslash sum\_^\backslash infty\; e^\; H\_n(x)\; \backslash frac.$$

The Fourier transform of the left side is given by

The Fourier transform of the right side is given by$$\backslash mathcal\; \backslash left\backslash \; =\; \backslash sum\_^\backslash infty\; \backslash mathcal\; \backslash left\; \backslash \; \backslash frac.$$

Equating like powers of in the transformed versions of the left and right sides finally yields$$\backslash mathcal\; \backslash left\backslash \; =\; (-i)^n\; e^\; H\_n(k).$$

The Hermite functions are thus an orthonormal basis of, which diagonalizes the Fourier transform operator.^[13]

Wigner distributions of Hermite functions

The Wigner distribution function of the th-order Hermite function is related to the th-order Laguerre polynomial. The Laguerre polynomials are $$L\_n(x)\; :=\; \backslash sum\_^n\; \backslash binom\; \backslash fracx^k,$$leading to the oscillator Laguerre functions$$l\_n\; (x)\; :=\; e^\; L\_n(x).$$For all natural integers, it is straightforward to see that$$W\_(t,f)\; =\; (-1)^n\; l\_n\; \backslash big(4\backslash pi\; (t^2\; +\; f^2)\; \backslash big),$$where the Wigner distribution of a function is defined as$$W\_x(t,f)\; =\; \backslash int\_^\backslash infty\; x\backslash left(t\; +\; \backslash frac\backslash right)\; \backslash ,\; x\backslash left(t\; -\; \backslash frac\backslash right)^*\; \backslash ,\; e^\; \backslash ,d\backslash tau.$$This is a fundamental result for the quantum harmonic oscillator discovered by Hip Groenewold in 1946 in his PhD thesis.^[14] 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 of variance 1, the absolute value of the coefficient of is the number of (unordered) partitions of an -element set into singletons and (unordered) pairs. Equivalently, it is the number of involutions of an -element set with precisely fixed points, or in other words, the number of matchings in the complete graph on vertices that leave 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,... .

This combinatorial interpretation can be related to complete exponential Bell polynomials as$$\backslash operatorname\_n(x)\; =\; B\_n(x,\; -1,\; 0,\; \backslash ldots,\; 0),$$where for all .

These numbers may also be expressed as a special value of the Hermite polynomials:$$T(n)\; =\; \backslash frac.$$

Completeness relation

The Christoffel–Darboux formula for Hermite polynomials reads$$\backslash sum\_^n\; \backslash frac\; =\; \backslash frac\backslash ,\backslash frac.$$

Moreover, the following completeness identity for the above Hermite functions holds in the sense of distributions:$$\backslash sum\_^\backslash infty\; \backslash psi\_n(x)\; \backslash psi\_n(y)\; =\; \backslash delta(x\; -\; y),$$where is the Dirac delta function, the Hermite functions, and represents the Lebesgue measure on the line in, normalized so that its projection on the horizontal axis is the usual Lebesgue measure.

This distributional identity follows by taking in Mehler's formula, valid when :$$E(x,\; y;\; u)\; :=\; \backslash sum\_^\backslash infty\; u^n\; \backslash ,\; \backslash psi\_n\; (x)\; \backslash ,\; \backslash psi\_n\; (y)\; =\; \backslash frac\; \backslash ,\; \backslash exp\backslash left(-\backslash frac\; \backslash ,\; \backslash frac\; -\; \backslash frac\; \backslash ,\; \backslash frac\backslash right),$$which is often stated equivalently as a separable kernel,^{[15] [16]} $$\backslash sum\_^\backslash infty\; \backslash frac\; \backslash left(\backslash frac\; u\; 2\backslash right)^n\; =\; \backslash frac\; e^.$$

The function is the bivariate Gaussian probability density on, which is, when is close to 1, very concentrated around the line, and very spread out on that line. It follows that$$\backslash sum\_^\backslash infty\; u^n\; \backslash langle\; f,\; \backslash psi\_n\; \backslash rangle\; \backslash langle\; \backslash psi\_n,\; g\; \backslash rangle\; =\; \backslash iint\; E(x,\; y;\; u)\; f(x)\; \backslash overline\; \backslash ,dx\; \backslash ,dy\; \backslash to\; \backslash int\; f(x)\; \backslash overline\; \backslash ,dx\; =\; \backslash langle\; f,\; g\; \backslash rangle$$when and are continuous and compactly supported.

This yields that can be expressed in Hermite functions as the sum of a series of vectors in, namely,$$f\; =\; \backslash sum\_^\backslash infty\; \backslash langle\; f,\; \backslash psi\_n\; \backslash rangle\; \backslash psi\_n.$$

In order to prove the above equality for, the Fourier transform of Gaussian functions is used repeatedly:$$\backslash rho\; \backslash sqrt\; e^\; =\; \backslash int\; e^\; \backslash ,ds\; \backslash quad\; \backslash text\backslash rho\; >\; 0.$$

The Hermite polynomial is then represented as$$H\_n(x)\; =\; (-1)^n\; e^\; \backslash frac\; \backslash left(\backslash frac\; \backslash int\; e^\; \backslash ,ds\; \backslash right)\; =\; (-1)^n\; e^\backslash frac\; \backslash int\; (is)^n\; e^\; \backslash ,ds.$$

With this representation for and, it is evident thatand this yields the desired resolution of the identity result, using again the Fourier transform of Gaussian kernels under the substitution$$s\; =\; \backslash frac,\; \backslash quad\; t\; =\; \backslash frac.$$

See also

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

References

• • • • • • Oeuvres complètes 12, pp.357-412, English translation .
• - 2000 references of Bibliography on Hermite polynomials.

External links

• • • GNU Scientific Library — includes C version of Hermite polynomials, functions, their derivatives and zeros (see also GNU Scientific Library)

Notes and References

1. Laplace . Mémoire sur les intégrales définies et leur application aux probabilités, et spécialement a la recherche du milieu qu'il faut choisir entre les resultats des observations . Mémoires de la Classe des Sciences Mathématiques et Physiques de l'Institut Impérial de France . 1811 . 11 . 297–347 . Memoire on definite integrals and their application to probabilities, and especially to the search for the mean which must be chosen among the results of observations . French.
2. Collected in Œuvres complètes VII.
3. P. . Tchébychef . Sur le développement des fonctions à une seule variable . On the development of single-variable functions . Bulletin de l'Académie impériale des sciences de St.-Pétersbourg . 1 . 1860 . 193–200 . French . Collected in Œuvres I, 501–508.
4. C. . Hermite . Sur un nouveau développement en série de fonctions . On a new development in function series . C. R. Acad. Sci. Paris . 58 . 1864 . 93–100, 266–273 . French . Collected in Œuvres II, 293–308.
5. and Abramowitz & Stegun.
6. 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.
7. Web site: 18. Orthogonal Polynomials, Classical Orthogonal Polynomials, Sums . Digital Library of Mathematical Functions . National Institute of Standards and Technology . 30 January 2015 . DLMF.
8. , 13.6.38 and 13.5.16.
9. Web site: MATHEMATICA tutorial, part 2.5: Hermite expansion . 2023-12-24 . www.cfm.brown.edu.
10. Askey . Richard . Wainger . Stephen . 1965 . Mean Convergence of Expansions in Laguerre and Hermite Series . American Journal of Mathematics . 87 . 3 . 695–708 . 10.2307/2373069 . 0002-9327.
11. .
12. .
13. In this case, we used the unitary version of the Fourier transform, so the eigenvalues are . 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.
14. Groenewold . H. J. . 1946 . On the Principles of elementary quantum mechanics . Physica . 12 . 7. 405–460 . 10.1016/S0031-8914(46)80059-4 . 1946Phy....12..405G.
15. . See p. 174, eq. (18) and p. 173, eq. (13).
16. , 10.13 (22).