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Wikipedia / Hermite polynomials

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

The polynomials arise in:

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 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"

  • while the "physicist's Hermite polynomials"

These equations have the form of a Rodrigues' formula

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

is that used in the standard references. is the probability density function for the normal distribution with expected value 0 and standard deviation 1. The probabilist's Hermite polynomials are also called the monic Hermite polynomials, because they are monic.

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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, with the same parity as

Orthogonality

and are th-degree polynomials for . These polynomials are orthogonal with respect to the weight function (measure

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 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 for every , then .

One possible way to do this is to appreciate that the entire function 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 Sturm–Liouville differential equation 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 denotes a constant.

Rewriting the differential equation as an eigenvalue problem may be understood as eigenfunctions . This eigenvalue problem is called the Hermite equation denotes a constant, after imposing the boundary condition that should be polynomially bounded at infinity.

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 and , , .

and , , .

The Hermite polynomials constitute an Appell sequence

An integral recurrence that is deduced and demonstrated in

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

In consequence, for the

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

Moreover, the following multiplication theorem

Explicit expression

These two equations may be combined into one using the floor function

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

Inverse explicit expression

The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials

The corresponding expressions for the physicist's Hermite polynomials follow directly by properly scaling this:

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

one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.

A slight generalization states

Expected values

If is a random variable with a normal distribution with standard deviation 1 and expected value

where is the double factorial of mean 0 and variance 1:

Integral representations

From the generating-function representation above, we see that the Hermite polynomials have a representation in terms of a contour integral with the contour encircling the origin.

Other properties

The discriminant is expressed as a hyperfactorial:

The addition theorem, or the summation theorem, states that.

The multiplication theorem states that.

Feldheim formula has a positive real part. As a special case,

Asymptotics

As , which, using Stirling's approximationThis 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 corresponds to the probability of finding a classical particle in a potential well. This is a general method in semiclassical analysis, the location where the classical particle would be turned back. This is a fold catastrophe, at which point the Airy function is needed.

The Plancherel–Rotach asymptotics method, applied to Hermite polynomials, takes into account the uneven spacing of the zeros near the edges.

with 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 .

Kibble–Slepian formula

symmetric matrix, then the Kibble–Slepian formula is the trace. This gives Mehler's formula.

is a positive semidefinite matrixEquivalently stated in a form closer to the boson quantum mechanics of the harmonic oscillator:The Kibble–Slepian formula was proposed by Kibble in 1945 and proven by Slepian in 1972 using Fourier analysis. Foata gave a combinatorial proof while Louck gave a proof via boson quantum mechanics. It has a generalization for complex-argument Hermite polynomials.

Zeroes

-th zero of the Airy function, we need only consider the positive half of its roots.

We have,.

See also, and the formulas involving the zeroes of Laguerre polynomials.

be the cumulative distribution function, then we have the semicircle law The Stieltjes relation states that to the real line, and connecting each particle to the origin by a spring. This is also called the electrostatic model, and relates to the Coulomb gas interpretation of the eigenvalues of gaussian ensembles.

As the zeroes specify the polynomial up to scaling, the Stieltjes relation provides an alternative way to uniquely characterize the Hermite polynomials.

Similarly, we have

Relations to other functions

Laguerre polynomials

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

Hypergeometric functions

The physicist's Hermite polynomials can be expressed as a special case of the parabolic cylinder functions in the right half-plane, where is Tricomi's confluent hypergeometric function where is Kummer's confluent hypergeometric functionThere is also

Limit relations

The Hermite polynomials can be obtained as the limit of various other polynomials.

As a limit of Jacobi polynomials

Hermite polynomial expansion

, then it has an expansion in the physicist's Hermite polynomials.

.

.

Differential-operator representation

The probabilist's Hermite polynomials satisfy the identity 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.

Generalizations

Variance

which has expected value 0 and variance 1.

Scaling, one may analogously speak of generalized Hermite polynomials of variance , where

then the polynomial sequence whose is called the umbral composition 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 #Recursion relations.)

"Negative variance"

Since polynomial sequences form a group under the operation of umbral composition 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 .

These arise as moments of normal probability distributions: The th moment of the normal distribution with expected value and variance where

Hermite functions

Definition

One can define the Hermite functions

Since these functions contain the square root of the weight function and have been scaled appropriately, they are orthonormal 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 and thereby to other parabolic cylinder functions.

This equation is equivalent to the Schrödinger equation for a harmonic oscillator in quantum mechanics, so these functions are the eigenfunctions.

Recursion relation

Extending the first relation to the arbitrary th derivatives for any positive integer

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 and Jack Indritz:

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

Equating like powers of

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

Wigner distribution functions

The Wigner distribution function of the th-order Hermite function is related to the th-order Laguerre polynomial For all natural integers , one can prove that where the Wigner distribution of a function 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

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

This combinatorial interpretation can be related to complete exponential Bell polynomials where for all .

These numbers may also be expressed as a special value of the Hermite polynomials:

Completeness relation

The Christoffel–Darboux formula

Moreover, the following completeness identity for the above Hermite functions holds in the sense of distributions 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 which is often stated equivalently as a separable kernel,

The function is the bivariate Gaussian probability density on , which is, when is close to 1, very concentrated around the line 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

In order to prove the above equality for , the Fourier transform of Gaussian functions

With this representation for and

See also

Notes

References