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 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, Collected in Œuvres complètes VII. though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Collected in Œuvres I, 501–508. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new. Collected in Œuvres II, 293–308. 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" are given by while the "physicist's Hermite polynomials" are given by
These equations have the form of a Rodrigues' formula and can also be written as,
The two definitions are not exactly identical; each is a rescaling of the other:
These are Hermite polynomial sequences of different variances; see the material on variances below.
The notation and is that used in the standard references. and Abramowitz & Stegun. The polynomials are sometimes denoted by , especially in probability theory, because
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. The first eleven probabilist's Hermite polynomials are: The first eleven physicist's Hermite polynomials are: {| class="wikitable" |+Quick reference table ! !physicist's !probabilist's |- |symbol
|head coefficient
|differential operator
|orthogonal to
|inner product
|generating function
|Rodrigues' formula
|recurrence relation
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) or i.e., we have
Furthermore,
and
where 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
in which the inner product is given by the integral
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
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, and the solution is uniquely given by , where denotes a constant.
Rewriting the differential equation as an eigenvalue problem
the Hermite polynomials may be understood as eigenfunctions of the differential operator . This eigenvalue problem is called the Hermite equation, although the term is also used for the closely related equation
whose solution is uniquely given in terms of physicist's Hermite polynomials in the form , where 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
the general solution takes the form
where and are constants, are physicist's Hermite polynomials (of the first kind), and are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as where 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
Individual coefficients are related by the following recursion formula:
and , , .
For the physicist's polynomials, assuming
we have
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 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. is as follows:
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
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:
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 are
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 to write the Hermite polynomials as
Using this in the sum
one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.
A slight generalization states(Rainville 1971), p. 198
Expected values If is a random variable with a normal distribution with standard deviation 1 and expected value , then
The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices:
where is the double factorial. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as a Gaussian integral (or equivalently, a Gaussian expectation) with a standard Gaussian random variable 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, as
with the contour encircling the origin.
Using the Fourier transform of the gaussian , we have
Other properties The discriminant is expressed as a hyperfactorial:
The addition theorem, or the summation theorem, states thatfor any nonzero vector .
The multiplication theorem states thatfor any nonzero .
Feldheim formulaFeldheim, 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, IIIwhere has a positive real part. As a special case,
Asymptotics As ,, 13.6.38 and 13.5.16. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude:
which, using Stirling's approximation, can be further simplified, in the limit, to 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. The term corresponds to the probability of finding a classical particle in a potential well of shape at location , if its total energy is . This is a general method in semiclassical analysis. The semiclassical approximation breaks down near , the location where the classical particle would be turned back. This is a fold catastrophe, at which point the Airy function is needed.
A better approximation, which accounts for the variation in frequency, is given by
The Plancherel–Rotach asymptotics method, applied to Hermite polynomials, takes into account the uneven spacing of the zeros near the edges. It makes use of the substitution
with which one has the uniform approximation
Similar approximations hold for the monotonic and transition regions. Specifically, if
then
while for with complex and bounded, the approximation is
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 . Equivalently, .
In terms of the probabilist's polynomials this translates to
Kibble–Slepian formula
Let be a real symmetric matrix, then the Kibble–Slepian formula states that where is the -fold summation over all symmetric matrices with non-negative integer entries, is the trace of , and is defined as . This gives Mehler's formula when .