HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Hermite polynomials are a classical
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
polynomial sequence In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in ...
. The polynomials arise in: *
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
as Hermitian wavelets for wavelet transform analysis *
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
, such as the Edgeworth series, as well as in connection with
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
; *
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
, as an example of an Appell sequence, obeying the umbral calculus; *
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
as
Gaussian quadrature In numerical analysis, an -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the nodes and weights for . Th ...
; *
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, where they give rise to the eigenstates of the
quantum harmonic oscillator The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
; and they also occur in some cases of the
heat equation In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...
(when the term \beginxu_\end is present); *
systems theory Systems theory is the Transdisciplinarity, transdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or artificial. Every system has causal boundaries, is influenced by its context, de ...
in connection with nonlinear operations on
Gaussian noise Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by
Pierre-Simon Laplace Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
in 1810, though in scarcely recognizable form, and studied in detail by
Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebysh ...
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 In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev 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 \operatorname_n(x) = (-1)^n e^\frace^, * while the "physicist's Hermite polynomials" are given by H_n(x) = (-1)^n e^\frace^. These equations have the form of a Rodrigues' formula and can also be written as, \operatorname_n(x) = \left(x - \frac \right)^n \cdot 1, \quad H_n(x) = \left(2x - \frac \right)^n \cdot 1. The two definitions are not exactly identical; each is a rescaling of the other: H_n(x)=2^\frac \operatorname_n\left(\sqrt \,x\right), \quad \operatorname_n(x)=2^ H_n\left(\frac \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. The polynomials are sometimes denoted by , especially in probability theory, because \frace^ is the
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
for the
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
with
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
0 and
standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
1. * The first eleven probabilist's Hermite polynomials are: \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: \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 File:Hermite poly.svg, The first six probabilist's Hermite polynomials \operatorname_n(x) File:Hermite poly phys.svg, The first six physicist's Hermite polynomials H_n(x)


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),\quad \operatorname_n(-x)=(-1)^n\operatorname_n(x).


Orthogonality

and are th-degree polynomials for . These polynomials are orthogonal with respect to the ''weight function'' ( measure) w(x) = e^ \quad (\text\operatorname) or w(x) = e^ \quad (\text H), i.e., we have \int_^\infty H_m(x) H_n(x)\, w(x) \,dx = 0 \quad \textm \neq n. Furthermore, \int_^\infty H_m(x) H_n(x)\, e^ \,dx = \sqrt\, 2^n n!\, \delta_, and \int_^\infty \operatorname_m(x) \operatorname_n(x)\, e^ \,dx = \sqrt\, n!\, \delta_, where \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. 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 In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
of functions satisfying \int_^\infty \bigl, f(x)\bigr, ^2\, w(x) \,dx < \infty, in which the inner product is given by the integral \langle f,g\rangle = \int_^\infty f(x) \overline\, w(x) \,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 In mathematics, the linear span (also called the linear hull or just span) of a set S of elements of a vector space V is the smallest linear subspace of V that contains S. It is the set of all finite linear combinations of the elements of , and ...
of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that if satisfies \int_^\infty f(x) x^n e^ \,dx = 0 for every , then . One possible way to do this is to appreciate that the
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...
F(z) = \int_^\infty f(x) e^ \,dx = \sum_^\infty \frac \int f(x) x^n e^ \,dx = 0 vanishes identically. The fact then that for every real means that the
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of is 0, hence is 0
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. Variants of the above completeness proof apply to other weights with
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda Lambda (; uppe ...
. 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 \left(e^u'\right)' + \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_\lambda(x) , where C_ denotes a constant. Rewriting the differential equation as an
eigenvalue problem In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
L = u'' - x u' = -\lambda u, the Hermite polynomials \operatorname_\lambda(x) may be understood as
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of the differential operator L /math> . 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_\lambda(x) , where C_ 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_\lambda(x) + C_2 h_\lambda(x), where C_ and C_ 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) = _1F_1(-\tfrac;\tfrac;x^2) 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 In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
, the Hermite polynomials can be generalized to obtain more general
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s 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 In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
\operatorname_(x) = x \operatorname_n(x) - \operatorname_n'(x). Individual coefficients are related by the following recursion formula: a_ = \begin - (k+1) a_ & k = 0, \\ a_ - (k+1) a_ & k > 0, \end and , , . For the physicist's polynomials, assuming H_n(x) = \sum^n_ a_ x^k, we have H_(x) = 2xH_n(x) - H_n'(x). Individual coefficients are related by the following recursion formula: a_ = \begin - a_ & k = 0, \\ 2 a_ - (k+1)a_ & k > 0, \end and , , . The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity \begin \operatorname_n'(x) &= n\operatorname_(x), \\ H_n'(x) &= 2nH_(x). \end An integral recurrence that is deduced and demonstrated in is as follows: \operatorname_(x) = (n+1)\int_0^x \operatorname_n(t)dt - He'_n(0), H_(x) = 2(n+1)\int_0^x H_n(t)dt - H'_n(0). Equivalently, by Taylor-expanding, \begin \operatorname_n(x+y) &= \sum_^n \binomx^ \operatorname_(y) &&= 2^ \sum_^n \binom \operatorname_\left(x\sqrt 2\right) \operatorname_k\left(y\sqrt 2\right), \\ H_n(x+y) &= \sum_^n \binomH_(x) (2y)^ &&= 2^\cdot\sum_^n \binom H_\left(x\sqrt 2\right) H_k\left(y\sqrt 2\right). \end These umbral identities are self-evident and included in the differential operator representation detailed below, \begin \operatorname_n(x) &= e^ x^n, \\ H_n(x) &= 2^n e^ x^n. \end In consequence, for the th derivatives the following relations hold: \begin \operatorname_n^(x) &= \frac \operatorname_(x) &&= m! \binom \operatorname_(x), \\ H_n^(x) &= 2^m \frac H_(x) &&= 2^m m! \binom H_(x). \end It follows that the Hermite polynomials also satisfy the
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
\begin \operatorname_(x) &= x\operatorname_n(x) - n\operatorname_(x), \\ H_(x) &= 2xH_n(x) - 2nH_(x). \end These last relations, together with the initial polynomials and , can be used in practice to compute the polynomials quickly. Turán's inequalities are \mathit_n(x)^2 - \mathit_(x) \mathit_(x) = (n-1)! \sum_^ \frac\mathit_i(x)^2 > 0. Moreover, the following multiplication theorem holds: \begin H_n(\gamma x) &= \sum_^ \gamma^(\gamma^2 - 1)^i \binom \frac H_(x), \\ \operatorname_n(\gamma x) &= \sum_^ \gamma^(\gamma^2 - 1)^i \binom \frac2^ \operatorname_(x). \end


Explicit expression

The physicist's Hermite polynomials can be written explicitly as H_n(x) = \begin \displaystyle n! \sum_^ \frac (2x)^ & \text n, \\ \displaystyle n! \sum_^ \frac (2x)^ & \text n. \end These two equations may be combined into one using the
floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...
: H_n(x) = n! \sum_^ \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 : \operatorname_n(x) = n! \sum_^ \frac \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! \sum_^ \frac \operatorname_(x). The corresponding expressions for the physicist's Hermite polynomials follow directly by properly scaling this: x^n = \frac \sum_^ \frac H_(x).


Generating function

The Hermite polynomials are given by the exponential generating function \begin e^ &= \sum_^\infty \operatorname_n(x) \frac, \\ e^ &= \sum_^\infty H_n(x) \frac. \end This equality is valid for all
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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 In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
to write the Hermite polynomials as H_n(x) = (-1)^n e^ \frac e^ = (-1)^n e^ \frac \oint_\gamma \frac \,dz. Using this in the sum \sum_^\infty H_n(x) \frac , one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function. A slight generalization statese^ H_k(x-t) = \sum_^ \frac


Expected values

If is a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
with a
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
with standard deviation 1 and expected value , then \operatorname\left operatorname_n(X)\right= \mu^n. The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices: \operatorname\left ^\right= (-1)^n \operatorname_(0) = (2n-1)!!, where is the
double factorial In mathematics, the double factorial of a number , denoted by , is the product of all the positive integers up to that have the same Parity (mathematics), parity (odd or even) as . That is, n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. Restated ...
. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments: \operatorname_n(x) = \frac \int_^\infty (x + iy)^n e^ \,dy.


Integral representations

From the generating-function representation above, we see that the Hermite polynomials have a representation in terms of a contour integral, as \begin \operatorname_n(x) &= \frac \oint_C \frac\,dt, \\ H_n(x) &= \frac \oint_C \frac\,dt, \end with the contour encircling the origin. Using the Fourier transform of the gaussian e^=\frac \int e^ dt , we have\begin H_n(x) &= (-1)^n e^ \frac e^ = \frac \int t^n e^ d t \\ \operatorname_n(x) &= \frac \int t^n \, e^\, dt. \end


Other properties

The addition theorem, or the summation theorem, states that\frac H_n\left(\frac\right)=\sum_ \prod_^r\left\ for any nonzero vector a_. The multiplication theorem states thatH_\left(\lambda x\right)=\lambda^\sum_^\frac(1-\lambda^)^H_\left(x\right)for any nonzero \lambda. 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). Par
IIIIII
/ref>\begin \frac & \int_^ e^ H_m\left(\frac\right) H_n\left(\frac\right) d x \\ & = \left(1-\frac\right)^\left(1-\frac\right)^ \sum_^ r!\binom\binom \left(\frac\right)^r H_\left(\frac\right) H_\left(\frac\right) \endwhere a \in \mathbb C has a positive real part. As a special case,\frac \int_^ e^ H_m(t \sin \theta+v \cos \theta) H_n(t \cos \theta-v \sin \theta) d t =(-1)^n \cos ^m \theta \sin ^n \theta H_(v)


Asymptotic expansion

Asymptotically, as , the expansion e^\cdot H_n(x) \sim \frac\Gamma\left(\frac2\right) \cos \left(x \sqrt- \frac \right) holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude: e^\cdot H_n(x) \sim \frac\Gamma\left(\frac2\right) \cos \left(x \sqrt- \frac \right)\left(1-\frac\right)^=\frac \cos \left(x \sqrt- \frac \right)\left(1-\frac\right)^, which, using
Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related ...
, can be further simplified, in the limit, to e^\cdot H_n(x) \sim \left(\frac\right)^ \sqrt \cos \left(x \sqrt- \frac \right)\left(1-\frac\right)^. This expansion is needed to resolve the
wavefunction In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
of a
quantum harmonic oscillator The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
such that it agrees with the classical approximation in the limit of the
correspondence principle In physics, a correspondence principle is any one of several premises or assertions about the relationship between classical and quantum mechanics. The physicist Niels Bohr coined the term in 1920 during the early development of quantum theory; ...
. A better approximation, which accounts for the variation in frequency, is given by e^\cdot H_n(x) \sim \left(\frac\right)^ \sqrt \cos \left(x \sqrt- \frac \right)\left(1-\frac\right)^. A finer approximation, which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution x = \sqrt\cos(\varphi), \quad 0 < \varepsilon \leq \varphi \leq \pi - \varepsilon, with which one has the uniform approximation e^\cdot H_n(x) = 2^\sqrt(\pi n)^(\sin \varphi)^ \cdot \left(\sin\left(\frac + \left(\frac + \frac\right)\left(\sin 2\varphi-2\varphi\right) \right)+O\left(n^\right) \right). Similar approximations hold for the monotonic and transition regions. Specifically, if x = \sqrt \cosh(\varphi), \quad 0 < \varepsilon \leq \varphi \leq \omega < \infty, then e^\cdot H_n(x) = 2^\sqrt(\pi n)^(\sinh \varphi)^ \cdot e^\left(1+O\left(n^\right) \right), while for x = \sqrt + t with complex and bounded, the approximation is e^\cdot H_n(x) =\pi^2^\sqrt\, n^\left( \operatorname\left(2^n^t\right)+ O\left(n^\right) \right), where is the
Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear in ...
of the first kind.


Special values

The physicist's Hermite polynomials evaluated at zero argument are called Hermite numbers. H_n(0) = \begin 0 & \textn, \\ (-2)^\frac (n-1)!! & \textn, \end which satisfy the recursion relation . Equivalently, H_(0) = (-2)^n (2n-1)!!. In terms of the probabilist's polynomials this translates to \operatorname_n(0) = \begin 0 & \textn, \\ (-1)^\frac (n-1)!! & \textn. \end


Kibble–Slepian formula

Let M be a real n\times n symmetric matrix, then the Kibble–Slepian formula states that\det(I+M)^ e^ = \sum_K \left prod_ \frac\right2^ H_(x_1) \cdots H_(x_n) where \sum_K is the \frac-fold summation over all n \times n symmetric matrices with non-negative integer entries, tr(K) is the trace of K, and k_i is defined as k_ + \sum_^n k_. This gives Mehler's formula when M = \begin 0 & u \\ u & 0\end. Equivalently stated, if T is a
positive semidefinite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. M ...
, then set M = -T(I+T)^, we have M(I+M)^ = -T, so e^ = \det(I+T)^ \sum_K \left prod_ \frac\right2^ H_(x_1) \dots H_(x_n) Equivalently stated in a form closer to the
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
of the
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive const ...
: \pi^\det(I+M)^e^= \sum_K\left prod_ M_^ / k_!\rightleft prod_ k_!\right 2^ \psi_\left(x_1\right) \cdots \psi_\left(x_n\right) . where each \psi_n(x) is the n-th eigenfunction of the harmonic oscillator, defined as \psi_n(x) := \frac\left(\frac\right)^ e^ H_n(x) 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.


Relations to other functions


Laguerre polynomials

The Hermite polynomials can be expressed as a special case of the Laguerre polynomials: \begin H_(x) &= (-4)^n n! L_n^(x^2) &&= 4^n n! \sum_^n (-1)^ \binom \frac, \\ H_(x) &= 2(-4)^n n! x L_n^(x^2) &&= 2\cdot 4^n n!\sum_^n (-1)^ \binom \frac. \end


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\left(-\tfrac12 n, \tfrac12, x^2\right) in the right half-plane, where is Tricomi's confluent hypergeometric function. Similarly, \begin H_(x) &= (-1)^n \frac \,_1F_1\big(-n, \tfrac12; x^2\big), \\ H_(x) &= (-1)^n \frac\,2x \,_1F_1\big(-n, \tfrac32; x^2\big), \end where is Kummer's confluent hypergeometric function. There is alsoH_\left(x\right)=(2x)^\left(;-\frac\right).


Limit relations

The Hermite polynomials can be obtained as the limit of various other polynomials. As a limit of Jacobi polynomials:\lim_\alpha^P^_\left(\alpha^x\right)=\frac.As a limit of ultraspherical polynomials:\lim_\lambda^C^_\left(\lambda^x\right)=\frac.As a limit of associated Laguerre polynomials:\lim_\left(\frac\right)^L^_\left((2\alpha)^x+\alpha\right)=\fracH_\left(x\right).


Hermite polynomial expansion

Similar to Taylor expansion, some functions are expressible as an infinite sum of Hermite polynomials. Specifically, if \int e^f(x)^2 dx < \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.x^n = \frac \,\sum_^ \frac \, H_ (x) = n! \sum_^ \frac \, \operatorname_ (x) , \qquad n \in \mathbb_ . e^ = e^ \sum_ \frac \, H_n (x) , \qquad a\in \mathbb, \quad x\in \mathbb .e^ = \sum_ \frac\, H_ (x) .\operatorname(x)=\frac \int_0^x e^ ~dt=\frac \sum_ \frac H_(x) .\cosh (2x) = e \sum_ \frac\, H_ (x) , \qquad \sinh (2x) = e \sum_ \frac \, H_ (x) .\cos (x) = e^ \,\sum_ \frac \, H_ (x) \quad \sin (x) = e^ \,\sum_ \frac \, H_ (x)


Differential-operator representation

The probabilist's Hermite polynomials satisfy the identity \operatorname_n(x) = e^x^n, where represents differentiation with respect to , and the exponential is interpreted by expanding it as a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
. 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

The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is \frac e^, which has expected value 0 and variance 1. Scaling, one may analogously speak of generalized Hermite polynomials \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 \frac e^. They are given by \operatorname_n^(x) = \alpha^\operatorname_n\left(\frac\right) = \left(\frac\right)^ H_n\left( \frac\right) = e^ \left(x^n\right). Now, if \operatorname_n^(x) = \sum_^n h^_ x^k, then the polynomial sequence whose th term is \left(\operatorname_n^ \circ \operatorname^\right)(x) \equiv \sum_^n h^_\,\operatorname_k^(x) is called the umbral composition of the two polynomial sequences. It can be shown to satisfy the identities \left(\operatorname_n^ \circ \operatorname^\right)(x) = \operatorname_n^(x) and \operatorname_n^(x + y) = \sum_^n \binom \operatorname_k^(x) \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 #Recursion relations.)


"Negative variance"

Since polynomial sequences form a group under the operation of umbral composition, one may denote by \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_n^(x) are just the absolute values of the corresponding coefficients of \operatorname_n^(x). These arise as moments of normal probability distributions: The th moment of the normal distribution with expected value and variance is E ^n= \operatorname_n^(\mu), where is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that \sum_^n \binom \operatorname_k^(x) \operatorname_^(y) = \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: \psi_n(x) = \left (2^n n! \sqrt \right )^ e^ H_n(x) = (-1)^n \left (2^n n! \sqrt \right)^ e^\frac e^. Thus, \sqrt~~\psi_(x)= \left ( x- \right ) \psi_n(x). Since these functions contain the square root of the weight function and have been scaled appropriately, they are orthonormal: \int_^\infty \psi_n(x) \psi_m(x) \,dx = \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) = \left(n! \sqrt\right)^ \psi_n\left(\frac\right) = (-1)^n e^\frac \frac e^\frac and thereby to other parabolic cylinder functions. The Hermite functions satisfy the differential equation \psi_n''(x) + \left(2n + 1 - x^2\right) \psi_n(x) = 0. This equation is equivalent to the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
for a harmonic oscillator in quantum mechanics, so these functions are the
eigenfunctions In mathematics, an eigenfunction of a linear map, linear operator ''D'' defined on some function space is any non-zero function (mathematics), function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor calle ...
. \begin \psi_0(x) &= \pi^ \, e^, \\ \psi_1(x) &= \sqrt \, \pi^ \, x \, e^, \\ \psi_2(x) &= \left(\sqrt \, \pi^\right)^ \, \left(2x^2-1\right) \, e^, \\ \psi_3(x) &= \left(\sqrt \, \pi^\right)^ \, \left(2x^3-3x\right) \, e^, \\ \psi_4(x) &= \left(2 \sqrt \, \pi^\right)^ \, \left(4x^4-12x^2+3\right) \, e^, \\ \psi_5(x) &= \left(2 \sqrt \, \pi^\right)^ \, \left(4x^5-20x^3+15x\right) \, e^. \end


Recursion relation

Following recursion relations of Hermite polynomials, the Hermite functions obey \psi_n'(x) = \sqrt\,\psi_(x) - \sqrt\psi_(x) and x\psi_n(x) = \sqrt\,\psi_(x) + \sqrt\psi_(x). Extending the first relation to the arbitrary th derivatives for any positive integer leads to \psi_n^(x) = \sum_^m \binom (-1)^k 2^\frac \sqrt \psi_(x) \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 Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statis ...
and Jack Indritz: \bigl, \psi_n(x)\bigr, \le \pi^.


Hermite functions as eigenfunctions of the Fourier transform

The Hermite functions are a set of
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of the
continuous Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function (mathematics), function as input then outputs another function that describes the extent to which various Frequency, frequencies are present in the origin ...
. To see this, take the physicist's version of the generating function and multiply by . This gives e^ = \sum_^\infty e^ H_n(x) \frac. The Fourier transform of the left side is given by \begin \mathcal \left\(k) &= \frac\int_^\infty e^e^\, dx \\ &= e^ \\ &= \sum_^\infty e^ H_n(k) \frac. \end The Fourier transform of the right side is given by \mathcal \left\ = \sum_^\infty \mathcal \left \ \frac. Equating like powers of in the transformed versions of the left and right sides finally yields \mathcal \left\ = (-i)^n e^ H_n(k). The Hermite functions are thus an orthonormal basis of , which ''diagonalizes the Fourier transform operator''. In short, we have:\frac \int e^ \psi_n(x) dx = (-i)^n \psi_n(k), \quad \frac \int e^ \psi_n(k) dk = i^n \psi_n(x)


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) := \sum_^n \binom \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 \big(4\pi (t^2 + f^2) \big), where the Wigner distribution of a function is defined as W_x(t,f) = \int_^\infty x\left(t + \frac\right) \, x\left(t - \frac\right)^* \, e^ \,d\tau. This is a fundamental result for the
quantum harmonic oscillator The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
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\neq \ell ) over a given interval has the exact result: \int_^\psi_(x) \psi_(x)\,dx =\frac\left(\psi_k'(x_2)\psi_\ell(x_2)-\psi_\ell'(x_2)\psi_k(x_2)-\psi_k'(x_1)\psi_\ell(x_1)+\psi_\ell'(x_1)\psi_k(x_1)\right).


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 In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices i ...
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 \operatorname_n(x) = B_n(x, -1, 0, \ldots, 0), where for all . These numbers may also be expressed as a special value of the Hermite polynomials: T(n) = \frac.


Completeness relation

The Christoffel–Darboux formula for Hermite polynomials reads \sum_^n \frac = \frac\,\frac. Moreover, the following completeness identity for the above Hermite functions holds in the sense of distributions: \sum_^\infty \psi_n(x) \psi_n(y) = \delta(x - y), where is the
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
, the Hermite functions, and represents the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
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) := \sum_^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac \, \exp\left(-\frac \, \frac - \frac \, \frac\right), which is often stated equivalently as a separable kernel,, 10.13 (22). \sum_^\infty \frac \left(\frac u 2\right)^n = \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 \sum_^\infty u^n \langle f, \psi_n \rangle \langle \psi_n, g \rangle = \iint E(x, y; u) f(x) \overline \,dx \,dy \to \int f(x) \overline \,dx = \langle f, g \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 = \sum_^\infty \langle f, \psi_n \rangle \psi_n. In order to prove the above equality for , the
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of
Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real number, rea ...
s is used repeatedly: \rho \sqrt e^ = \int e^ \,ds \quad \text\rho > 0. The Hermite polynomial is then represented as H_n(x) = (-1)^n e^ \frac \left( \frac \int e^ \,ds \right) = (-1)^n e^\frac \int (is)^n e^ \,ds. With this representation for and , it is evident that \begin E(x, y; u) &= \sum_^\infty \frac \, H_n(x) H_n(y) e^ \\ &= \frac\iint\left( \sum_^\infty \frac (-ust)^n \right ) e^\, ds\,dt \\ & =\frac\iint e^ \, e^\, ds\,dt, \end and this yields the desired resolution of the identity result, using again the Fourier transform of Gaussian kernels under the substitution s = \frac, \quad t = \frac.


See also

* Hermite transform *
Legendre polynomials In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
*
Mehler kernel The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator. It was first discovered by Mehler in 1866, and since then, as Einar Hille remarked in 1932, "has been rediscovered by almost everybody ...
* Parabolic cylinder function * Romanovski polynomials * Turán's inequalities


Notes


References

* * * * *
Oeuvres complètes 12, pp.357-412English 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 The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
) {{DEFAULTSORT:Hermite Polynomials Orthogonal polynomials Polynomials Special hypergeometric functions