Hermite polynomials

Hermite polynomials

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Hermite polynomials are a classical orthogonal
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...

polynomial sequence
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...

that arise in probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, such as the Edgeworth series
Edgeworth series
The Gram–Charlier A series , and the Edgeworth series are series that approximate a probability distribution in terms of its cumulants...

; in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

, as an example of an Appell sequence, obeying the umbral calculus
Umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them. These techniques were introduced by and are sometimes called Blissard's symbolic method...

; in numerical analysis as Gaussian quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration....

; and in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, where they give rise to the eigenstates of the quantum harmonic oscillator
Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

. They are also used in systems theory in connection with nonlinear operations on Gaussian noise
Gaussian noise
Gaussian noise is statistical noise that has its probability density function equal to that of the normal distribution, which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian-distributed. A special case is white Gaussian noise, in which...

. They are named after Charles Hermite
Charles Hermite
Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....

(1864) although they were studied earlier by and Chebyshev (1859).

There are two different standard ways of normalizing Hermite polynomials:

(the "probabilists' Hermite polynomials"), and

(the "physicists' Hermite polynomials").
Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Hermite polynomials are a classical orthogonal
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...

polynomial sequence
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...

that arise in probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, such as the Edgeworth series
Edgeworth series
The Gram–Charlier A series , and the Edgeworth series are series that approximate a probability distribution in terms of its cumulants...

; in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

, as an example of an Appell sequence, obeying the umbral calculus
Umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them. These techniques were introduced by and are sometimes called Blissard's symbolic method...

; in numerical analysis as Gaussian quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration....

; and in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, where they give rise to the eigenstates of the quantum harmonic oscillator
Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

. They are also used in systems theory in connection with nonlinear operations on Gaussian noise
Gaussian noise
Gaussian noise is statistical noise that has its probability density function equal to that of the normal distribution, which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian-distributed. A special case is white Gaussian noise, in which...

. They are named after Charles Hermite
Charles Hermite
Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....

(1864) although they were studied earlier by and Chebyshev (1859).

Definition

There are two different standard ways of normalizing Hermite polynomials:

(the "probabilists' Hermite polynomials"), and

(the "physicists' Hermite polynomials"). These two definitions are not exactly equivalent; either is a rescaling of the other, to wit

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

The notation He and H is that used in the standard references and Abramowitz & Stegun.
The polynomials Hen are sometimes denoted by Hn, especially in probability theory, because

is the probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

for the normal distribution with expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

0 and standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

1.

The first eleven probabilists' Hermite polynomials are:

and the first eleven physicists' Hermite polynomials are:

Properties

Hn is a polynomial of degree n. The probabilists' version He has leading coefficient 1, while the physicists' version H has leading coefficient 2n.

Orthogonality

Hn(x) and Hen(x) are nth-degree polynomials for n = 0, 1, 2, 3, .... These polynomials are orthogonal
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...

with respect to the weight function (measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

)
(He)

or
(H)

i.e., we have
when m ≠ n. Furthermore,
(probabilist)
or
(physicist).
The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.

Completeness

The Hermite polynomials (probabilist or physicist) form an orthogonal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

of the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

of functions satisfying

in which the inner product is given by the integral including the Gaussian weight function w(x) defined in the preceding section,

An orthogonal basis for L2(R, w(x) dx)
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

is a complete orthogonal system. For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function ƒ ∈ L2(R, w(x) dx) 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 ƒ satisfies

for every n ≥ 0, then ƒ = 0. One possible way to do it is to see that the entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

vanishes identically. The fact that F(it) = 0 for every t real means that the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

of ƒ(x) exp(−x2) is 0, hence ƒ 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 "Completeness relation" below).

An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for L2(R, w(x) dx) consists in introducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis for L2(R).

Hermite's differential equation

The probabilists' Hermite polynomials are solutions of the differential equation
where λ is a constant, with the boundary conditions that u should be polynomially bounded at infinity. With these boundary conditions, the equation has solutions only if λ is a non-negative integer, and up to an overall scaling, the solution is uniquely given by u(x) = Hλ(x). Rewriting the differential equation as an eigenvalue problem
solutions are the eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...

s of the differential operator L. This eigenvalue problem is called the Hermite equation, although the term is also used for the closely related equation
whose solutions are the physicists' Hermite polynomials.

With more general boundary conditions, the Hermite polynomials can be generalized to obtain more general analytic function
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, categories that are similar in some ways, but different in others...

s Hλ(z) for λ a complex index. An explicit formula can be given in terms of a contour integral .

Recursion relation

The sequence of Hermite polynomials also satisfies the recursion
Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...

(probabilist) (physicist)

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

or equivalently,
(probabilist) (physicist)

(the equivalence of these last two identities may not be obvious, but its proof is a routine exercise).

It follows that the Hermite polynomials also satisfy the recurrence relation
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....

(probabilist) (physicist)

These last relations, together with the initial polynomials H0(x) and H1(x), can be used in practice to compute the polynomials quickly.

Turán's inequalities
Turán's inequalities
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by . There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors....

are

Moreover, the following multiplication theorem
Multiplication theorem
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name...

holds:

Explicit expression

The physicists' Hermite polynomials can be written explicitly as

for even values of n and

for odd values of n. These two equations may be combined into one using the floor
Floor function
In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...

function:

The probabilists' Hermite polynomials He have similar formulas, which may be obtained from these by replacing
the power of 2x with the corresponding power of (√2)x, and multiplying the entire sum by 2-n/2.

Generating function

The Hermite polynomials are given by the exponential generating function
(probabilist)
(physicist).

This equality is valid for all x, t complex, and can be obtained by writing the Taylor expansion at x of the entire function z → exp(−z2) (in 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.

Expected value

If X is a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

with a normal distribution with standard deviation 1 and expected value μ then
(probabilist)

Asymptotic expansion

Asymptotically, as tends to infinity, the expansion (physicist)
holds true.
For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude
Which, using Stirling's approximation
Stirling's approximation
In mathematics, Stirling's approximation is an approximation for large factorials. It is named after James Stirling.The formula as typically used in applications is\ln n! = n\ln n - n +O\...

, can be further simplified, in the limit, to
This expansion is needed to resolve the wave-function of a quantum harmonic oscillator
Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

such that it agrees with the classical approximation in the limit of the correspondence principle
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

.

A finer approximation, which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution , for , with which one has the uniform approximation

Similar approximations hold for the monotonic and transition regions. Specifically, if for then
while for with complex and bounded then
where is the Airy function
Airy function
In the physical sciences, the Airy function Ai is a special function named after the British astronomer George Biddell Airy...

of the first kind.

Laguerre polynomials

The Hermite polynomials can be expressed as a special case of the Laguerre polynomials
Laguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:x\,y + \,y' + n\,y = 0\,which is a second-order linear differential equation....

.
(physicist) (physicist)

Relation to confluent hypergeometric functions

The Hermite polynomials can be expressed as a special case of the parabolic cylinder functions.
(physicist)

where is Whittaker's confluent hypergeometric function
Confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity...

. Similarly,
(physicist)
(physicist)

where is Kummer's confluent hypergeometric function
Confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity...

.

Differential operator representation

The probabilists' Hermite polynomials satisfy the identity

where D represents differentiation with respect to x, and the exponential
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

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 xn can be written down explicitly, this differential operator representation gives rise to a concrete formula for the coefficients of Hn that can be used to quickly compute these polynomials.

Since the formal expression for the Weierstrass transform
Weierstrass transform
In mathematics, the Weierstrass transform of a function f : R → R, named after Karl Weierstrass, is the function F defined by...

W is eD2, we see that the Weierstrass transform of (√2)nHen(x/√2) is xn.
Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series.

The existence of some formal power series g(D), with nonzero constant coefficient, such that Hen(x) = g(D)xn, 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

The Hermite polynomials have a representation in terms of a contour integral, as
(probabilist)
(physicist)

with the contour encircling the origin.

Generalizations

The (probabilists') Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is

which has expected value 0 and variance 1. One may speak of Hermite polynomials

of variance α, where α is any positive number. These are orthogonal with respect to the normal probability distribution whose density function is

They are given by

In particular, the physicists' Hermite polynomials are

If

then the polynomial sequence whose nth term is

is the umbral composition of the two polynomial sequences, and it can be
shown to satisfy the identities

and

The last identity is expressed by saying that this parameterized family of polynomial sequences is a cross-sequence.

"Negative variance"

Since polynomial sequences form a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

under the operation of umbral composition, one may denote by

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 α > 0, the coefficients of Hen[−α](x) are just the absolute values of the corresponding coefficients of Hen[α](x).

These arise as moments of normal probability distributions: The nth moment of the normal distribution with expected value μ and variance σ2 is

where X is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that

Hermite functions

One can define the Hermite functions from the physicists' polynomials:

Since these functions contain the square root of the weight function, and have been scaled
appropriately, they are orthonormal:

and form an orthonormal basis of L2(R). This fact is equivalent to the corresponding statement for Hermite polynomials (see above).

The Hermite functions are closely related to the Whittaker function
Whittaker function
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric...

(Whittaker and Watson, 1962) :

and thereby to other parabolic cylinder functions. The Hermite functions satisfy the differential equation:

This equation is equivalent to the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

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

Cramér's inequality

The Hermite functions satisfy the following bound due to Harald Cramér
Harald Cramér
Harald Cramér was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. He was once described by John Kingman as "one of the giants of statistical theory".-Early life:Harald Cramér was born in Stockholm, Sweden on September...

for x real, where the constant K is less than 1.086435.

Hermite functions as eigenfunctions of the Fourier transform

The Hermite functions are a set of eigenfunctions of the continuous Fourier transform
Continuous Fourier transform
The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes of the individual notes that make...

. To see this, take the physicist's version of the generating function and multiply by exp(−x 2/2). This gives

Choosing the unitary representation of the Fourier transform, the Fourier transform of the left hand side is given by

The Fourier transform of the right hand side is given by

Equating like powers of t in the transformed versions of the left- and right-hand sides gives

The Hermite functions are therefore an orthonormal basis of L2(R) which diagonalizes the Fourier transform operator. In this case, we chose the unitary version of the Fourier transform, so the eigenvalues are (−i) n.

Combinatorial interpretation of coefficients

In the Hermite polynomial Hn(x) of variance 1, the absolute value of the coefficient of xk is the number of (unordered) partitions of an n-member set into k singletons and (n − k)/2 (unordered) pairs.

Completeness relation

The Christoffel–Darboux formula
Christoffel–Darboux formula
In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by and . It states thatwhere f_j is the jth term of a set of orthogonal polynomials of norm hj and leading coefficient kj....

Moreover, the following identity holds in the sense of distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

where δ is the Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

, (ψn) the Hermite functions, and δ(x − y) represents the Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

on the line y = x in R2, normalized so that its projection on the horizontal axis is the usual Lebesgue measure. This distributional identity follows by letting u → 1 in Mehler's formula, valid when −1 < u < 1:

which is often stated equivalently as

The function (x, y) → E(x, y; u) is the density for a Gaussian measure on R2 which is, when u is close to 1, very concentrated around the line y = x, and very spread out on that line. It follows that

when ƒ, g are continuous and compactly supported. This yields that ƒ can be expressed from the Hermite functions, as sum of a series of vectors in L2(R), namely

In order to prove the equality above for E(x, y; u), the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

of Gaussian functions will be used several times,

The Hermite polynomial is then represented as

With this representation for Hn(x) and Hn(y), one sees that

and this implies the desired result, using again the Fourier transform of Gaussian kernels after performing the substitution