In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; and in physics, as the eigenstates of the quantum harmonic oscillator. They are named in honor of Charles Hermite. For other meanings of mathematics or math, see mathematics (disambiguation). ... In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials... 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. ... Probability is the extent to which something is likely to happen or be the case[1]. Probability theory is used extensively in areas such as statistics, mathematics, science, philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. ... The Edgeworth series or Gram-Charlier A series, named in honor of Francis Ysidro Edgeworth, are series that approximate a probability distribution in terms of its cumulants. ... Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ... In mathematics, a polynomial sequence, i. ... In mathematics, before the 1970s, the term umbral calculus was understood to mean the surprising similarities between otherwise unrelated polynomial equations, and certain shadowy techniques that can be used to prove them. ... Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... Charles Hermite (pronounced in IPA, , or phonetically air-meet) (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ...

Definition

The Hermite polynomials are defined either by

(the "probabilists' Hermite polynomials"), or sometimes by

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

.

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

Below, we usually follow the first convention. That convention is often preferred by probabilists because

is the probability density function for the normal distribution with expected value 0 and standard deviation 1. In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... The normal distribution, also called Gaussian distribution (named after Carl Friedrich Gauss, a German mathematician, although Gauss was not the first to work with it), is a probability distribution of great importance in many fields. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is defined as the square root of the variance. ...

The first five (probabilists') Hermite polynomials.

The first several Hermite polynomials are: Image File history File links Hermite_poly. ... Image File history File links Hermite_poly. ...

in probabilists' notation, or

in physicists' notation.

Properties

Orthogonality

H_n(x) is an nth-degree polynomial for n = 0, 1, 2, 3, .... These polynomials are orthogonal with respect to the weight function (measure) In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p0(x), p1(x), p2(x) ... , in which each pn(x) has degree n, and such that any two different polynomials in the sequence are orthogonal to each other in the following sense: One can define an inner... In mathematics, a measure is a function that assigns a number, e. ...

(probabilist)

or

(physicist)

i.e., we have

(probabilist)

or

(physicist)

where δ_ij is the Kronecker delta, which equals unity when n = m and zero otherwise. The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

They form an orthogonal basis of the Hilbert space of functions satisfying In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...

in which the inner product is given by the integral including a gaussian function Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ...

Hermite's differential equation

The nth Hermite polynomial satisfies Hermite's differential equation:

(probabilist)

(physicist)

Recursion relation

The sequence of Hermite polynomials also satisfies the recursion A Sierpinski triangle —a confined recursion of triangles to form a geometric lattice. ...

(probabilist)

(physicist)

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

(probabilist)

(physicist)

or equivalently,

(probabilist)

(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 In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

(probabilist)

(physicist)

Generating function

The Hermite polynomials are given by the exponential generating function In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

(probabilist)

(physicist)

Expected value

If X is a random variable with a normal distribution with standard deviation 1 and expected value μ then A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...

(probabilist)

Relations to other functions

Laguerre polynomials

The Hermite polynomials can be expressed as a special case of the Laguerre polynomials. In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by These polynomials are orthogonal to each other with respect to the inner product given by Also, for each n, Ln(x) is a solution of Laguerres equation which is a second-order...

(physicist)

(physicist)

Relation to confluent hypergeometric functions

The Hermite polynomials can be expressed as a special case of the parabolic cylinder functions. In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This can be brought into two distinct forms (A) and (B) by change of variable, to (A) and (B) If f(a,z) is a solution, then so are f(a,−z), f...

(physicist)

where U(a,b;z) is Whittaker's confluent hypergeometric function. Similarly, In mathematics, the confluent hypergeometric function is formed from hypergeometric series. ...

(physicist)

(physicist)

where is Kummer's confluent hypergeometric function. In mathematics, the confluent hypergeometric function is formed from hypergeometric series. ...

Differential operator representation

The Hermite polynomials satisfy the identity

, (probabilist)

where D represents differentiation with respect to x, 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. The existence of some formal power series g(D), with nonzero constant coefficient, such that H_n(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. In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, a polynomial sequence, i. ...

Contour integral representation

The Hermite polynomials have a representation in terms of a contour integral, as This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...

(physicist)

with the contour encircling the origin.

Generalization

The (probabilists') Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution

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

They are given by

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 under the operation of umbral composition, one may denote by In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

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 H_n^[−α](x) are just the absolute values of the corresponding coefficients of H_n^[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

Applications

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 othonormal:

(physicist)

They satisfy the differential equation:

This equation is equivalent to the Schrödinger equation for a harmonic oscillator in quantum mechanics, so these functions are the eigenfunctions. In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the space- and time-dependence of quantum mechanical systems. ...

Hermite functions 0 (black), 1 (red), 2 (blue), 3 (yellow), 4 (green), and 5 (magenta).

Hermite functions 0 (black), 2 (blue), 4 (green), and 50 (magenta).

The Hermite functions ψ_n(x) are eigenfunctions of the continuous Fourier transform, with eigenvalues − iⁿ. Image File history File links Herm5. ... Image File history File links Herm5. ... Image File history File links Herm50. ... Image File history File links Herm50. ... In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

Combinatorial coefficients

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

References

• Milton Abramowitz and Irene A. Stegun, eds. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4. (See chapter 22)
• Norbert Wiener, The Fourier Integral and Certain of its Applications, (1958) Dover Publications, New York. ISBN 0-486-60272-9 .