Hermite polynomials

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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; in numerical analysis as Gaussian quadrature; in finite element methods as shape functions for beams; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator. They are also used in systems theory in connection with nonlinear operations on Gaussian noise. They were defined by Template:Harvtxt [1] though in scarcely recognizable form, and studied in detail by Chebyshev (1859).[2] Chebyshev's work was overlooked and they were named later after Charles Hermite who wrote on the polynomials in 1864 describing them as new.[3] They were consequently not new although in later 1865 papers Hermite was the first to define the multidimensional polynomials.

Definition

There are two different ways of standardizing the Hermite polynomials:

  • The "probabilists' Hermite polynomials" are given by
,
  • while the "physicists' Hermite polynomials" are given by
.

These two definitions are not exactly identical; each one is a rescaling of the other,

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 Template:Harvs and Abramowitz & Stegun. The polynomials Hen are sometimes denoted by Hn, especially in probability theory, because

is the probability density function for the normal distribution with expected value 0 and standard deviation 1.

The first six probabilists' Hermite polynomials Hen(x).

The first eleven probabilists' Hermite polynomials are:

The first six (physicists') Hermite polynomials Hn(x).

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 with respect to the weight function (measure)

      (He)

or

      (H)

i.e., we have

Furthermore,

     (probabilists')

or

      (physicists').

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 of the Hilbert space 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(Rw(x) dx) 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(Rw(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 Template:Mvar satisfies

for every Template:Mvar ≥ 0, then Template:Mvar = 0.

One possible way to do this is to appreciate that the entire function

vanishes identically. The fact then that F(it) = 0 for every Template:Mvar real means that the Fourier transform of f(x) exp(−x2) is 0, hence Template:Mvar 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 L2(Rw(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 Template:Mvar is a constant, with the boundary conditions that Template:Mvar 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) = Heλ(x).

Rewriting the differential equation as an eigenvalue problem

solutions are the eigenfunctions of the differential operator Template:Mvar. 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 functions Heλ(z) for Template:Mvar a complex index. An explicit formula can be given in terms of a contour integral Template:Harv.

Recursion relation

The sequence of Hermite polynomials also satisfies the recursion

   (probabilists')

Individual coefficients are related by the following recursion formula:

and Template:Mvar[0,0]=1, Template:Mvar[1,0]=0, Template:Mvar[1,1]=1.

(Assuming :)

  (physicists')

Individual coefficients are related by the following recursion formula:

and a[0,0]=1, a[1,0]=0, a[1,1]=2.

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

   (probabilists')
   (physicists')

or, equivalently, by Taylor expanding,

   (probabilists')
   (physicists')

In consequence, for the m-th derivatives the following relations hold:

   (probabilists')
   (physicists')

It follows that the Hermite polynomials also satisfy the recurrence relation

   (probabilists')
   (physicists')

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 are

Moreover, the following multiplication theorem holds:

Explicit expression

The physicists' Hermite polynomials can be written explicitly as

for even value of Template:Mvar and

for odd values of Template:Mvar.

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

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

Generating function

The Hermite polynomials are given by the exponential generating function

   (probabilists')


   (physicists').

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 values

If X is a random variable with a normal distribution with standard deviation 1 and expected value Template:Mvar, then

   (probabilists')

The moments of the standard normal 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 probabilists' Hermite polynomials as moments

Asymptotic expansion

Asymptotically, as n → ∞, the expansion

  (physicist[4])

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, can be further simplified, in the limit, to

This expansion is needed to resolve the wave-function of a quantum harmonic oscillator such that it agrees with the classical approximation in the limit of the correspondence principle.

A finer approximation,[5] which takes into account the uneven spacing of the zeros near the edges, 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 Template:Mvar complex and bounded, then

where Ai(·) is the Airy function of the first kind.

Special Values

The Hermite polynomials evaluated at zero argument are called Hermite numbers.

which satisfy the recursion relation . In terms of the probabilist's polynomials this translates to

Relations to other functions

Laguerre polynomials

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

   (physicists')
   (physicists')

Relation to confluent hypergeometric functions

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

   (physicists')

where is Whittaker's confluent hypergeometric function. Similarly,

   (physicists')
   (physicists')

where is Kummer's confluent hypergeometric function.

Differential operator representation

The probabilists' Hermite polynomials satisfy the identity

where Template:Mvar represents differentiation with respect to Template:Mvar, and the exponential is interpreted by expanding it as a power series. There are no delicate questions of convergence of this series when it operates on polynomials, since all but finitely many terms vanish.

Since the power series coefficients of the exponential are well known, and higher order derivatives of the monomial 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 Template:Mvar 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−−cf. [[Weierstrass transform#The inverse|Template:Mvar]]. Since they are an Appell sequence, they are a fortiori a Sheffer sequence.

Contour integral representation

From the generating function representation above, we see that the Hermite polynomials have a representation in terms of a contour integral, as

   (probabilists')
   (physicists')

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.

Scaling, one may analogously speak of generalized Hermite polynomials

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

They are given by

In particular, the physicists' Hermite polynomials are thus

Now, if

then the polynomial sequence whose Template:Mvarth term is

is called the umbral composition of the two polynomial sequences. 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. (See the above section on Appel sequences and on the #Differential operator representation, which leads to a ready derivation of it. This binomial type identity, for Template:Mvar = Template:Mvar = 1/2, 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

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 Template:Mvar > 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 n-th moment of the normal distribution with expected value Template:Mvar and variance Template:Mvar2 is

where Template:Mvar 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 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 and Watson, 1962) Dn(z),

and thereby to other parabolic cylinder functions.

The Hermite functions 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.

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).

Recursion relation

Following recursion relations of Hermite polynomials, the Hermite functions obey

as well as

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

This formula can be used in connection with the recurrence relations for Hen and ψn to calculate any derivative of the Hermite functions efficiently.

Cramér's inequality

The Hermite functions satisfy the following bound due to Harald Cramér[6][7]

for Template:Mvar real, where the constant Template:Mvar is less than 1.086435.

Hermite functions as eigenfunctions of the Fourier transform

The Hermite functions ψn(x) are a set of eigenfunctions of the continuous Fourier transform. To see this, take the physicist's version of the generating function and multiply by exp(−x2/2). This gives