Hermite polynomials

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev 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 in his later 1865 publications.
Definition.
Like the other classical orthogonal polynomials, the Hermite polynomials can be defined from several different starting points. Noting from the outset that there are two different standardizations in common use, one convenient method is as follows:
These equations have the form of a Rodrigues' formula and can also be written as,
The two definitions are not exactly identical; each is a rescaling of the other:
These are Hermite polynomial sequences of different variances; see the material on variances below.
The notation and is that used in the standard references.
The polynomials are sometimes denoted by , especially in probability theory, because
is the probability density function for the normal distribution with expected value 0 and standard deviation 1.
As an alternative to calculating th-order derivatives of and , an easier, less computationally-intensive method of sequentially deriving individual terms of the th-order Hermite polynomials is to consider the combination of coefficients in the corresponding terms in the th-order Hermite polynomial. For the probabilist's notation, follow the following rules:
Thus, for , and hence the first term is , with a coefficient of 1. The second term has a power of equal to 6 − 2 = 4. The coefficient is obtained by taking the coefficient of the second term in (which is 10) and adding to it the product of the power of and its coefficient in the first term of (which are 5 and 1, respectively). Thus, 10 + 5 × 1 = 15. Make this coefficient negative, since this is an even-numbered term. The third term in has a power of equal to 2 (which is 2less than the power of in the second term), and its coefficient is 15 + 3 × 10 = 45, where 15 is the coefficient of the third term in ; 3 is the power of in the second term of the polynomial; and 10 is the coefficient of the second term in . This coefficient (45) is positive, since this is an odd-numbered term. Finally, the fourth term in is the zeroth power in (which is 2less than the power of in the third term), and its coefficient is 0 + 1 × 15 = 15, where 0 is the coefficient of the (nonexistent) fourth term in the polynomial; 1 is the power of in the third term of , and 15 is the coefficient of the third term in . This coefficient (15) is made negative, since this is an even-numbered term. Thus, .
For , the first term is ; the second coefficient is 15 + 6 × 1 = 21 (negative, i.e. ). The third coefficient is 45 + 4 × 15 = 105 (positive, i.e. ). The fourth coefficient is 15 + 2 × 45 = 105 (negative, i.e. −105"x"). Thus, .
For the physicist's notation, follow the following rules:
Thus, for , the first coefficient is 26 = 64 (i.e. ). The second coefficient is 2 × 160 + 5 × 32 = 480 (negative, i.e. ). The third coefficient is 2 × 120 + 3 × 160 = 720 (positive, i.e. ). The fourth coefficient is 2 × 0 + 1 × 120 = 120 (negative, i.e., −120). Thus, .
For , the first coefficient is 27 = 128 (i.e., ). The second coefficient is 2 × 480 + 6 × 64 = 1344 (negative, i.e. ). The third coefficient is 2 × 720 + 4 × 480 = 3360 (positive, i.e. ). The fourth coefficient is 2 × 120 + 2 × 720 = 1680 (negative, i.e. \quad (\text{for }\mathit{He})</math>
or
i.e., we have
Furthermore,
or
where formula_13 is the Kronecker delta.
The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.
Completeness.
The Hermite polynomials (probabilist's or physicist's) form an orthogonal basis of the Hilbert space of functions satisfying
in which the inner product is given by the integral
including the Gaussian weight function defined in the preceding section
An orthogonal basis for is a "complete" orthogonal system. For an orthogonal system, "completeness" is equivalent to the fact that the 0 function is the only function orthogonal to "all" functions in the system.
Since the linear span of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that if satisfies
for every , then .
One possible way to do this is to appreciate that the entire function
vanishes identically. The fact then that for every real means that the Fourier transform of is 0, hence is 0 almost everywhere. Variants of the above completeness proof apply to other weights with exponential decay.
In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see section on the Completeness relation below).
An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for consists in introducing Hermite "functions" (see below), and in saying that the Hermite functions are an orthonormal basis for .
Hermite's differential equation.
The probabilist's Hermite polynomials are solutions of the differential equation
where is a constant. Imposing the boundary condition that should be polynomially bounded at infinity, the equation has solutions only if is a non-negative integer, and the solution is uniquely given by formula_19, where formula_20 denotes a constant.
Rewriting the differential equation as an eigenvalue problem
the Hermite polynomials formula_22 may be understood as eigenfunctions of the differential operator formula_23 . This eigenvalue problem is called the Hermite equation, although the term is also used for the closely related equation
whose solution is uniquely given in terms of physicist's Hermite polynomials in the form formula_25, where formula_20 denotes a constant, after imposing the boundary condition that should be polynomially bounded at infinity.
The general solutions to the above second-order differential equations are in fact linear combinations of both Hermite polynomials and confluent hypergeometric functions of the first kind. For example, for the physicist's Hermite equation
the general solution takes the form
where formula_20 and formula_30 are constants, formula_31 are physicist's Hermite polynomials (of the first kind), and formula_32 are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as formula_33 where formula_34 are Confluent hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions, see below.
With more general boundary conditions, the Hermite polynomials can be generalized to obtain more general analytic functions for complex-valued . An explicit formula of Hermite polynomials in terms of contour integrals is also possible.
Recurrence relation.
The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation
Individual coefficients are related by the following recursion formula:
and , , .
For the physicist's polynomials, assuming 
we have
Individual coefficients are related by the following recursion formula:
and , , .
The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity
Equivalently, by Taylor-expanding,
These umbral identities are self-evident and included in the differential operator representation detailed below, 
In consequence, for the th derivatives the following relations hold:
It follows that the Hermite polynomials also satisfy the recurrence relation
These last relations, together with the initial polynomials and , can be used in practice to compute the polynomials quickly.
Turán's inequalities are
Moreover, the following multiplication theorem holds:
Explicit expression.
The physicist's Hermite polynomials can be written explicitly as
These two equations may be combined into one using the floor function:
The probabilist's Hermite polynomials have similar formulas, which may be obtained from these by replacing the power of with the corresponding power of and multiplying the entire sum by :
Inverse explicit expression.
The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials are
The corresponding expressions for the physicist's Hermite polynomials follow directly by properly scaling this:
Generating function.
The Hermite polynomials are given by the exponential generating function
This equality is valid for all complex values of and , and can be obtained by writing the Taylor expansion at of the entire function (in the physicist's case). One can also derive the (physicist's) generating function by using Cauchy's integral formula to write the Hermite polynomials as
Using this in the sum 
one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.
Expected values.
If is a random variable with a normal distribution with standard deviation 1 and expected value , then
The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices:
where is the double factorial. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments:
Asymptotic expansion.
Asymptotically, as , the expansion
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 wavefunction of a quantum harmonic oscillator such that it agrees with the classical approximation in the limit of the correspondence principle.
A better approximation, which accounts for the variation in frequency, is given by
A finer approximation, 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 complex and bounded, the approximation is
where is the Airy function of the first kind.
Special values.
The physicist's Hermite polynomials evaluated at zero argument are called Hermite numbers.
which satisfy the recursion relation .
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:
Relation to confluent hypergeometric functions.
The physicist's Hermite polynomials can be expressed as a special case of the parabolic cylinder functions:
in the right half-plane, where is Tricomi's confluent hypergeometric function. Similarly,
where is Kummer's confluent hypergeometric function.
Differential-operator representation.
The probabilist's Hermite polynomials satisfy the identity
where represents differentiation with respect to , 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 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.
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
with the contour encircling the origin.
Generalizations.
The probabilist's 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 , where is any positive number. These are then orthogonal with respect to the normal probability distribution whose density function is
They are given by
Now, if 
then the polynomial sequence whose th 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 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
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 formula_83 are just the absolute values of the corresponding coefficients of formula_76.
These arise as moments of normal probability distributions: The th moment of the normal distribution with expected value and variance is
where 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 (often called Hermite-Gaussian functions) from the physicist's polynomials:
Thus, 
Since these functions contain the square root of the weight function and have been scaled appropriately, they are orthonormal:
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 :
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.
Recursion relation.
Following recursion relations of Hermite polynomials, the Hermite functions obey
and
Extending the first relation to the arbitrary th derivatives for any positive integer leads to
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 and Jack Indritz:
Hermite functions as eigenfunctions of the Fourier transform.
The Hermite functions 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 . This gives
The Fourier transform of the left side is given by
The Fourier transform of the right side is given by
Equating like powers of in the transformed versions of the left and right sides finally yields
The Hermite functions are thus an orthonormal basis of , which "diagonalizes the Fourier transform operator".
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 
leading to the oscillator Laguerre functions
For all natural integers , it is straightforward to see that
where the Wigner distribution of a function is defined as
This is a fundamental result for the quantum harmonic oscillator 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.
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 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
This combinatorial interpretation can be related to complete exponential Bell polynomials as
where for all .
These numbers may also be expressed as a special value of the Hermite polynomials:
Completeness relation.
The Christoffel–Darboux formula for Hermite polynomials reads
Moreover, the following completeness identity for the above Hermite functions holds in the sense of distributions:
where is the Dirac delta function, the Hermite functions, and represents the Lebesgue measure 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 :
which is often stated equivalently as a separable kernel,
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
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,
In order to prove the above equality for , the Fourier transform of Gaussian functions is used repeatedly:
The Hermite polynomial is then represented as
With this representation for and , it is evident that
and this yields the desired resolution of the identity result, using again the Fourier transform of Gaussian kernels under the substitution