This article is about the family of orthogonal polynomials on the real line. For polynomial interpolation on a segment using derivatives, see
Hermite interpolation. For integral transform of Hermite polynomials, see
Hermite transform.
Hermite polynomials were defined by
Pierre-Simon Laplace in 1810,^{
[1]}^{
[2]} though in scarcely recognizable form, and studied in detail by
Pafnuty Chebyshev in 1859.^{
[3]} Chebyshev's work was overlooked, and they were named later after
Charles Hermite, who wrote on the polynomials in 1864, describing them as new.^{
[4]} 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:
The "probabilist's Hermite polynomials" are given by
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.^{
[5]}
The polynomials He_{n} are sometimes denoted by H_{n}, especially in probability theory, because
The nth-order Hermite polynomial is a polynomial of degree n. The probabilist's version He_{n} has leading coefficient 1, while the physicist's version H_{n} has leading coefficient 2^{n}.
Symmetry
From the Rodrigues formulae given above, we can see that H_{n}(x) and He_{n}(x) are
even or odd functions depending on n:
H_{n}(x) and He_{n}(x) are nth-degree polynomials for n = 0, 1, 2, 3,.... These
polynomials are orthogonal with respect to the weight function (
measure)
including the
Gaussian weight function w(x) defined in the preceding section
An orthogonal basis for L^{2}(R, w(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 f ∈ L^{2}(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 f satisfies
vanishes identically. The fact then that F(it) = 0 for every real t means that the
Fourier transform of f(x)e^{−x2} is 0, hence f 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 L^{2}(R, w(x) dx) consists in introducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis for L^{2}(R).
where λ is a constant. Imposing the boundary condition that u should be polynomially bounded at infinity, the equation has solutions only if λ is a non-negative integer, and the solution is uniquely given by $u(x)=C_{1}He_{\lambda }(x)$, where $C_{1}$ denotes a constant.
the Hermite polynomials $He_{\lambda }(x)$ may be understood as
eigenfunctions of the differential operator $L[u]$ . 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 $C_{1}$ denotes a constant, after imposing the boundary condition that u 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 $C_{1}$ and $C_{2}$ 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 $h_{\lambda }(x)={}_{1}F_{1}(-{\tfrac {\lambda }{2}};{\tfrac {1}{2}};x^{2})$ where ${}_{1}F_{1}(a;b;z)$ are
Confluent hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions, see below.
The probabilist's Hermite polynomials He have similar formulas, which may be obtained from these by replacing the power of 2x with the corresponding power of √2x and multiplying the entire sum by 2^{−.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}n/2}:
This equality is valid for all
complex values of x and t, and can be obtained by writing the Taylor expansion at x of the entire function z → e^{−z2} (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
where (2n − 1)!! is the
double factorial. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments:
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.
Since the power-series coefficients of the exponential are well known, and higher-order derivatives of the monomial x^{n} can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients of H_{n} that can be used to quickly compute these polynomials.
Since the formal expression for the
Weierstrass transformW is e^{D2}, we see that the Weierstrass transform of (√2)^{n}He_{n}(x/√2) is x^{n}. 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 He_{n}(x) = g(D)x^{n}, 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.
The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is
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 α = β = 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
${\mathit {He}}_{n}^{[-\alpha ]}(x)$
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 α > 0, the coefficients of ${\mathit {He}}_{n}^{[-\alpha ]}(x)$ are just the absolute values of the corresponding coefficients of ${\mathit {He}}_{n}^{[\alpha ]}(x)$.
These arise as moments of normal probability distributions: The nth moment of the normal distribution with expected value μ and variance σ^{2} is
This formula can be used in connection with the recurrence relations for He_{n} and ψ_{n} to calculate any derivative of the Hermite functions efficiently.
Cramér's inequality
For real x, the Hermite functions satisfy the following bound due to
Harald Cramér^{
[10]}^{
[11]} and Jack Indritz:^{
[12]}
Hermite functions as eigenfunctions of the Fourier transform
The Hermite functions ψ_{n}(x) are a set of
eigenfunctions of the
continuous Fourier transformF. To see this, take the physicist's version of the generating function and multiply by e^{−1/2x2}. This gives
There are
further relations between the two families of polynomials.
Combinatorial interpretation of coefficients
In the Hermite polynomial He_{n}(x) of variance 1, the absolute value of the coefficient of x^{k} is the number of (unordered) partitions of an n-element set into k singletons and n − k/2 (unordered) pairs. Equivalently, it is the number of involutions of an n-element set with precisely k fixed points, or in other words, the number of matchings in the
complete graph on n vertices that leave k 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,... (sequence A000085 in the
OEIS).
This combinatorial interpretation can be related to complete exponential
Bell polynomials as
${\mathit {He}}_{n}(x)=B_{n}(x,-1,0,\ldots ,0),$
where x_{i} = 0 for all i > 2.
These numbers may also be expressed as a special value of the Hermite polynomials:^{
[16]}
where δ is the
Dirac delta function, ψ_{n} the Hermite functions, and δ(x − y) represents the
Lebesgue measure on the line y = x in R^{2}, normalized so that its projection on the horizontal axis is the usual Lebesgue measure.
The function (x, y) → E(x, y; u) is the bivariate Gaussian probability density on R^{2}, 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
^Laplace, P.-S. (1812), Théorie analytique des probabilités [Analytic Probability Theory], vol. 2, pp. 194–203 Collected in
Œuvres complètesVII.
^Chebyshev, P. L. (1859). "Sur le développement des fonctions à une seule variable" [On the development of single-variable functions]. Bull. Acad. Sci. St. Petersb. 1: 193–200. Collected in ŒuvresI, 501–508.
^Hermite, C. (1864). "Sur un nouveau développement en série de fonctions" [On a new development in function series]. C. R. Acad. Sci. Paris. 58: 93–100. Collected in ŒuvresII, 293–303.
^Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al. (
2010) and
Abramowitz & Stegun.
^In this case, we used the unitary version of the Fourier transform, so the
eigenvalues are (−i)^{n}. The ensuing resolution of the identity then serves to define powers, including fractional ones, of the Fourier transform, to wit a
Fractional Fourier transform generalization, in effect a
Mehler kernel.
^Folland, G. B. (1989), Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122, Princeton University Press,
ISBN978-0-691-08528-9
Laplace, P. S. (1810), "Mémoire sur les intégrales définies et leur application aux probabilités, et spécialement a la recherche du milieu qu'il faut choisir entre les résultats des observations", Mémoires de l'Académie des Sciences: 279–347
Oeuvres complètes 12, pp.357-412,
English translationArchived 2016-03-04 at the
Wayback Machine.
Shohat, J.A.; Hille, Einar; Walsh, Joseph L. (1940), A bibliography on orthogonal polynomials, Bulletin of the National Research Council, vol. Number 103, Washington D.C.: National Academy of Sciences - 2000 references of Bibliography on Hermite polynomials.