Editing Hermite polynomials (section)

==Hermite functions==

===Definition===
One can define the '''Hermite functions''' (often called Hermite-Gaussian functions) from the physicist's polynomials:
<math display="block">\psi_n(x) = \left (2^n n! \sqrt{\pi} \right )^{-\frac12} e^{-\frac{x^2}{2}} H_n(x) = (-1)^n \left (2^n n! \sqrt{\pi} \right)^{-\frac12} e^{\frac{x^2}{2}}\frac{d^n}{dx^n} e^{-x^2}.</math>
Thus, 
<math display="block">\sqrt{2(n+1)}~~\psi_{n+1}(x)=    \left ( x- {d\over dx}\right ) \psi_n(x).</math>

Since these functions contain the square root of the [[weight function]] and have been scaled appropriately, they are [[Orthonormality|orthonormal]]:
<math display="block">\int_{-\infty}^\infty \psi_n(x) \psi_m(x) \,dx = \delta_{nm},</math>
and they form an orthonormal basis of {{math|''L''<sup>2</sup>('''R''')}}. This fact is equivalent to the corresponding statement for Hermite polynomials (see above).

The Hermite functions are closely related to the [[Whittaker function]] {{Harv|Whittaker|Watson|1996}} {{math|''D''<sub>''n''</sub>(''z'')}}:
<math display="block">D_n(z) = \left(n! \sqrt{\pi}\right)^{\frac12} \psi_n\left(\frac{z}{\sqrt 2}\right) = (-1)^n e^\frac{z^2}{4} \frac{d^n}{dz^n} e^\frac{-z^2}{2}</math>
and thereby to other [[parabolic cylinder function]]s.

The Hermite functions satisfy the differential equation
<math display="block">\psi_n''(x) + \left(2n + 1 - x^2\right) \psi_n(x) = 0.</math>
This equation is equivalent to the [[Schrödinger equation]] for a harmonic oscillator in quantum mechanics, so these functions are the [[eigenfunctions]].

[[Image:Herm5.svg|thumb|center|450px|Hermite functions: 0 (blue, solid), 1 (orange, dashed), 2 (green, dot-dashed), 3 (red, dotted), 4 (purple, solid), and 5 (brown, dashed)]]
<math display="block">\begin{align}
 \psi_0(x) &= \pi^{-\frac14} \, e^{-\frac12 x^2}, \\
 \psi_1(x) &= \sqrt{2} \, \pi^{-\frac14} \, x \, e^{-\frac12 x^2}, \\
 \psi_2(x) &= \left(\sqrt{2} \, \pi^{\frac14}\right)^{-1} \, \left(2x^2-1\right) \, e^{-\frac12 x^2}, \\
 \psi_3(x) &= \left(\sqrt{3} \, \pi^{\frac14}\right)^{-1} \, \left(2x^3-3x\right) \, e^{-\frac12 x^2}, \\
 \psi_4(x) &= \left(2 \sqrt{6} \, \pi^{\frac14}\right)^{-1} \, \left(4x^4-12x^2+3\right) \, e^{-\frac12 x^2}, \\
 \psi_5(x) &= \left(2 \sqrt{15} \, \pi^{\frac14}\right)^{-1} \, \left(4x^5-20x^3+15x\right) \, e^{-\frac12 x^2}.
\end{align}</math>
[[Image:Herm50.svg|thumb|center|680px|Hermite functions: 0 (blue, solid), 2 (orange, dashed), 4 (green, dot-dashed), and 50 (red, solid)]]

=== Recursion relation ===
Following recursion relations of Hermite polynomials, the Hermite functions obey
<math display="block">\psi_n'(x) = \sqrt{\frac{n}{2}}\,\psi_{n-1}(x) - \sqrt{\frac{n+1}{2}}\psi_{n+1}(x)</math>
and
<math display="block">x\psi_n(x) = \sqrt{\frac{n}{2}}\,\psi_{n-1}(x) + \sqrt{\frac{n+1}{2}}\psi_{n+1}(x).</math>

Extending the first relation to the arbitrary {{mvar|m}}th derivatives for any positive integer {{mvar|m}} leads to
<math display="block">\psi_n^{(m)}(x) = \sum_{k=0}^m \binom{m}{k} (-1)^k 2^\frac{m-k}{2} \sqrt{\frac{n!}{(n-m+k)!}} \psi_{n-m+k}(x) \operatorname{He}_k(x).</math>

This formula can be used in connection with the recurrence relations for {{math|''He<sub>n</sub>''}} and {{math|''ψ''<sub>''n''</sub>}} to calculate any derivative of the Hermite functions efficiently.
===Cramér's inequality===
For real {{mvar|x}}, the Hermite functions satisfy the following bound due to [[Harald Cramér]]<ref>{{harvnb|Erdélyi|Magnus|Oberhettinger|Tricomi|1955|p=207}}.</ref><ref>{{harvnb|Szegő|1955}}.</ref> and Jack Indritz:<ref name="indritz">{{citation
 | last1 = Indritz | first1 = Jack
 | doi = 10.1090/S0002-9939-1961-0132852-2
 | issue = 6
 | journal = [[Proceedings of the American Mathematical Society]]
 | mr = 0132852
 | pages = 981–983
 | title = An inequality for Hermite polynomials
 | volume = 12
 | year = 1961| doi-access = free
 }}</ref>
<math display="block"> \bigl|\psi_n(x)\bigr| \le \pi^{-\frac14}.</math>

===Hermite functions as eigenfunctions of the Fourier transform===
The Hermite functions {{math|''ψ''<sub>''n''</sub>(''x'')}} are a set of [[eigenfunction]]s of the [[continuous Fourier transform]] {{mathcal|F}}. To see this, take the physicist's version of the generating function and multiply by {{math|''e''<sup>−{{sfrac|1|2}}''x''<sup>2</sup></sup>}}. This gives
<math display="block">e^{-\frac12 x^2 + 2xt - t^2} = \sum_{n=0}^\infty e^{-\frac12 x^2} H_n(x) \frac{t^n}{n!}.</math>

The Fourier transform of the left side is given by
<math display="block">\begin{align}
 \mathcal{F} \left\{ e^{ -\frac12 x^2 + 2xt - t^2 } \right\}(k)
  &= \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^\infty e^{-ixk}e^{-\frac12 x^2 + 2xt - t^2}\, dx \\
  &= e^{-\frac12 k^2 - 2kit + t^2 } \\
  &= \sum_{n=0}^\infty e^{ -\frac12 k^2 } H_n(k) \frac{(-it)^n}{n!}.
\end{align}</math>

The Fourier transform of the right side is given by
<math display="block">\mathcal{F} \left\{ \sum_{n=0}^\infty e^{-\frac12 x^2} H_n(x) \frac {t^n}{n!} \right\} = \sum_{n=0}^\infty \mathcal{F} \left \{ e^{-\frac12 x^2} H_n(x) \right\} \frac{t^n}{n!}.</math>

Equating like powers of {{mvar|t}} in the transformed versions of the left and right sides finally yields
<math display="block">\mathcal{F} \left\{ e^{-\frac12 x^2} H_n(x) \right\} = (-i)^n e^{-\frac12 k^2} H_n(k).</math>

The Hermite functions {{math|''ψ<sub>n</sub>''(''x'')}} are thus an orthonormal basis of {{math|''L''<sup>2</sup>('''R''')}}, which ''diagonalizes the Fourier transform operator''.<ref>In this case, we used the unitary version of the Fourier transform, so the [[eigenvalue]]s are {{math|(−''i'')<sup>''n''</sup>}}. 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]].</ref> In short, we have:<math display="block">\frac{1}{\sqrt{2\pi}} \int e^{-ikx} \psi_n(x) dx = (-i)^n \psi_n(k), \quad \frac{1}{\sqrt{2\pi}} \int e^{+ikx} \psi_n(k) dk = i^n \psi_n(x)</math>

===Wigner distributions of Hermite functions===
The [[Wigner distribution function]] of the {{mvar|n}}th-order Hermite function is related to the {{mvar|n}}th-order [[Laguerre polynomial]]. The Laguerre polynomials are 
<math display="block">L_n(x) := \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{k!}x^k,</math>
leading to the oscillator Laguerre functions
<math display="block">l_n (x) := e^{-\frac{x}{2}} L_n(x).</math>
For all natural integers {{mvar|n}}, it is straightforward to see<ref>{{Citation |author-link=Gerald Folland |first=G. B. |last=Folland |title=Harmonic Analysis in Phase Space | series=Annals of Mathematics Studies |volume=122 |publisher=Princeton University Press |date=1989 |isbn=978-0-691-08528-9}}</ref> that
<math display="block">W_{\psi_n}(t,f) = (-1)^n l_n \big(4\pi (t^2 + f^2) \big),</math>
where the Wigner distribution of a function {{math|''x'' ∈ ''L''<sup>2</sup>('''R''', '''C''')}} is defined as
<math display="block"> W_x(t,f) = \int_{-\infty}^\infty x\left(t + \frac{\tau}{2}\right) \, x\left(t - \frac{\tau}{2}\right)^* \, e^{-2\pi i\tau f} \,d\tau.</math>
This is a fundamental result for the [[quantum harmonic oscillator]] discovered by [[Hilbrand J. Groenewold|Hip Groenewold]] in 1946 in his PhD thesis.<ref name="Groenewold1946">{{cite journal | last1 = Groenewold | first1 = H. J. | year = 1946 | title = On the Principles of elementary quantum mechanics | journal = Physica | volume = 12 | issue = 7| pages = 405–460 | doi = 10.1016/S0031-8914(46)80059-4 | bibcode=1946Phy....12..405G}}</ref> It is the standard paradigm of [[Phase-space formulation#Simple harmonic oscillator|quantum mechanics in phase space]].

There are [[Laguerre function#Relation to Hermite polynomials|further relations]] between the two families of polynomials.

===Partial Overlap Integrals===
It can be shown<ref>{{cite arXiv |last=Mawby|first=Clement|title=Tests of Macrorealism in Discrete and Continuous Variable Systems |date=2024 |class=quant-ph |eprint=2402.16537}}</ref><ref>{{cite arXiv |last=Moriconi|first=Marco|title=Nodes of Wavefunctions |date=2007 |eprint=quant-ph/0702260}}</ref> that the overlap between two different Hermite functions (<math> k\neq \ell </math>) over a given interval has the exact result:
<math display="block">\int_{x_1}^{x_2}\psi_{k}(x) \psi_{\ell}(x)\,dx =\frac{1}{2(\ell - k)}\left(\psi_k'(x_2)\psi_\ell(x_2)-\psi_\ell'(x_2)\psi_k(x_2)-\psi_k'(x_1)\psi_\ell(x_1)+\psi_\ell'(x_1)\psi_k(x_1)\right).
</math>

===Combinatorial interpretation of coefficients===
In the Hermite polynomial {{math|''He''<sub>''n''</sub>(''x'')}} of variance 1, the absolute value of the coefficient of {{math|''x''<sup>''k''</sup>}} is the number of (unordered) partitions of an {{mvar|n}}-element set into {{mvar|k}} singletons and {{math|{{sfrac|''n'' − ''k''|2}}}} (unordered) pairs. Equivalently, it is the number of involutions of an {{mvar|n}}-element set with precisely {{mvar|k}} fixed points, or in other words, the number of matchings in the [[complete graph]] on {{mvar|n}} vertices that leave {{mvar|k}} vertices uncovered (indeed, the Hermite polynomials are the [[matching polynomial]]s 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 number (mathematics)|telephone numbers]]
: 1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496,... {{OEIS|A000085}}.

This combinatorial interpretation can be related to complete exponential [[Bell polynomials]] as
<math display="block">\operatorname{He}_n(x) = B_n(x, -1, 0, \ldots, 0),</math>
where {{math|1=''x''<sub>''i''</sub> = 0}} for all {{math|''i'' > 2}}.

These numbers may also be expressed as a special value of the Hermite polynomials:<ref name="gfgt">{{citation
 | last1 = Banderier | first1 = Cyril
 | last2 = Bousquet-Mélou | first2 = Mireille | author2-link = Mireille Bousquet-Mélou
 | last3 = Denise | first3 = Alain
 | last4 = Flajolet | first4 = Philippe | author4-link = Philippe Flajolet
 | last5 = Gardy | first5 = Danièle
 | last6 = Gouyou-Beauchamps | first6 = Dominique
 | arxiv = math/0411250 | doi = 10.1016/S0012-365X(01)00250-3 | issue = 1–3 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]] | mr = 1884885 | pages = 29–55 | title = Generating functions for generating trees | volume = 246 | year = 2002| s2cid = 14804110
 }}</ref>
<math display="block">T(n) = \frac{\operatorname{He}_n(i)}{i^n}.</math>

=== Completeness relation ===
The [[Christoffel–Darboux formula]] for Hermite polynomials reads
<math display="block">\sum_{k=0}^n \frac{H_k(x) H_k(y)}{k!2^k} = \frac{1}{n!2^{n+1}}\,\frac{H_n(y) H_{n+1}(x) - H_n(x) H_{n+1}(y)}{x - y}.</math>

Moreover, the following [[Borel functional calculus#Resolution of the identity|completeness identity]] for the above Hermite functions holds in the sense of [[distribution (mathematics)|distributions]]:
<math display="block">\sum_{n=0}^\infty \psi_n(x) \psi_n(y) = \delta(x - y),</math>
where {{mvar|δ}} is the [[Dirac delta function]], {{math|''ψ''<sub>''n''</sub>}} the Hermite functions, and {{math|''δ''(''x'' − ''y'')}} represents the [[Lebesgue measure]] on the line {{math|1=''y'' = ''x''}} in {{math|'''R'''<sup>2</sup>}}, normalized so that its projection on the horizontal axis is the usual Lebesgue measure.

This distributional identity follows {{harvtxt|Wiener|1958}} by taking {{math|''u'' → 1}} in [[Mehler's formula]], valid when {{math|−1 < ''u'' < 1}}:
<math display="block">E(x, y; u) := \sum_{n=0}^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac{1}{\sqrt{\pi (1 - u^2)}} \, \exp\left(-\frac{1 - u}{1 + u} \, \frac{(x + y)^2}{4} - \frac{1 + u}{1 - u} \, \frac{(x - y)^2}{4}\right),</math>
which is often stated equivalently as a separable kernel,<ref>{{Citation | last1=Mehler | first1=F. G. | title=Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002152975 | language=de |trans-title=On the development of a function of arbitrarily many variables according to higher-order Laplace functions |id={{ERAM|066.1720cj}} | year=1866 | journal=Journal für die Reine und Angewandte Mathematik | issn=0075-4102 | issue=66 | pages=161–176}}. See p. 174, eq. (18) and p. 173, eq. (13).</ref><ref>{{harvnb|Erdélyi|Magnus|Oberhettinger|Tricomi|1955|page=194}}, 10.13 (22).</ref>
<math display="block">\sum_{n=0}^\infty \frac{H_n(x) H_n(y)}{n!} \left(\frac u 2\right)^n = \frac{1}{\sqrt{1 - u^2}} e^{\frac{2u}{1 + u}xy - \frac{u^2}{1 - u^2}(x - y)^2}.</math>

The function {{math|(''x'', ''y'') → ''E''(''x'', ''y''; ''u'')}} is the bivariate Gaussian probability density on {{math|'''R'''<sup>2</sup>}}, which is, when {{mvar|u}} is close to 1, very concentrated around the line {{math|1=''y'' = ''x''}}, and very spread out on that line. It follows that
<math display="block">\sum_{n=0}^\infty u^n \langle f, \psi_n \rangle \langle \psi_n, g \rangle = \iint E(x, y; u) f(x) \overline{g(y)} \,dx \,dy \to \int f(x) \overline{g(x)} \,dx = \langle f, g \rangle</math>
when {{math|''f''}} and {{math|''g''}} are continuous and compactly supported.

This yields that {{mvar|f}} can be expressed in Hermite functions as the sum of a series of vectors in {{math|''L''<sup>2</sup>('''R''')}}, namely,
<math display="block">f = \sum_{n=0}^\infty \langle f, \psi_n \rangle \psi_n.</math>

In order to prove the above equality for {{math|''E''(''x'',''y'';''u'')}}, the [[Fourier transform]] of [[Gaussian function]]s is used repeatedly:
<math display="block">\rho \sqrt{\pi} e^{-\frac{\rho^2 x^2}{4}} = \int e^{isx - \frac{s^2}{\rho^2}} \,ds \quad \text{for }\rho > 0.</math>

The Hermite polynomial is then represented as
<math display="block"> H_n(x) = (-1)^n e^{x^2} \frac {d^n}{dx^n} \left( \frac {1}{2\sqrt{\pi}} \int e^{isx - \frac{s^2}{4}} \,ds \right) = (-1)^n e^{x^2}\frac{1}{2\sqrt{\pi}} \int (is)^n e^{isx - \frac{s^2}{4}} \,ds.</math>

With this representation for {{math|''H<sub>n</sub>''(''x'')}} and {{math|''H<sub>n</sub>''(''y'')}}, it is evident that
<math display="block">\begin{align}
 E(x, y; u) &= \sum_{n=0}^\infty \frac{u^n}{2^n n! \sqrt{\pi}} \, H_n(x) H_n(y) e^{-\frac{x^2+y^2}{2}} \\
  &= \frac{e^{\frac{x^2+y^2}{2}}}{4\pi\sqrt{\pi}}\iint\left( \sum_{n=0}^\infty \frac{1}{2^n n!} (-ust)^n \right ) e^{isx+ity - \frac{s^2}{4} - \frac{t^2}{4}}\, ds\,dt \\
  & =\frac{e^{\frac{x^2+y^2}{2}}}{4\pi\sqrt{\pi}}\iint e^{-\frac{ust}{2}} \, e^{isx+ity - \frac{s^2}{4} - \frac{t^2}{4}}\, ds\,dt,
\end{align}</math>
and this yields the desired resolution of the identity result, using again the Fourier transform of Gaussian kernels under the substitution
<math display="block">s = \frac{\sigma + \tau}{\sqrt 2}, \quad t = \frac{\sigma - \tau}{\sqrt 2}.</math>