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{{Short description|Polynomial sequence}}
{{About|the family of orthogonal polynomials on the real line|polynomial interpolation on a segment using derivatives|Hermite interpolation|integral transform of Hermite polynomials|Hermite transform}}
{{Use American English|date = March 2019}}

In [[mathematics]], the '''Hermite polynomials''' are a classical [[orthogonal polynomials|orthogonal]] [[polynomial sequence]].

The polynomials arise in:
* [[signal processing]] as [[Hermitian wavelet]]s for [[wavelet transform]] analysis
* [[probability]], such as the [[Edgeworth series]], as well as in connection with [[Brownian motion]];
* [[combinatorics]], as an example of an [[Appell sequence]], obeying the [[umbral calculus]];
* [[numerical analysis]] as [[Gaussian quadrature]]; 
* [[physics]], where they give rise to the [[eigenstate]]s of the [[quantum harmonic oscillator]]; and they also occur in some cases of the [[heat equation]] (when the term <math>\begin{align}xu_{x}\end{align}</math> is present);
* [[systems theory]] in connection with nonlinear operations on [[Gaussian noise]]. 
* [[random matrix theory]] in [[Gaussian ensemble]]s.

Hermite polynomials were defined by [[Pierre-Simon Laplace]] in 1810,<ref>{{cite journal |last1=Laplace |title=Mémoire sur les intégrales définies et leur application aux probabilités, et spécialement a la recherche du milieu qu'il faut choisir entre les resultats des observations |journal=Mémoires de la Classe des Sciences Mathématiques et Physiques de l'Institut Impérial de France |date=1811 |volume=11 |pages=297–347 |url=https://www.biodiversitylibrary.org/item/55081#page/293/mode/1up |trans-title=Memoire on definite integrals and their application to probabilities, and especially to the search for the mean which must be chosen among the results of observations |language=French}}</ref><ref>{{Citation |first=P.-S. |last=Laplace |title=Théorie analytique des probabilités |trans-title=Analytic Probability Theory |date=1812 |volume=2 |pages=194–203}} Collected in [https://gallica.bnf.fr/ark:/12148/bpt6k775950.r=Oeuvres%20complètes%20de%20Laplace.%20Tome%207?rk=21459;2 ''Œuvres complètes'' '''VII'''].</ref> though in scarcely recognizable form, and studied in detail by [[Pafnuty Chebyshev]] in 1859.<ref>{{cite journal |first=P. |last=Tchébychef |title=Sur le développement des fonctions à une seule variable |trans-title=On the development of single-variable functions |journal=Bulletin de l'Académie impériale des sciences de St.-Pétersbourg |volume=1 |date=1860 |pages=193–200 |url=https://www.biodiversitylibrary.org/item/104584#page/129/mode/1up |language=French }} Collected in [https://archive.org/details/117744684_001/page/n511/mode/2up ''Œuvres'' '''I''', 501–508.]</ref> Chebyshev's work was overlooked, and they were named later after [[Charles Hermite]], who wrote on the polynomials in 1864, describing them as new.<ref>{{cite journal |first=C. |last=Hermite |title=Sur un nouveau développement en série de fonctions |trans-title=On a new development in function series |journal=C. R. Acad. Sci. Paris |volume=58 |date=1864 |pages=93–100, 266–273 |url=https://www.biodiversitylibrary.org/item/23663#page/99/mode/1up |language=French }} Collected in ''Œuvres'' '''II''', 293–308.</ref> They were consequently not new, although Hermite was the first to define the multidimensional polynomials.

==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 <math display="block">\operatorname{He}_n(x) = (-1)^n e^{\frac{x^2}{2}}\frac{d^n}{dx^n}e^{-\frac{x^2}{2}},</math>
* while the '''"physicist's Hermite polynomials"''' are given by <math display="block">H_n(x) = (-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}.</math>

These equations have the form of a [[Rodrigues' formula]] and can also be written as,
<math display="block">\operatorname{He}_n(x) = \left(x - \frac{d}{dx} \right)^n \cdot 1, \quad H_n(x) = \left(2x - \frac{d}{dx} \right)^n \cdot 1.</math>

The two definitions are not exactly identical; each is a rescaling of the other:
<math display="block">H_n(x)=2^\frac{n}{2} \operatorname{He}_n\left(\sqrt{2} \,x\right), \quad \operatorname{He}_n(x)=2^{-\frac{n}{2}} H_n\left(\frac {x}{\sqrt 2} \right).</math>

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

The notation {{mvar|He}} and {{mvar|H}} is that used in the standard references.<ref>{{harvs|txt|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek|first4=René F. |last4=Swarttouw|year=2010}} and [[Abramowitz & Stegun]].</ref>
The polynomials {{mvar|He<sub>n</sub>}} are sometimes denoted by {{mvar|H<sub>n</sub>}}, especially in probability theory, because
<math display="block">\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}</math>
is the [[probability density function]] for the [[normal distribution]] with [[expected value]] 0 and [[standard deviation]] 1.

[[Image:Hermite poly.svg|thumb|right|390px|The first six probabilist's Hermite polynomials {{math|''He''<sub>''n''</sub>(''x'')}}]]
[[Image:Hermite poly phys.svg|thumb|right|390px|The first six (physicist's) Hermite polynomials {{math|''H''<sub>''n''</sub>(''x'')}}]]
* The first eleven probabilist's Hermite polynomials are: <math display="block">\begin{align}
\operatorname{He}_0(x) &= 1, \\
\operatorname{He}_1(x) &= x, \\
\operatorname{He}_2(x) &= x^2 - 1, \\
\operatorname{He}_3(x) &= x^3 - 3x, \\
\operatorname{He}_4(x) &= x^4 - 6x^2 + 3, \\
\operatorname{He}_5(x) &= x^5 - 10x^3 + 15x, \\
\operatorname{He}_6(x) &= x^6 - 15x^4 + 45x^2 -  15, \\
\operatorname{He}_7(x) &= x^7 - 21x^5 + 105x^3 - 105x, \\
\operatorname{He}_8(x) &= x^8 - 28x^6 + 210x^4 - 420x^2 + 105, \\
\operatorname{He}_9(x) &= x^9 - 36x^7 + 378x^5 - 1260x^3 + 945x, \\
\operatorname{He}_{10}(x) &= x^{10} - 45x^8 + 630x^6 - 3150x^4 + 4725x^2 - 945.
\end{align}</math>
* The first eleven physicist's Hermite polynomials are: <math display="block">\begin{align}
H_0(x) &= 1, \\
H_1(x) &= 2x, \\
H_2(x) &= 4x^2 - 2, \\
H_3(x) &= 8x^3 - 12x, \\
H_4(x) &= 16x^4 - 48x^2 + 12, \\
H_5(x) &= 32x^5 - 160x^3 + 120x, \\
H_6(x) &= 64x^6 - 480x^4 + 720x^2 - 120, \\
H_7(x) &= 128x^7 - 1344x^5 + 3360x^3 - 1680x, \\
H_8(x) &= 256x^8 - 3584x^6 + 13440x^4 - 13440x^2 + 1680, \\
H_9(x) &= 512x^9 - 9216x^7 + 48384x^5 - 80640x^3 + 30240x, \\
H_{10}(x) &= 1024x^{10} - 23040x^8 + 161280x^6 - 403200x^4 + 302400x^2 - 30240.
\end{align}</math>
<!--
As an alternative to calculating {{mvar|n}}th-order derivatives of {{math|''e''<sup>−{{sfrac|''x''<sup>2</sup>|2}}</sup>}} and {{math|''e''<sup>−''x''<sup>2</sup></sup>}}, an easier, less computationally-intensive method of sequentially deriving individual terms of the {{mvar|n}}th-order Hermite polynomials is to consider the combination of coefficients in the corresponding terms in the {{math|(''n'' − 1)}}th-order Hermite polynomial. For the probabilist's notation, follow the following rules:
# For the starting point in the sequence, the zeroth-order polynomial ({{math|''He''<sub>0</sub>}}) is equal to 1.
# The first term has a power of {{mvar|x}} equal to the given {{mvar|n}}th-order polynomial being derived, and the coefficient of this term is 1.
# The power of {{mvar|x}} of each successive term is two less than the preceding term.
# The coefficient of each term after the first term is calculated by taking the coefficient of the same-numbered term in the {{math|(''n'' − 1)}}th polynomial, and adding to it the product of the power of {{mvar|x}} and the corresponding coefficient of the immediately preceding term in the {{math|(''n'' − 1)}}th polynomial.
# All even-numbered terms in each polynomial are negative, and all odd-numbered terms are positive.

Thus, for {{math|''He''<sub>6</sub>}}, {{math|1=''n'' = 6}} and hence the first term is {{math|''x''<sup>6</sup>}}, with a coefficient of 1. The second term has a power of {{mvar|x}} equal to {{math|1=6− 2 = 4}}. The coefficient is obtained by taking the coefficient of the second term in {{math|''He''<sub>5</sub>}} (which is 10) and adding to it the product of the power of {{mvar|x}} and its coefficient in the first term of {{math|''He''<sub>5</sub>}} (which are 5 and 1, respectively). Thus, {{math|1=10 + 5 × 1 = 15}}. Make this coefficient negative, since this is an even-numbered term. The third term in {{math|''He''<sub>6</sub>}} has a power of {{mvar|x}} equal to 2 (which is 2less than the power of {{mvar|x}} in the second term), and its coefficient is {{math|1=15 + 3 × 10 = 45}}, where 15 is the coefficient of the third term in {{math|''He''<sub>5</sub>}}; 3 is the power of {{mvar|x}} in the second term of the {{math|''He''<sub>5</sub>}} polynomial; and 10 is the coefficient of the second term in {{math|''H''<sub>5</sub>}}. This coefficient (45) is positive, since this is an odd-numbered term. Finally, the fourth term in {{math|''He''<sub>6</sub>}} is the zeroth power in {{mvar|x}} (which is 2less than the power of {{mvar|x}} in the third term), and its coefficient is {{math|1=0 + 1 × 15 = 15}}, where 0 is the coefficient of the (nonexistent) fourth term in the {{math|''He''<sub>5</sub>}} polynomial; 1 is the power of {{mvar|x}} in the third term of {{math|''He''<sub>5</sub>}}, and 15 is the coefficient of the third term in {{math|''H''<sub>5</sub>}} . This coefficient (15) is made negative, since this is an even-numbered term. Thus, {{math|1=''He''<sub>6</sub>(''x'') = ''x''<sup>6</sup> − 15''x''<sup>4</sup> + 45''x''<sup>2</sup> − 15}}.

For {{math|''He''<sub>7</sub>}}, the first term is {{math|''x''<sup>7</sup>}}; the second coefficient is {{math|1=15 + 6 × 1 = 21}} (negative, i.e. {{math|−21''x''<sup>5</sup>}}). The third coefficient is {{math|1=45 + 4 × 15 = 105}} (positive, i.e. {{math|105''x''<sup>3</sup>}}). The fourth coefficient is {{math|1=15 + 2 × 45 = 105}} (negative, i.e. −105''x''). Thus, {{math|1=''He''<sub>7</sub>(''x'') = ''x''<sup>7</sup> − 21''x''<sup>5</sup> + 105''x''<sup>3</sup> − 105''x''}}.

For the physicist's notation, follow the following rules:
# For the starting point in the sequence, the zeroth-order polynomial ({{math|''H''<sub>0</sub>}}) is equal to 1.
# The first term has a power of {{mvar|x}} equal to the given {{mvar|n}}th-order polynomial being derived, and the coefficient of this term is {{math|2<sup>''n''</sup>}}.
# The power of {{mvar|x}} of each successive term is two less than the preceding term.
# The coefficient of each term after the first term is calculated by taking the coefficient of the same-numbered term in the {{math|(''n'' − 1)}}th polynomial, multiplying it by 2, and then adding to it the product of the power of {{mvar|x}} and the corresponding coefficient of the immediately preceding term in the {{math|(''n'' − 1)}}th polynomial.
# All even-numbered terms in each polynomial are negative, and all odd-numbered terms are positive.

Thus, for {{math|''H''<sub>6</sub>}}, the first coefficient is {{math|1=2<sup>6</sup> = 64}} (i.e. {{math|64''x''<sup>6</sup>}}). The second coefficient is {{math|1=2 × 160 + 5 × 32 = 480}} (negative, i.e. {{math|−480''x''<sup>4</sup>}}). The third coefficient is {{math|1=2 × 120 + 3 × 160 = 720}} (positive, i.e. {{math|720''x''<sup>2</sup>}}). The fourth coefficient is {{math|1=2 × 0 + 1 × 120 = 120}} (negative, i.e., −120). Thus, {{math|1=''H''<sub>6</sub>(x) = 64''x''<sup>6</sup> − 480''x''<sup>4</sup> + 720''x''<sup>2</sup> − 120}}.

For {{math|''H''<sub>7</sub>}}, the first coefficient is 2<sup>7</sup> = 128 (i.e., {{math|128''x''<sup>7</sup>}}). The second coefficient is {{math|1=2 × 480 + 6 × 64 = 1344}} (negative, i.e. {{math|−1344''x''<sup>5</sup>}}). The third coefficient is {{math|1=2 × 720 + 4 × 480 = 3360}} (positive, i.e. {{math|3360''x''<sup>3</sup>}}). The fourth coefficient is {{math|1=2 × 120 + 2 × 720 = 1680}} (negative, i.e. {{math|−1680''x''}}. Thus, {{math|''H''<sub>7</sub>(''x'') = 128''x''<sup>7</sup> − 1344''x''<sup>5</sup> + 3360''x''<sup>3</sup> − 1680''x''}}.

Recognizing that {{math|1=''H''<sub>0</sub> = 1}}, these rules can be followed to sequentially derive all {{mvar|n}}th-order Hermite polynomials from {{math|1=''n'' = 1}} towards infinity, and can be computer-coded relatively easily for practical applications. -->

==Properties==
The {{mvar|n}}th-order Hermite polynomial is a polynomial of degree {{mvar|n}}. The probabilist's version {{mvar|He<sub>n</sub>}} has leading coefficient 1, while the physicist's version {{mvar|H<sub>n</sub>}} has leading coefficient {{math|2<sup>''n''</sup>}}.

===Symmetry===
From the Rodrigues formulae given above, we can see that {{math|''H<sub>n</sub>''(''x'')}} and {{math|''He<sub>n</sub>''(''x'')}} are [[Even and odd functions|even or odd functions]] depending on {{mvar|n}}:
<math display="block">H_n(-x)=(-1)^nH_n(x),\quad \operatorname{He}_n(-x)=(-1)^n\operatorname{He}_n(x).</math>

===Orthogonality===
{{math|''H<sub>n</sub>''(''x'')}} and {{math|''He<sub>n</sub>''(''x'')}} are {{mvar|n}}th-degree polynomials for {{math|''n'' {{=}} 0, 1, 2, 3,...}}. These [[orthogonal polynomials|polynomials are orthogonal]] with respect to the ''weight function'' ([[measure (mathematics)|measure]])
<math display="block">w(x) = e^{-\frac{x^2}{2}} \quad (\text{for }\operatorname{He})</math> or <math display="block">w(x) = e^{-x^2} \quad (\text{for } H),</math>
i.e., we have
<math display="block">\int_{-\infty}^\infty H_m(x) H_n(x)\, w(x) \,dx = 0 \quad \text{for all }m \neq n.</math>

Furthermore,
<math display="block">\int_{-\infty}^\infty H_m(x) H_n(x)\, e^{-x^2} \,dx = \sqrt{\pi}\, 2^n n!\, \delta_{nm},</math>
and
<math display="block">\int_{-\infty}^\infty \operatorname{He}_m(x) \operatorname{He}_n(x)\, e^{-\frac{x^2}{2}} \,dx = \sqrt{2 \pi}\, n!\, \delta_{nm},</math>
where <math>\delta_{nm}</math> 
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 [[orthonormal basis|orthogonal basis]] of the [[Hilbert space]] of functions satisfying
<math display="block">\int_{-\infty}^\infty \bigl|f(x)\bigr|^2\, w(x) \,dx < \infty,</math>
in which the inner product is given by the integral
<math display="block">\langle f,g\rangle = \int_{-\infty}^\infty f(x) \overline{g(x)}\, w(x) \,dx</math>
including the [[gaussian function|Gaussian]] weight function {{math|''w''(''x'')}} defined in the preceding section.

An orthogonal basis for {{math|[[Lp space|''L''<sup>2</sup>('''R''', ''w''(''x'') ''dx'')]]}} is a [[Hilbert space#Orthonormal bases|''complete'' orthogonal system]]. For an orthogonal system, ''completeness'' is equivalent to the fact that the 0 function is the only function {{math|''f'' ∈ ''L''<sup>2</sup>('''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 {{mvar|f}} satisfies
<math display="block">\int_{-\infty}^\infty f(x) x^n e^{- x^2} \,dx = 0</math>
for every {{math|''n'' ≥ 0}}, then {{math|1=''f'' = 0}}.

One possible way to do this is to appreciate that the [[entire function]]
<math display="block">F(z) = \int_{-\infty}^\infty f(x) e^{z x - x^2} \,dx = \sum_{n=0}^\infty \frac{z^n}{n!} \int f(x) x^n e^{- x^2} \,dx = 0</math>
vanishes identically. The fact then that {{math|1=''F''(''it'') = 0}} for every real {{mvar|t}} means that the [[Fourier transform]] of {{math|''f''(''x'')''e''<sup>−''x''<sup>2</sup></sup>}} is 0, hence {{mvar|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|Completeness relation]] below).

An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for {{math|''L''<sup>2</sup>('''R''', ''w''(''x'') ''dx'')}} consists in introducing Hermite ''functions'' (see below), and in saying that the Hermite functions are an orthonormal basis for {{math|''L''<sup>2</sup>('''R''')}}.

===Hermite's differential equation===
The probabilist's Hermite polynomials are solutions of the [[differential equation]]
<math display="block">\left(e^{-\frac12 x^2}u'\right)' + \lambda e^{-\frac 1 2 x^2}u = 0,</math>
where {{mvar|λ}} is a constant. Imposing the boundary condition that {{mvar|u}} should be polynomially bounded at infinity, the equation has solutions only if {{mvar|λ}} is a non-negative integer, and the solution is uniquely given by <math>u(x) = C_1 \operatorname{He}_\lambda(x) </math>, where <math>C_{1}</math> denotes a constant.

Rewriting the differential equation as an [[Eigenvalue|eigenvalue problem]]
<math display="block">L[u] = u'' - x u' = -\lambda u,</math>
the Hermite polynomials <math>\operatorname{He}_\lambda(x) </math> may be understood as [[eigenfunction]]s of the differential operator <math>L[u]</math> . This eigenvalue problem is called the '''Hermite equation''', although the term is also used for the closely related equation
<math display="block">u'' - 2xu' = -2\lambda u.</math> 
whose solution is uniquely given in terms of physicist's Hermite polynomials in the form <math>u(x) = C_1 H_\lambda(x) </math>, where <math>C_{1}</math> denotes a constant, after imposing the boundary condition that {{mvar|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
<math display="block">u'' - 2xu' + 2\lambda u = 0,</math>
the general solution takes the form
<math display="block">u(x) = C_1   H_\lambda(x) + C_2  h_\lambda(x),</math>
where <math>C_{1}</math> and <math>C_{2}</math> are constants, <math>H_\lambda(x)</math> are physicist's Hermite polynomials (of the first kind), and  <math>h_\lambda(x)</math> are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as <math> h_\lambda(x) = {}_1F_1(-\tfrac{\lambda}{2};\tfrac{1}{2};x^2)</math> where  <math>{}_1F_1(a;b;z)</math> are [[Confluent hypergeometric function|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 function]]s for complex-valued {{mvar|λ}}. An explicit formula of Hermite polynomials in terms of [[contour integral]]s {{harv|Courant|Hilbert|1989}} is also possible.

===Recurrence relation===
The sequence of probabilist's Hermite polynomials also satisfies the [[recurrence relation]]
<math display="block">\operatorname{He}_{n+1}(x) = x \operatorname{He}_n(x) - \operatorname{He}_n'(x).</math>
Individual coefficients are related by the following recursion formula:
<math display="block">a_{n+1,k} = \begin{cases}
 - (k+1) a_{n,k+1} &  k = 0, \\
 a_{n,k-1} - (k+1) a_{n,k+1} & k > 0,
\end{cases}</math> 
and {{math|1=''a''<sub>0,0</sub> = 1}}, {{math|1=''a''<sub>1,0</sub> = 0}}, {{math|1=''a''<sub>1,1</sub> = 1}}.

For the physicist's polynomials, assuming 
<math display="block">H_n(x) = \sum^n_{k=0} a_{n,k} x^k,</math>
we have
<math display="block">H_{n+1}(x) = 2xH_n(x) - H_n'(x).</math>
Individual coefficients are related by the following recursion formula:
<math display="block">a_{n+1,k} = \begin{cases}
 - a_{n,k+1} & k = 0, \\ 
 2 a_{n,k-1} - (k+1)a_{n,k+1} & k > 0,
\end{cases}</math>
and {{math|1=''a''<sub>0,0</sub> = 1}}, {{math|1=''a''<sub>1,0</sub> = 0}}, {{math|1=''a''<sub>1,1</sub> = 2}}.

The Hermite polynomials constitute an [[Appell sequence]], i.e., they are a polynomial sequence satisfying the identity
<math display="block">\begin{align}
 \operatorname{He}_n'(x) &= n\operatorname{He}_{n-1}(x), \\
 H_n'(x) &= 2nH_{n-1}(x).
\end{align}</math>

An integral recurrence that is deduced and demonstrated in <ref>Hurtado Benavides, Miguel Ángel. (2020). De las sumas de potencias a las sucesiones de Appell y su caracterización a través de funcionales. [Tesis de maestría]. Universidad Sergio Arboleda.</ref> is as follows:
<math display="block">\operatorname{He}_{n+1}(x) = (n+1)\int_0^x \operatorname{He}_n(t)dt - He'_n(0),</math>

<math display="block">H_{n+1}(x) = 2(n+1)\int_0^x H_n(t)dt - H'_n(0).</math>

Equivalently, by [[Taylor series|Taylor-expanding]],
<math display="block">\begin{align}
 \operatorname{He}_n(x+y) &= \sum_{k=0}^n \binom{n}{k}x^{n-k} \operatorname{He}_{k}(y)
  &&= 2^{-\frac n 2} \sum_{k=0}^n \binom{n}{k} \operatorname{He}_{n-k}\left(x\sqrt 2\right) \operatorname{He}_k\left(y\sqrt 2\right), \\
 H_n(x+y) &= \sum_{k=0}^n \binom{n}{k}H_{k}(x) (2y)^{n-k}
  &&= 2^{-\frac n 2}\cdot\sum_{k=0}^n \binom{n}{k} H_{n-k}\left(x\sqrt 2\right) H_k\left(y\sqrt 2\right).
\end{align}</math>
These [[umbral calculus|umbral]] identities are self-evident and [[#Generalizations|included]] in the [[Hermite polynomials#Differential-operator representation|differential operator representation]] detailed below, 
<math display="block">\begin{align}
 \operatorname{He}_n(x) &= e^{-\frac{D^2}{2}} x^n, \\
 H_n(x) &= 2^n e^{-\frac{D^2}{4}} x^n.
\end{align}</math>

In consequence, for the {{mvar|m}}th derivatives the following relations hold:
<math display="block">\begin{align}
 \operatorname{He}_n^{(m)}(x) &= \frac{n!}{(n-m)!} \operatorname{He}_{n-m}(x)
  &&= m! \binom{n}{m} \operatorname{He}_{n-m}(x), \\
 H_n^{(m)}(x) &= 2^m \frac{n!}{(n-m)!} H_{n-m}(x)
  &&= 2^m m! \binom{n}{m} H_{n-m}(x).
\end{align}</math>

It follows that the Hermite polynomials also satisfy the [[recurrence relation]]
<math display="block">\begin{align}
 \operatorname{He}_{n+1}(x) &= x\operatorname{He}_n(x) - n\operatorname{He}_{n-1}(x), \\
 H_{n+1}(x) &= 2xH_n(x) - 2nH_{n-1}(x).
\end{align}</math>

These last relations, together with the initial polynomials {{math|''H''<sub>0</sub>(''x'')}} and {{math|''H''<sub>1</sub>(''x'')}}, can be used in practice to compute the polynomials quickly.

[[Turán's inequalities]] are
<math display="block">\mathit{H}_n(x)^2 - \mathit{H}_{n-1}(x) \mathit{H}_{n+1}(x) = (n-1)! \sum_{i=0}^{n-1} \frac{2^{n-i}}{i!}\mathit{H}_i(x)^2 > 0.</math>

Moreover, the following [[multiplication theorem]] holds:
<math display="block">\begin{align}
 H_n(\gamma x) &= \sum_{i=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \gamma^{n-2i}(\gamma^2 - 1)^i \binom{n}{2i} \frac{(2i)!}{i!} H_{n-2i}(x), \\
 \operatorname{He}_n(\gamma x) &= \sum_{i=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \gamma^{n-2i}(\gamma^2 - 1)^i \binom{n}{2i} \frac{(2i)!}{i!}2^{-i} \operatorname{He}_{n-2i}(x).
\end{align}</math>

===Explicit expression===
The physicist's Hermite polynomials can be written explicitly as
<math display="block">H_n(x) = \begin{cases}
\displaystyle n! \sum_{l = 0}^{\frac{n}{2}} \frac{(-1)^{\tfrac{n}{2} - l}}{(2l)! \left(\tfrac{n}{2} - l \right)!} (2x)^{2l} & \text{for even } n, \\
\displaystyle n! \sum_{l = 0}^{\frac{n-1}{2}} \frac{(-1)^{\frac{n-1}{2} - l}}{(2l + 1)! \left (\frac{n-1}{2} - l \right )!} (2x)^{2l + 1}  & \text{for odd } n.
\end{cases}</math>

These two equations may be combined into one using the [[floor function]]:
<math display="block">H_n(x) = n! \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{(-1)^m}{m!(n - 2m)!} (2x)^{n - 2m}.</math>

The probabilist's Hermite polynomials {{mvar|He}} have similar formulas, which may be obtained from these by replacing the power of {{math|2''x''}} with the corresponding power of {{math|{{sqrt|2}} ''x''}} and multiplying the entire sum by {{math|2<sup>−{{sfrac|''n''|2}}</sup>}}:
<math display="block">\operatorname{He}_n(x) = n! \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{(-1)^m}{m!(n - 2m)!} \frac{x^{n - 2m}}{2^m}.</math>

===Inverse explicit expression===
The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials {{mvar|He}} are
<math display="block">x^n = n! \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{1}{2^m  m!(n-2m)!} \operatorname{He}_{n-2m}(x).</math>

The corresponding expressions for the physicist's Hermite polynomials {{mvar|H}} follow directly by properly scaling this:<ref>{{cite web |title=18. Orthogonal Polynomials, Classical Orthogonal Polynomials, Sums |url=http://dlmf.nist.gov/18.18.E20 |website=Digital Library of Mathematical Functions |publisher=National Institute of Standards and Technology |access-date=30 January 2015 |ref=DLMF}}</ref>
<math display="block">x^n = \frac{n!}{2^n} \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{1}{m!(n-2m)! } H_{n-2m}(x).</math>

===Generating function===
The Hermite polynomials are given by the [[exponential generating function]]
<math display="block">\begin{align}
 e^{xt - \frac12 t^2} &= \sum_{n=0}^\infty \operatorname{He}_n(x) \frac{t^n}{n!}, \\
 e^{2xt - t^2} &= \sum_{n=0}^\infty  H_n(x) \frac{t^n}{n!}.
\end{align}</math>

This equality is valid for all [[complex number|complex]] values of {{mvar|x}} and {{mvar|t}}, and can be obtained by writing the Taylor expansion at {{mvar|x}} of the entire function {{math|''z'' → ''e''<sup>−''z''<sup>2</sup></sup>}} (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
<math display="block">H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} = (-1)^n e^{x^2} \frac{n!}{2\pi i} \oint_\gamma \frac{e^{-z^2}}{(z-x)^{n+1}} \,dz.</math>

Using this in the sum 
<math display="block">\sum_{n=0}^\infty H_n(x) \frac {t^n}{n!},</math> 
one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.

===Expected values===
If {{mvar|X}} is a [[random variable]] with a [[normal distribution]] with standard deviation 1 and expected value {{mvar|μ}}, then
<math display="block">\operatorname{\mathbb E}\left[\operatorname{He}_n(X)\right] = \mu^n.</math>

The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices:
<math display="block">\operatorname{\mathbb E}\left[X^{2n}\right] = (-1)^n \operatorname{He}_{2n}(0) = (2n-1)!!,</math>
where {{math|(2''n'' &minus; 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:
<math display="block">\operatorname{He}_n(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty (x + iy)^n e^{-\frac{y^2}{2}} \,dy.</math>

===Asymptotic expansion===
Asymptotically, as {{math|''n'' → ∞}}, the expansion<ref>{{harvnb|Abramowitz|Stegun|1983|page=508–510}}, [http://www.math.sfu.ca/~cbm/aands/page_508.htm 13.6.38 and 13.5.16].</ref>
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \frac{2^n}{\sqrt \pi}\Gamma\left(\frac{n+1}2\right) \cos \left(x \sqrt{2 n}- \frac{n\pi}{2} \right)</math> 
holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude:
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \frac{2^n}{\sqrt \pi}\Gamma\left(\frac{n+1}2\right) \cos \left(x \sqrt{2 n}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}=\frac{\Gamma(n)}{\Gamma\left(\frac{n}2\right)} \cos \left(x \sqrt{2 n}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14},</math>
which, using [[Stirling's approximation]], can be further simplified, in the limit, to
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \left(\frac{2n}{e}\right)^{\frac{n}{2}} \sqrt{2} \cos \left(x \sqrt{2n}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}.</math>

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
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \left(\frac{2n}{e}\right)^{\frac{n}{2}} \sqrt{2} \cos \left(x \sqrt{2n+1-\frac{x^2}{3}}- \frac {n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}.</math>

A finer approximation,<ref>{{harvnb|Szegő|1955|p=201}}</ref> which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution 
<math display="block">x = \sqrt{2n + 1}\cos(\varphi), \quad 0 < \varepsilon \leq \varphi \leq \pi - \varepsilon,</math> 
with which one has the uniform approximation
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) = 2^{\frac{n}{2}+\frac14}\sqrt{n!}(\pi n)^{-\frac14}(\sin \varphi)^{-\frac12} \cdot \left(\sin\left(\frac{3\pi}{4} + \left(\frac{n}{2} + \frac{1}{4}\right)\left(\sin 2\varphi-2\varphi\right) \right)+O\left(n^{-1}\right) \right).</math>

Similar approximations hold for the monotonic and transition regions. Specifically, if 
<math display="block">x = \sqrt{2n+1} \cosh(\varphi), \quad 0 < \varepsilon \leq \varphi \leq \omega < \infty,</math> 
then
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) = 2^{\frac{n}{2}-\frac34}\sqrt{n!}(\pi n)^{-\frac14}(\sinh \varphi)^{-\frac12} \cdot e^{\left(\frac{n}{2}+\frac{1}{4}\right)\left(2\varphi-\sinh 2\varphi\right)}\left(1+O\left(n^{-1}\right) \right),</math>
while for <math display="block">x = \sqrt{2n + 1} + t</math> with {{mvar|t}} complex and bounded, the approximation is
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) =\pi^{\frac14}2^{\frac{n}{2}+\frac14}\sqrt{n!}\, n^{-\frac{1}{12}}\left( \operatorname{Ai}\left(2^{\frac12}n^{\frac16}t\right)+ O\left(n^{-\frac23}\right) \right),</math>
where {{math|Ai}} is the [[Airy function]] of the first kind.

===Special values===
The physicist's Hermite polynomials evaluated at zero argument {{math|''H<sub>n</sub>''(0)}} are called [[Hermite number]]s.

<math display="block">H_n(0) = \begin{cases} 
 0 & \text{for odd }n, \\
 (-2)^\frac{n}{2} (n-1)!! & \text{for even }n,
\end{cases}</math>
which satisfy the recursion relation {{math|1=''H<sub>n</sub>''(0) = −2(''n'' − 1)''H''<sub>''n'' − 2</sub>(0)}}.

In terms of the probabilist's polynomials this translates to
<math display="block">\operatorname{He}_n(0) = \begin{cases} 
 0 & \text{for odd }n, \\
 (-1)^\frac{n}{2} (n-1)!! & \text{for even }n.
\end{cases}</math>

==Relations to other functions==

===Laguerre polynomials===
The Hermite polynomials can be expressed as a special case of the [[Laguerre polynomials]]:
<math display="block">\begin{align}
 H_{2n}(x) &= (-4)^n n! L_n^{\left(-\frac12\right)}(x^2)
  &&= 4^n n! \sum_{k=0}^n (-1)^{n-k} \binom{n-\frac12}{n-k} \frac{x^{2k}}{k!}, \\
 H_{2n+1}(x) &= 2(-4)^n n! x L_n^{\left(\frac12\right)}(x^2)
  &&= 2\cdot 4^n n!\sum_{k=0}^n (-1)^{n-k} \binom{n+\frac12}{n-k} \frac{x^{2k+1}}{k!}.
\end{align}</math>

===Relation to confluent hypergeometric functions===
The physicist's Hermite polynomials can be expressed as a special case of the [[parabolic cylinder function]]s:
<math display="block">H_n(x) = 2^n U\left(-\tfrac12 n, \tfrac12, x^2\right)</math>
in the [[right half-plane]], where {{math|''U''(''a'', ''b'', ''z'')}} is [[confluent hypergeometric function|Tricomi's confluent hypergeometric function]]. Similarly,
<math display="block">\begin{align}
 H_{2n}(x) &= (-1)^n \frac{(2n)!}{n!} \,_1F_1\big(-n, \tfrac12; x^2\big), \\
 H_{2n+1}(x) &= (-1)^n \frac{(2n+1)!}{n!}\,2x \,_1F_1\big(-n, \tfrac32; x^2\big),
\end{align}</math>
where {{math|<sub>1</sub>''F''<sub>1</sub>(''a'', ''b''; ''z'') {{=}} ''M''(''a'', ''b''; ''z'')}} is [[confluent hypergeometric function|Kummer's confluent hypergeometric function]].

== Hermite polynomial expansion ==
Similar to Taylor expansion, some functions are expressible as an infinite sum of Hermite polynomials. Specifically, if <math>\int e^{-x^2}f(x)^2 dx < \infty</math>, then it has an expansion in the physicist's Hermite polynomials.<ref>{{Cite web |title=MATHEMATICA tutorial, part 2.5: Hermite expansion |url=https://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch5/hermite.html |access-date=2023-12-24 |website=www.cfm.brown.edu}}</ref>

Given such <math>f</math>, the partial sums of the Hermite expansion of <math>f</math> converges to in the <math>L^p</math> norm if and only if <math>4 / 3<p<4</math>.<ref>{{Cite journal |last=Askey |first=Richard |last2=Wainger |first2=Stephen |date=1965 |title=Mean Convergence of Expansions in Laguerre and Hermite Series |url=https://www.jstor.org/stable/2373069 |journal=American Journal of Mathematics |volume=87 |issue=3 |pages=695–708 |doi=10.2307/2373069 |issn=0002-9327}}</ref><math display="block">x^n = \frac{n!}{2^n} \,\sum_{k= 0}^{\left\lfloor n/2 \right\rfloor} \frac{1}{k! \,(n-2k)!} \, H_{n-2k} (x) = n! \sum_{k= 0}^{\left\lfloor n/2 \right\rfloor} \frac{1}{k! \,2^k \,(n-2k)!} \, \operatorname{He}_{n-2k} (x) , \qquad n \in \mathbb{Z}_{+} .
</math><math display="block">e^{ax} = e^{a^2 /4} \sum_{n\ge 0} \frac{a^n}{n! \,2^n} \, H_n (x) , \qquad a\in \mathbb{C}, \quad x\in \mathbb{R} .</math><math display="block">e^{-a^2 x^2} = \sum_{n\ge 0} \frac{(-1)^n a^{2n}}{n! \left( 1 + a^2 \right)^{n + 1/2} 2^{2n}}\, H_{2n} (x) .</math><math display="block">\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} ~dt=\frac{1}{\sqrt{2 \pi}} \sum_{k \geq 0} \frac{(-1)^k}{k !(2 k+1) 2^{3 k}} H_{2 k}(x) .</math><math display="block">\cosh (2x) = e \sum_{k\ge 0} \frac{1}{(2k)!}\, H_{2k} (x) , \qquad \sinh (2x) = e \sum_{k\ge 0} \frac{1}{(2k+1)!} \, H_{2k+1} (x) .</math><math display="block">\cos (x) = e^{-1/4} \,\sum_{k\ge 0} \frac{(-1)^k}{2^{2k} \, (2k)!} \, H_{2k} (x) 
\quad \sin (x) = e^{-1/4} \,\sum_{k\ge 0} \frac{(-1)^k}{2^{2k+1} \, (2k+1)!} \, H_{2k+1} (x) </math>

==Differential-operator representation==
The probabilist's Hermite polynomials satisfy the identity<ref>{{cite book |last1=Rota |first1=Gian-Carlo |last2=Doubilet |first2=P. |title=Finite operator calculus |date=1975 |publisher=Academic Press |location=New York |isbn=9780125966504 |page=44}}</ref> <math display="block">\operatorname{He}_n(x) = e^{-\frac{D^2}{2}}x^n,</math> where {{mvar|D}} represents differentiation with respect to {{mvar|x}}, and the [[exponential function|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 {{math|''x''<sup>''n''</sup>}} can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients of {{math|''H<sub>n</sub>''}} that can be used to quickly compute these polynomials.

Since the formal expression for the [[Weierstrass transform]] {{mvar|W}} is {{math|''e''<sup>''D''<sup>2</sup></sup>}}, we see that the Weierstrass transform of {{math|({{sqrt|2}})<sup>''n''</sup>''He<sub>n</sub>''({{sfrac|''x''|{{sqrt|2}}}})}} is {{math|''x<sup>n</sup>''}}. Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding [[Maclaurin series]].

The existence of some formal power series {{math|''g''(''D'')}} with nonzero constant coefficient, such that {{math|1=''He<sub>n</sub>''(''x'') = ''g''(''D'')''x<sup>n</sup>''}}, 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]].

{{Further|Weierstrass transform#The inverse transform}}

==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
<math display="block">\begin{align}
 \operatorname{He}_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{tx-\frac{t^2}{2}}}{t^{n+1}}\,dt, \\
 H_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{2tx-t^2}}{t^{n+1}}\,dt,
\end{align}</math>
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
<math display="block">\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}},</math>
which has expected value 0 and variance 1.

Scaling, one may analogously speak of '''generalized Hermite polynomials'''<ref>{{Citation |last=Roman |first=Steven |date=1984 |title=The Umbral Calculus |series=Pure and Applied Mathematics |volume=111 |publisher=Academic Press |edition=1st |isbn=978-0-12-594380-2 |pages=87–93}}</ref>
<math display="block">\operatorname{He}_n^{[\alpha]}(x)</math>
of variance {{mvar|α}}, where {{mvar|α}} is any positive number. These are then orthogonal with respect to the normal probability distribution whose density function is
<math display="block">\frac{1}{\sqrt{2 \pi \alpha}} e^{-\frac{x^2}{2\alpha}}.</math>
They are given by
<math display="block">\operatorname{He}_n^{[\alpha]}(x) = \alpha^{\frac{n}{2}}\operatorname{He}_n\left(\frac{x}{\sqrt{\alpha}}\right) = \left(\frac{\alpha}{2}\right)^{\frac{n}{2}} H_n\left( \frac{x}{\sqrt{2 \alpha}}\right) = e^{-\frac{\alpha D^2}{2}} \left(x^n\right).</math>

Now, if 
<math display="block">\operatorname{He}_n^{[\alpha]}(x) = \sum_{k=0}^n h^{[\alpha]}_{n,k} x^k,</math>
then the polynomial sequence whose {{mvar|n}}th term is
<math display="block">\left(\operatorname{He}_n^{[\alpha]} \circ \operatorname{He}^{[\beta]}\right)(x) \equiv \sum_{k=0}^n h^{[\alpha]}_{n,k}\,\operatorname{He}_k^{[\beta]}(x)</math>
is called the [[Binomial type#Umbral composition of polynomial sequences|umbral composition]] of the two polynomial sequences. It can be shown to satisfy the identities
<math display="block">\left(\operatorname{He}_n^{[\alpha]} \circ \operatorname{He}^{[\beta]}\right)(x) = \operatorname{He}_n^{[\alpha+\beta]}(x)</math>
and
<math display="block">\operatorname{He}_n^{[\alpha+\beta]}(x + y) = \sum_{k=0}^n \binom{n}{k} \operatorname{He}_k^{[\alpha]}(x) \operatorname{He}_{n-k}^{[\beta]}(y).</math>
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|differential-operator representation]], which leads to a ready derivation of it. This [[binomial type]] identity, for {{math|1=''α'' = ''β'' = {{sfrac|1|2}}}}, has already been encountered in the above section on [[#Recursion relation]]s.)

==="Negative variance"===
Since polynomial sequences form a [[group (mathematics)|group]] under the operation of [[Binomial type#Umbral composition of polynomial sequences|umbral composition]], one may denote by
<math display="block">\operatorname{He}_n^{[-\alpha]}(x)</math>
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 {{math|α > 0}}, the coefficients of <math>\operatorname{He}_n^{[-\alpha]}(x)</math> are just the absolute values of the corresponding coefficients of <math>\operatorname{He}_n^{[\alpha]}(x)</math>.

These arise as moments of normal probability distributions: The {{mvar|n}}th moment of the normal distribution with expected value {{mvar|μ}} and variance {{math|''σ''<sup>2</sup>}} is
<math display="block">E[X^n] = \operatorname{He}_n^{[-\sigma^2]}(\mu),</math>
where {{mvar|X}} is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that
<math display="block">\sum_{k=0}^n \binom{n}{k} \operatorname{He}_k^{[\alpha]}(x) \operatorname{He}_{n-k}^{[-\alpha]}(y) = \operatorname{He}_n^{[0]}(x + y) = (x + y)^n.</math>

==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>

===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 |eprint=2402.16537}}</ref><ref>{{cite arXiv |last=Moriconi|first=Marco|title=Nodes of Wavefunctions |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>

==See also==
{{Div col}}
*[[Hermite transform]]
*[[Legendre polynomials]]
*[[Mehler kernel]]
*[[Parabolic cylinder function]]
*[[Romanovski polynomials]]
*[[Turán's inequalities]]
{{Div col end}}

==Notes==
{{Reflist|30em}}

==References==
{{refbegin|30em}}
*{{Abramowitz Stegun ref|22|773}}
*{{citation |last1=Courant |first1=Richard |author-link1=Richard Courant |last2=Hilbert |first2=David |author-link2=David Hilbert |title=Methods of Mathematical Physics |volume=1 |publisher=Wiley-Interscience |orig-year=1953 |year=1989 |isbn=978-0-471-50447-4}}
*{{citation |last1=Erdélyi |first1=Arthur |author-link1=Arthur Erdélyi |last2=Magnus |first2=Wilhelm |author-link2=Wilhelm Magnus |last3=Oberhettinger |first3=Fritz |last4=Tricomi |first4=Francesco G. |author-link4=Francesco Tricomi |title=Higher transcendental functions |volume=II |publisher=McGraw-Hill |year=1955 |url=http://apps.nrbook.com/bateman/Vol2.pdf |isbn=978-0-07-019546-2 |access-date=2014-07-17 |archive-date=2011-07-14 |archive-url=https://web.archive.org/web/20110714210423/http://apps.nrbook.com/bateman/Vol2.pdf |url-status=dead }}
*{{eom|first=M.V.|last=Fedoryuk|title=Hermite function}}
*{{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|authorlink1 = Tom H. Koornwinder|first2=Roderick S. C.|last2= Wong| first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
*{{citation |last=Laplace |first=P. S. |title=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 |journal=Mémoires de l'Académie des Sciences |year=1810 |pages=279–347}} [https://gallica.bnf.fr/ark:/12148/bpt6k77600r/f362.image.r=oeuvres%20completes%20de%20laplace Oeuvres complètes 12, pp.357-412], [http://cerebro.xu.edu/math/Sources/Laplace/defint.pdf English translation] {{Webarchive|url=https://web.archive.org/web/20160304220136/http://cerebro.xu.edu/math/Sources/Laplace/defint.pdf |date=2016-03-04 }}.
*{{Citation |last1=Shohat |first1=J.A.|last2=Hille |first2=Einar|last3=Walsh |first3=Joseph L. |title=A bibliography on orthogonal polynomials |series=Bulletin of the National Research Council |number=103 |publisher=National Academy of Sciences |location=Washington D.C. |date=1940}} - 2000 references of Bibliography on Hermite polynomials.
*{{eom|title=Hermite polynomials|first=P. K. |last=Suetin}}
*{{citation |last=Szegő |first=Gábor |author-link=Gábor Szegő |title=Orthogonal Polynomials |series=Colloquium Publications |volume=23 |publisher=American Mathematical Society |edition=4th |orig-year=1939 |year=1955 |isbn=978-0-8218-1023-1}}
*{{Citation|last=Temme |first=Nico |title=Special Functions: An Introduction to the Classical Functions of Mathematical Physics |publisher=Wiley |location=New York |date=1996 |isbn=978-0-471-11313-3}}
*{{citation |last=Wiener |first=Norbert |author-link=Norbert Wiener |title=The Fourier Integral and Certain of its Applications |orig-year=1933 |edition=revised |year=1958 |publisher=Dover Publications |location=New York |isbn=0-486-60272-9}}
*{{Citation |last1=Whittaker |first1=E. T. |author-link1=E. T. Whittaker |last2=Watson |first2=G. N.|author-link2=G. N. Watson |title=[[A Course of Modern Analysis]] |orig-year=1927 |year=1996 |publisher=Cambridge University Press |location=London |edition=4th |isbn=978-0-521-58807-2}}
{{refend}}

==External links==
*{{Commons category-inline}}
*{{MathWorld|urlname=HermitePolynomial|title=Hermite Polynomial}}
* [https://www.gnu.org/software/gsl/ GNU Scientific Library] — includes [[C (programming language)|C]] version of Hermite polynomials, functions, their derivatives and zeros (see also [[GNU Scientific Library]])

{{Authority control}}

{{DEFAULTSORT:Hermite Polynomials}}
[[Category:Orthogonal polynomials]]
[[Category:Polynomials]]
[[Category:Special hypergeometric functions]]