Hermite polynomials

{{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 polynomial sequence.

The polynomials arise in: * signal processing as Hermitian wavelets 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 eigenstates 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 ensembles.

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 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 <math>\operatorname{He}</math> and <math>H</math> 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 <math>\operatorname{He}_n</math> are sometimes denoted by <math>H_n</math>, 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. The probabilist's Hermite polynomials are also called the '''monic Hermite polynomials''', because they are monic. * 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''−{{sfrac|''x''2|2}}}} and {{math|''e''−''x''2}}, 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''0}}) 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''6}}, {{math|1=''n'' = 6}} and hence the first term is {{math|''x''6}}, 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''5}} (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''5}} (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''6}} 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''5}}; 3 is the power of {{mvar|x}} in the second term of the {{math|''He''5}} polynomial; and 10 is the coefficient of the second term in {{math|''H''5}}. This coefficient (45) is positive, since this is an odd-numbered term. Finally, the fourth term in {{math|''He''6}} 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''5}} polynomial; 1 is the power of {{mvar|x}} in the third term of {{math|''He''5}}, and 15 is the coefficient of the third term in {{math|''H''5}} . This coefficient (15) is made negative, since this is an even-numbered term. Thus, {{math|1=''He''6(''x'') = ''x''6 − 15''x''4 + 45''x''2 − 15}}.

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

For the physicist's notation, follow the following rules: # For the starting point in the sequence, the zeroth-order polynomial ({{math|''H''0}}) 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''n''}}. # 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''6}}, the first coefficient is {{math|1=26 = 64}} (i.e. {{math|64''x''6}}). The second coefficient is {{math|1=2 × 160 + 5 × 32 = 480}} (negative, i.e. {{math|−480''x''4}}). The third coefficient is {{math|1=2 × 120 + 3 × 160 = 720}} (positive, i.e. {{math|720''x''2}}). The fourth coefficient is {{math|1=2 × 0 + 1 × 120 = 120}} (negative, i.e., −120). Thus, {{math|1=''H''6(x) = 64''x''6 − 480''x''4 + 720''x''2 − 120}}.

For {{math|''H''7}}, the first coefficient is 27 = 128 (i.e., {{math|128''x''7}}). The second coefficient is {{math|1=2 × 480 + 6 × 64 = 1344}} (negative, i.e. {{math|−1344''x''5}}). The third coefficient is {{math|1=2 × 720 + 4 × 480 = 3360}} (positive, i.e. {{math|3360''x''3}}). The fourth coefficient is {{math|1=2 × 120 + 2 × 720 = 1680}} (negative, i.e. {{math|−1680''x''}}. Thus, {{math|''H''7(''x'') = 128''x''7 − 1344''x''5 + 3360''x''3 − 1680''x''}}.

Recognizing that {{math|1=''H''0 = 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. -->{| class="wikitable" |+Quick reference table ! !physicist's !probabilist's |- |symbol |<math>H_n</math> |<math>\operatorname{He}_n</math> |- |head coefficient |<math>2^n</math> |<math>1</math> |- |differential operator |<math>(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}</math> |<math>(-1)^n e^{\frac{x^2}{2}}\frac{d^n}{dx^n}e^{-\frac{x^2}{2}}</math> |- |orthogonal to |<math>e^{-x^2}</math> |<math>e^{-\frac 12 x^2}</math> |- |inner product |<math>\int H_m(x) H_n(x) \frac{e^{-x^2}}{\sqrt{\pi}}dx = 2^n n!\, \delta_{mn}</math> |<math>\int \operatorname{He}_m(x) \operatorname{He}_n(x)\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}} \,dx = n!\, \delta_{nm} </math> |- |generating function |<math>e^{2xt - t^2} = \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}</math> |<math>e^{xt - \frac12 t^2} = \sum_{n=0}^\infty \operatorname{He}_n(x) \frac{t^n}{n!} </math> |- |Rodrigues' formula |<math>\left(2x - \frac{d}{dx} \right)^n \cdot 1 </math> |<math>\left(x - \frac{d}{dx} \right)^n \cdot 1 </math> |- |recurrence relation |<math>H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)</math> |<math>\operatorname{He}_{n+1}(x) = x\operatorname{He}_n(x) - n\operatorname{He}_{n-1}(x)</math> |} <gallery widths="300" heights="300"> File:Hermite poly.svg|The first six probabilist's Hermite polynomials <math>\operatorname{He}_n(x)</math> File:Hermite poly phys.svg|The first six physicist's Hermite polynomials <math>H_n(x)</math> </gallery>

==Properties== The {{mvar|n}}th-order Hermite polynomial is a polynomial of degree {{mvar|n}}. The probabilist's version {{mvar|Hen}} has leading coefficient 1, while the physicist's version {{mvar|Hn}} has leading coefficient {{math|2''n''}}.

===Symmetry=== From the Rodrigues formulae given above, we can see that {{math|''Hn''(''x'')}} and {{math|''Hen''(''x'')}} are even or odd functions, with the same parity as {{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|''Hn''(''x'')}} and {{math|''Hen''(''x'')}} are {{mvar|n}}th-degree polynomials for {{math|''n'' {{=}} 0, 1, 2, 3,...}}. These polynomials are orthogonal with respect to the ''weight function'' (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 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 weight function {{math|''w''(''x'')}} defined in the preceding section.

An orthogonal basis for {{math|''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 {{math|''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 {{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''−''x''2}} 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 below).

An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for {{math|''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 {{math|''L''2('''R''')}}.

===Hermite's differential equation=== The probabilist's Hermite polynomials are solutions of the Sturm–Liouville 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 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 eigenfunctions 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 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 {{mvar|λ}}. An explicit formula of Hermite polynomials in terms of contour integrals {{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''0,0 = 1}}, {{math|1=''a''1,0 = 0}}, {{math|1=''a''1,1 = 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''0,0 = 1}}, {{math|1=''a''1,0 = 0}}, {{math|1=''a''1,1 = 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>

Equivalently, by 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 identities are self-evident and included in the 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''0(''x'')}} and {{math|''H''1(''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−{{sfrac|''n''|2}}}}: <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 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''−''z''2}} (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.

A slight generalization states<ref>(Rainville 1971), p. 198</ref><math display="block">e^{2 x t-t^2} H_k(x-t) = \sum_{n=0}^{\infty} \frac{H_{n+k}(x) t^n}{n!}</math>

===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'' − 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>

=== Integral representations === 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.

Using the Fourier transform of the gaussian <math>e^{-x^2}=\frac{1}{\sqrt{ \pi}} \int e^{-t^2+2 i x t} dt </math>, we have<math display="block">\begin{align} H_n(x) &= (-1)^n e^{x^2} \frac {d^n}{dx^n} e^{-x^2} = \frac{(-2 i)^n e^{x^2}}{\sqrt{\pi}} \int t^n e^{-t^2+2 i x t} d t \\ \operatorname{He}_n(x) &= \frac{(-i)^n e^{x^2/2}}{\sqrt{2\pi}} \int t^n \, e^{-t^2/2 + i x t}\, dt. \end{align}</math>

=== Other properties === The discriminant is expressed as a hyperfactorial:<ref name=":3">{{Cite web |title=DLMF: §18.16 Zeros ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.16 |access-date=2025-07-12 |website=dlmf.nist.gov}}</ref>

<math display="block">\begin{aligned} \operatorname{Disc}(H_n) &= 2^{\frac{3}{2} n(n-1)} \prod_{j=1}^n j^j \\ \operatorname{Disc}(\operatorname{He}_n) &= \prod_{j=1}^n j^j \end{aligned} </math>

The addition theorem, or the summation theorem, states that<ref name=":1">{{Cite web |title=DLMF: §18.18 Sums ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.18 |access-date=2025-03-18 |website=dlmf.nist.gov}}</ref><ref>{{Cite book |last1=Gradshteĭn |first1=I. S. |title=Table of integrals, series, and products |last2=Zwillinger |first2=Daniel |date=2015 |publisher=Elsevier, Academic Press is an imprint of Elsevier |isbn=978-0-12-384933-5 |edition=8 |location=Amsterdam ; Boston}}</ref>{{Pg|location=8.958}}<math display="block">\frac{\left(\sum_{k=1}^r a_k^2\right)^{\frac{n}{2}}}{n!} H_n\left(\frac{\sum_{k=1}^r a_k x_k}{\sqrt{\sum_{k=1}^r a_k^2}}\right)=\sum_{m_1+m_2+\ldots+m_r=n, m_i \geq 0} \prod_{k=1}^r\left\{\frac{a_k^{m_k}}{m_{k}!} H_{m_k}\left(x_k\right)\right\} </math>for any nonzero vector <math>a_{1:r}</math>.

The multiplication theorem states that<ref name=":1" /><math display="block">H_{n}\left(\lambda x\right)=\lambda^{n}\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{{\left(-n\right)_{2\ell}}}{\ell!}(1-\lambda^{-2})^{\ell}H_{n-2\ell}\left(x\right)</math>for any nonzero <math>\lambda</math>.

Feldheim formula<ref name=":2">Feldheim, Ervin. "Développements en série de polynômes d’Hermite et de Laguerrea l’aide des transformations de Gauss et de Hankel." Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 435 (1940). Part [https://dwc.knaw.nl/DL/publications/PU00017406.pdf I], [https://dwc.knaw.nl/DL/publications/PU00017407.pdf II], [https://dwc.knaw.nl/DL/publications/PU00017420.pdf III]</ref>{{Pg|location=Eq 46}}<math display="block">\begin{aligned} \frac{1}{\sqrt{a \pi}} & \int_{-\infty}^{+\infty} e^{-\frac{x^2}{a}} H_m\left(\frac{x+y}{\lambda}\right) H_n\left(\frac{x+z}{\mu}\right) d x \\ & = \left(1-\frac{a}{\lambda^2}\right)^{\frac{m}{2}}\left(1-\frac{a}{\mu^2}\right)^{\frac{n}{2}} \sum_{r=0}^{\min (m, n)} r!\binom{m}{r}\binom{n}{r} \left(\frac{2 a}{\sqrt{\left(\lambda^2-a\right)\left(\mu^2-a\right)}}\right)^r H_{m-r}\left(\frac{y}{\sqrt{\lambda^2-a}}\right) H_{n-r}\left(\frac{z}{\sqrt{\mu^2-a}}\right) \end{aligned}</math>where <math>a \in \mathbb C</math> has a positive real part. As a special case,<ref name=":2" />{{Pg|location=Eq 52}}<math display="block">\frac{1}{\sqrt{\pi}} \int_{-\infty}^{+\infty} e^{-t^2} H_m(t \sin \theta+v \cos \theta) H_n(t \cos \theta-v \sin \theta) d t =(-1)^n \cos ^m \theta \sin ^n \theta H_{m+n}(v)</math>

===Asymptotics=== As {{math|''n'' → ∞}},<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>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+1)}{\Gamma\left(\frac{n}{2} +1\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. The term <math>\left(1-\frac{x^2}{2n+1}\right)^{-\frac12}</math> corresponds to the probability of finding a classical particle in a potential well of shape <math>V(x) = \frac 12 x^2</math> at location <math>x</math>, if its total energy is <math>n + \frac 12</math>. This is a general method in semiclassical analysis. The semiclassical approximation breaks down near <math>\pm\sqrt{2n + 1}</math>, the location where the classical particle would be turned back. This is a fold catastrophe, at which point the Airy function is needed.<ref>{{Cite journal |last=Berry |first=M.V. |date=1976-01-01 |title=Waves and Thom's theorem |url=https://doi.org/10.1080/00018737600101342 |journal=Advances in Physics |volume=25 |issue=1 |pages=1–26 |doi=10.1080/00018737600101342 |bibcode=1976AdPhy..25....1B |issn=0001-8732|url-access=subscription }}</ref>

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>

The Plancherel–Rotach asymptotics method, applied to Hermite polynomials, takes into account the uneven spacing of the zeros near the edges.<ref>{{harvnb|Szegő|1975|p=201}}</ref> It 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|''Hn''(0)}} are called Hermite numbers.

<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=''Hn''(0) = −2(''n'' − 1)''H''''n'' − 2(0)}}. Equivalently, <math>H_{2n}(0) = (-2)^n (2n-1)!!</math>.

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>

=== Kibble–Slepian formula ===

Let <math display="inline">M</math> be a real <math display="inline">n\times n</math> symmetric matrix, then the '''Kibble–Slepian formula''' states that<math display="block">\det(I+M)^{-\frac 12 } e^{x^T M (I+M)^{-1}x} = \sum_K \left[\prod_{1\leq i \leq j \leq n} \frac{(M_{ij}/2)^{k_{ij}}}{k_{ij}!}\right] 2^{-tr(K)} H_{k_1}(x_1) \cdots H_{k_n}(x_n)</math> where <math display="inline">\sum_K</math> is the <math>\frac{n(n+1)}{2}</math>-fold summation over all <math display="inline">n \times n</math> symmetric matrices with non-negative integer entries, <math>tr(K)</math> is the trace of <math>K</math>, and <math display="inline">k_i</math> is defined as <math display="inline">k_{ii} + \sum_{j=1}^n k_{ij}</math>. This gives Mehler's formula when <math>M = \begin{bmatrix} 0 & u \\ u & 0\end{bmatrix}</math>.

Equivalently stated, if <math display="inline">T</math> is a positive semidefinite matrix, then set <math display="inline">M = -T(I+T)^{-1}</math>, we have <math display="inline">M(I+M)^{-1} = -T</math>, so <math display="block"> e^{-x^T T x} = \det(I+T)^{-\frac 12} \sum_K \left[\prod_{1\leq i \leq j \leq n} \frac{(M_{ij}/2)^{k_{ij}}}{k_{ij}!}\right] 2^{-tr(K)} H_{k_1}(x_1) \dots H_{k_n}(x_n) </math>Equivalently stated in a form closer to the boson quantum mechanics of the harmonic oscillator:<ref name=":0">{{Cite journal |last=Louck |first=J. D |date=1981-09-01 |title=Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods |url=https://dx.doi.org/10.1016/0196-8858%2881%2990005-1 |journal=Advances in Applied Mathematics |volume=2 |issue=3 |pages=239–249 |doi=10.1016/0196-8858(81)90005-1 |issn=0196-8858|url-access=subscription }}</ref><math display="block"> \pi^{-n/4}\det(I+M)^{-\frac 12 }e^{- \frac 12 x^T(I-M)(I+M)^{-1} x}= \sum_K\left[\prod_{1 \leq i \leq j \leq n} M_{i j}^{k_{i j}} / k_{i j}!\right]\left[\prod_{1 \leq i \leq n} k_{i}!\right]^{1 / 2} 2^{-\operatorname{tr} K} \psi_{k_1}\left(x_1\right) \cdots \psi_{k_n}\left(x_n\right) . </math> where each <math display="inline">\psi_n(x)</math> is the <math display="inline">n</math>-th eigenfunction of the harmonic oscillator, defined as <math display="block">\psi_n(x) := \frac{1}{\sqrt{2^n n!}}\left(\frac{1}{\pi}\right)^{\frac{1}{4}} e^{-\frac{1}{2} x^2} H_n(x) </math>The Kibble–Slepian formula was proposed by Kibble in 1945<ref>{{Cite journal |last=Kibble |first=W. F. |date=June 1945 |title=An extension of a theorem of Mehler's on Hermite polynomials |url=https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/an-extension-of-a-theorem-of-mehlers-on-hermite-polynomials/6CD265E3054D1595062F1CA83D492AC2 |journal=Mathematical Proceedings of the Cambridge Philosophical Society |language=en |volume=41 |issue=1 |pages=12–15 |doi=10.1017/S0305004100022313 |bibcode=1945PCPS...41...12K |issn=1469-8064|url-access=subscription }}</ref> and proven by Slepian in 1972 using Fourier analysis.<ref>{{Cite journal |last=Slepian |first=David |date=November 1972 |title=On the Symmetrized Kronecker Power of a Matrix and Extensions of Mehler's Formula for Hermite Polynomials |url=https://epubs.siam.org/doi/abs/10.1137/0503060 |journal=SIAM Journal on Mathematical Analysis |volume=3 |issue=4 |pages=606–616 |doi=10.1137/0503060 |issn=0036-1410|url-access=subscription }}</ref> Foata gave a combinatorial proof<ref>{{Cite journal |last=Foata |first=Dominique |date=1981-09-01 |title=Some Hermite polynomial identities and their combinatorics |url=https://dx.doi.org/10.1016/0196-8858%2881%2990006-3 |journal=Advances in Applied Mathematics |volume=2 |issue=3 |pages=250–259 |doi=10.1016/0196-8858(81)90006-3 |issn=0196-8858|url-access=subscription }}</ref> while Louck gave a proof via boson quantum mechanics.<ref name=":0" /> It has a generalization for complex-argument Hermite polynomials.<ref>{{Cite journal |last1=Ismail |first1=Mourad E.H. |last2=Zhang |first2=Ruiming |date=September 2016 |title=Kibble–Slepian formula and generating functions for 2D polynomials |url=https://doi.org/10.1016/j.aam.2016.05.003 |journal=Advances in Applied Mathematics |volume=80 |pages=70–92 |doi=10.1016/j.aam.2016.05.003 |issn=0196-8858|arxiv=1508.01816 }}</ref><ref>{{Cite journal |last1=Ismail |first1=Mourad E. H. |last2=Zhang |first2=Ruiming |date=2017-04-01 |title=A review of multivariate orthogonal polynomials |journal=Journal of the Egyptian Mathematical Society |volume=25 |issue=2 |pages=91–110 |doi=10.1016/j.joems.2016.11.001 |issn=1110-256X|doi-access=free }}</ref>

=== Zeroes === Let <math>x_{n,1} > \dots > x_{n,n} </math> be the roots of <math>H_n</math> in descending order. Let <math>a_m</math> be the <math>m</math>-th zero of the Airy function <math>\operatorname{Ai}(x)</math> in descending order: <math>0 > a_1 > a_2 > \cdots</math>. By the symmetry of <math>H_n</math>, we need only consider the positive half of its roots.

We have<ref name=":3" /><math display="block">(2 n+1)^{\frac{1}{2}}>x_{n, 1}>x_{n, 2}>\cdots>x_{n,\lfloor n / 2\rfloor}>0 .</math> For each <math>m</math>, asymptotically at <math>n \to\infty</math>,<ref name=":3" /><math display="block">x_{n, m}=(2 n+1)^{\frac{1}{2}}+2^{-\frac{1}{3}}(2 n+1)^{-\frac{1}{6}} a_m+\epsilon_{n, m},</math> where <math>\epsilon_{n, m}=O\left(n^{-\frac{5}{6}}\right)</math>, and <math>\epsilon_{n, m} < 0</math>.

See also,<ref>{{Harvard citation|Szegő|1975|loc=Section 6.21. Inequalities for the zeros of the classical polynomials}}</ref> and the formulas involving the zeroes of Laguerre polynomials.

Let <math>F_n(t) := \frac 1n \#\{i : x_{n, i} \leq t\}</math> be the cumulative distribution function for the roots of <math>H_n</math>, then we have the semicircle law<ref>{{Cite journal |last=Gawronski |first=Wolfgang |date=1987-07-01 |title=On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jonquière polynomials |url=https://dx.doi.org/10.1016/0021-9045%2887%2990020-7 |journal=Journal of Approximation Theory |volume=50 |issue=3 |pages=214–231 |doi=10.1016/0021-9045(87)90020-7 |issn=0021-9045|url-access=subscription }}</ref><math display="block">\lim_{n \to \infty} F_n(\sqrt{2n} t) = \frac 2\pi \int_{-1}^t \sqrt{1- s^2} ds \quad t \in (-1, +1) </math> The '''Stieltjes relation''' states that<ref>{{Cite journal |last1=Marcellán |first1=F. |last2=Martínez-Finkelshtein |first2=A. |last3=Martínez-González |first3=P. |date=2007-10-15 |title=Electrostatic models for zeros of polynomials: Old, new, and some open problems |url=https://www.sciencedirect.com/science/article/pii/S037704270600611X |journal=Journal of Computational and Applied Mathematics |series=Proceedings of The Conference in Honour of Dr. Nico Temme on the Occasion of his 65th birthday |volume=207 |issue=2 |pages=258–272 |doi=10.1016/j.cam.2006.10.020 |issn=0377-0427|arxiv=math/0512293 |hdl=10016/5921 }}</ref><ref>{{Harvard citation|Szegő|1975|p=|loc=Section 6.7. Electrostatic interpretation of the zeros of the classical polynomials}}</ref><math display="block">-x_{n,i} + \sum_{1 \leq j \leq n, i \neq j} \frac{1}{x_{n,i}-x_{n,j}} = 0</math> and can be physically interpreted as the equilibrium position of <math>n</math> particles on a line, such that each particle <math>i</math> is attracted to the origin by a linear force <math>-x_{n,i}</math>, and repelled by each other particle <math>j</math> by a reciprocal force <math>\frac{1}{x_{n,i} - x_{n,j}}</math>. This can be constructed by confining <math>n</math> positively charged particles in <math>\R^2</math> to the real line, and connecting each particle to the origin by a spring. This is also called the '''electrostatic model''', and relates to the Coulomb gas interpretation of the eigenvalues of gaussian ensembles.

As the zeroes specify the polynomial up to scaling, the Stieltjes relation provides an alternative way to uniquely characterize the Hermite polynomials.

Similarly, we have<ref>{{Cite journal |last1=Alıcı |first1=H. |last2=Taşeli |first2=H. |date=2015 |title=Unification of Stieltjes-Calogero type relations for the zeros of classical orthogonal polynomials |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.3285 |journal=Mathematical Methods in the Applied Sciences |language=en |volume=38 |issue=14 |pages=3118–3129 |doi=10.1002/mma.3285 |bibcode=2015MMAS...38.3118A |issn=1099-1476|hdl=11511/35468 |hdl-access=free |url-access=subscription }}</ref><math display="block">\begin{aligned} \sum_i x_{n,i}^2 &= \sum_{1 \leq i \leq n}^n \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{n,i} - x_{n,j})^2}\\ x_{n,i} &= \sum_{1 \leq j \leq n, i \neq j} \frac{1}{x_{n,i}-x_{n,j}}\\ \frac{2n - 2 - x_{n,i}^2}{3} &= \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{n,i}-x_{n,j})^2}\\ \frac 12 x_{n,i} &= \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{n,i}-x_{n,j})^3} \end{aligned}</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>

===Hypergeometric functions=== The physicist's Hermite polynomials can be expressed as a special case of the parabolic cylinder functions: <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 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|1''F''1(''a'', ''b''; ''z'') {{=}} ''M''(''a'', ''b''; ''z'')}} is Kummer's confluent hypergeometric function.<math display="block"> \begin{align} \mathrm{He}_{2n}(x)&=(-1)^n(2n-1)!!\;{}_1F_1\!\left(-n,\tfrac12;\tfrac{x^2}{2}\right),\\ \mathrm{He}_{2n+1}(x)&=(-1)^n(2n+1)!!\;x\;{}_1F_1\!\left(-n,\tfrac32;\tfrac{x^2}{2}\right). \end{align}

</math>There is also<ref>[https://dlmf.nist.gov/18.5#E13 DLMF Equation 18.5.13]</ref><math display="block">H_{n}\left(x\right)=(2x)^{n}{{}_{2}F_{0}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right).</math>

=== Limit relations === The Hermite polynomials can be obtained as the limit of various other polynomials.<ref>[https://dlmf.nist.gov/18.7#iii DLMF §18.7(iii) Limit Relations]</ref>

As a limit of Jacobi polynomials:<math display="block">\lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}P^{(\alpha,\alpha)}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{2^{n}n!}.</math> As a limit of ultraspherical polynomials:<math display="block">\lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}C^{(\lambda)}_{n}\left(\lambda^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{n!}.</math> As a limit of associated Laguerre polynomials:<math display="block">\lim_{\alpha\to\infty}\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}L^{(\alpha)}_{n}\left((2\alpha)^{\frac{1}{2}}x+\alpha\right)=\frac{(-1)^{n}}{n!}H_{n}\left(x\right).</math>

== 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>

For <math>f</math> that does not grow too fast, it has Hermite expansion <math>f(x) = \sum_k \frac{\mathbb E_{X \sim\mathcal N(0, 1)}[f^{(k)}(X)]}{k!}\operatorname{He}_k(x)</math>.<ref>{{Cite journal |last=Davis |first=Tom P. |date=2024-02-01 |title=A General Expression for Hermite Expansions with Applications |url=https://scholarworks.umt.edu/tme/vol21/iss1/6 |journal=The Mathematics Enthusiast |language=en |volume=21 |issue=1–2 |pages=71–87 |doi=10.54870/1551-3440.1618 |issn=1551-3440|url-access=subscription }}</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 |last1=Askey |first1=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 |jstor=2373069 |issn=0002-9327|url-access=subscription }}</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_{2k+1}(x) .</math><math display="block">\cosh(ax)=e^{a^{2}/2}\sum_{m=0}^{\infty}\frac{a^{2m}}{(2m)!}\,\mathrm{He}_{2m}(x), \quad \sinh(ax)=e^{a^{2}/2}\sum_{m=0}^{\infty}\frac{a^{2m+1}}{(2m+1)!}\,\mathrm{He}_{2m+1}(x) </math><math display="block">\cos(ax)=e^{-a^{2}/2}\sum_{m=0}^{\infty}\frac{(-1)^m a^{2m}}{(2m)!}\,\mathrm{He}_{2m}(x), \quad \sin(ax)=e^{-a^{2}/2}\sum_{m=0}^{\infty}\frac{(-1)^m a^{2m+1}}{(2m+1)!}\,\mathrm{He}_{2m+1}(x) </math><math display="block">\delta = \frac{1}{\sqrt{2\pi}}\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!! } \operatorname{He}_{2k} </math><math display="block">1_{x > 0} = \frac 12 \operatorname{He}_0 + \frac{1}{\sqrt{2\pi}}\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!!(2k+1) } \operatorname{He}_{2 k+1} </math>The probabilist's Hermite expansion for the power functions are the same as the power expansions for the probabilist's Hermite polynomials, except with positive signs. For example:<math display="block">\operatorname{He}_3(x) = x^3 - 3x, \quad x^3 = \operatorname{He}_3(x) + 3 \operatorname{He}_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 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''''n''}} can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients of {{math|''Hn''}} that can be used to quickly compute these polynomials.

Since the formal expression for the Weierstrass transform {{mvar|W}} is {{math|''e''''D''2}}, we see that the Weierstrass transform of {{math|({{sqrt|2}})''n''''Hen''({{sfrac|''x''|{{sqrt|2}}}})}} is {{math|''xn''}}. 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=''Hen''(''x'') = ''g''(''D'')''xn''}}, 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.

==Generalizations==

=== Variance === {{Anchor|Variance}}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 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, 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 relations.)

==="Negative variance"=== Since polynomial sequences form a group under the operation of 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''2}} 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 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''2('''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''''n''(''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 functions.

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.

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> 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|''Hen''}} and {{math|''ψ''''n''}} 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|1975}}.</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>

===As eigenfunctions of the Fourier transform=== The Hermite functions {{math|''ψ''''n''(''x'')}} are a set of eigenfunctions of the continuous Fourier transform {{mathcal|F}}. To see this, take the physicist's version of the generating function and multiply by {{math|''e''−{{sfrac|1|2}}''x''2}}. 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|''ψn''(''x'')}} are thus an orthonormal basis of {{math|''L''2('''R''')}}, which ''diagonalizes the Fourier transform operator''.<ref>In this case, we used the unitary version of the Fourier transform, so the eigenvalues are {{math|(−''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.</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 distribution 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}}, one can prove that<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) = 2\,(-1)^n\, l_n\big(4\pi (t^2 + f^2) \big),</math> where the Wigner distribution of a function {{math|''ψ'' ∈ ''L''2('''R''', '''C''')}} is defined as <math display="block"> W_\psi(t,f) = \int_{-\infty}^\infty \psi\left(t + \frac{\tau}{2}\right) \, \psi\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 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 quantum mechanics in phase space.

There are 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''''n''(''x'')}} of variance 1, the absolute value of the coefficient of {{math|''x''''k''}} 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 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,... {{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''''i'' = 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 | 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 completeness identity for the above Hermite functions holds in the sense of 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''''n''}} the Hermite functions, and {{math|''δ''(''x'' − ''y'')}} represents the Lebesgue measure on the line {{math|1=''y'' = ''x''}} in {{math|'''R'''2}}, 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'''2}}, 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''2('''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 functions 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|''Hn''(''x'')}} and {{math|''Hn''(''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 }}. *{{Cite book |last=Rainville |first=Earl David |title=Special functions |date=1971 |publisher=Chelsea Pub. Co |isbn=978-0-8284-0258-3 |location=Bronx, N.Y}} *{{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=1975 |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 version of Hermite polynomials, functions, their derivatives and zeros (see also GNU Scientific Library)

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