LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle u'' - 2xu' + 2\lambda u = 0,}

${\displaystyle u''-2xu'+2\lambda u=0,}$

... is translated to the CAS output ...

Semantic latex: u'' - 2xu' + 2\lambda u = 0

Confidence: 0

Mathematica

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(LaTeX -> Mathematica) The input LaTeX is invalid: Primes can only be translated behind semantic macros (differentiation primes) but not in other places.

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SymPy

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(LaTeX -> SymPy) The input LaTeX is invalid: Primes can only be translated behind semantic macros (differentiation primes) but not in other places.

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Maple

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(LaTeX -> Maple) The input LaTeX is invalid: Primes can only be translated behind semantic macros (differentiation primes) but not in other places.

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$u$

Description

physicists ' Hermite equation

constant

example

first kind

form

general solution

physicists ' hermite

physicists ' Hermite polynomial

second kind

Complete translation information:

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Wikitext context	{{Use American English\|date = March 2019}} {{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}} In [[mathematics]], the '''Hermite polynomials''' are a classical [[orthogonal polynomials\|orthogonal]] [[polynomial sequence]]. The polynomials arise in: * [[probability]], such as the [[Edgeworth series]]; * in [[combinatorics]], as an example of an [[Appell sequence]], obeying the [[umbral calculus]]; * in [[numerical analysis]] as [[Gaussian quadrature]]; * in [[physics]], where they give rise to the [[eigenstate]]s of the [[quantum harmonic oscillator]]; * in [[systems theory]] in connection with nonlinear operations on [[Gaussian noise]]. * in [[random matrix theory]] in Wigner–Dyson ensembles. Hermite polynomials were defined by [[Pierre-Simon Laplace]] in 1810,<ref>{{harvnb\|Laplace\|1810}} ([http://gallica.bnf.fr/ark:/12148/bpt6k775950/f369.image online]).</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. L. \|last=Chebyshev \|title=Sur le développement des fonctions à une seule variable \|trans-title=On the development of single-variable functions \|journal=Bull. Acad. Sci. St. Petersb. \|volume=1 \|date=1859 \|pages=193–200}} Collected in ''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}} Collected in ''Œuvres'' '''II''', 293–303.</ref> They were consequently not new, although Hermite was the first to define the multidimensional polynomials in his later 1865 publications. ==Definition== Like the other [[classical orthogonal polynomials]], the Hermite polynomials can be defined from several different starting points. Noting from the outset that there are two different standardizations in common use, one convenient method is as follows: * The '''"probabilists' Hermite polynomials"''' are given by :: <math>\mathit{He}_n(x) = (-1)^n e^{\frac{x^2}{2}}\frac{d^n}{dx^n}e^{-\frac{x^2}{2}},</math> * while the '''"physicists' Hermite polynomials"''' are given by :: <math>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>\mathit{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>H_n(x)=2^\frac{n}{2} \mathit{He}_n\left(\sqrt{2} \,x\right), \quad \mathit{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>\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 probabilists' Hermite polynomials {{math\|''He''<sub>''n''</sub>(''x'')}}]] * The first eleven probabilists' Hermite polynomials are: :<math>\begin{align} \mathit{He}_0(x) &= 1, \\ \mathit{He}_1(x) &= x, \\ \mathit{He}_2(x) &= x^2 - 1, \\ \mathit{He}_3(x) &= x^3 - 3x, \\ \mathit{He}_4(x) &= x^4 - 6x^2 + 3, \\ \mathit{He}_5(x) &= x^5 - 10x^3 + 15x, \\ \mathit{He}_6(x) &= x^6 - 15x^4 + 45x^2 - 15, \\ \mathit{He}_7(x) &= x^7 - 21x^5 + 105x^3 - 105x, \\ \mathit{He}_8(x) &= x^8 - 28x^6 + 210x^4 - 420x^2 + 105, \\ \mathit{He}_9(x) &= x^9 - 36x^7 + 378x^5 - 1260x^3 + 945x, \\ \mathit{He}_{10}(x) &= x^{10} - 45x^8 + 630x^6 - 3150x^4 + 4725x^2 - 945. \end{align}</math> [[Image:Hermite poly phys.svg\|thumb\|right\|390px\|The first six (physicists') Hermite polynomials {{math\|''H''<sub>''n''</sub>(''x'')}}]] * The first eleven physicists' Hermite polynomials are: :<math>\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 probabilists' 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\|''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 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, 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 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 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\|''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 15 + 6 × 1 = 21 (negative, i.e. {{math\|−21''x''<sup>5</sup>}}). The third coefficient is 45 + 4 × 15 = 105 (positive, i.e. {{math\|105''x''<sup>3</sup>}}). The fourth coefficient is 15 + 2 × 45 = 105 (negative, i.e. −105''x''). Thus, {{math\|''He''<sub>7</sub>(''x'') {{=}} ''x''<sup>7</sup> − 21''x''<sup>5</sup> + 105''x''<sup>3</sup> − 105''x''}}. For the physicists' 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 2<sup>6</sup> = 64 (i.e. {{math\|64''x''<sup>6</sup>}}). The second coefficient is 2 × 160 + 5 × 32 = 480 (negative, i.e. {{math\|−480''x''<sup>4</sup>}}). The third coefficient is 2 × 120 + 3 × 160 = 720 (positive, i.e. {{math\|720''x''<sup>2</sup>}}). The fourth coefficient is 2 × 0 + 1 × 120 = 120 (negative, i.e., −120). Thus, {{math\|''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 2 × 480 + 6 × 64 = 1344 (negative, i.e. {{math\|−1344''x''<sup>5</sup>}}). The third coefficient is 2 × 720 + 4 × 480 = 3360 (positive, i.e. {{math\|3360''x''<sup>3</sup>}}). The fourth coefficient is 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\|''H''<sub>0</sub> {{=}} 1}}, these rules can be followed to sequentially derive all {{mvar\|n}}th-order Hermite polynomials from {{math\|''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 probabilists' version {{mvar\|He<sub>n</sub>}} has leading coefficient 1, while the physicists' version {{mvar\|H<sub>n</sub>}} has leading coefficient {{math\|2<sup>''n''</sup>}}. ===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>w(x) = e^{-\frac{x^2}{2}} \quad (\text{for }\mathit{He})</math> or : <math>w(x) = e^{-x^2} \quad (\text{for } H),</math> i.e., we have : <math>\int_{-\infty}^\infty H_m(x) H_n(x)\, w(x) \,dx = 0 \quad \text{for all }m \neq n.</math> Furthermore, : <math>\int_{-\infty}^\infty \mathit{He}_m(x) \mathit{He}_n(x)\, e^{-\frac{x^2}{2}} \,dx = \sqrt{2 \pi}\, n!\, \delta_{nm},</math> or :<math>\int_{-\infty}^\infty H_m(x) H_n(x)\, e^{-x^2} \,dx = \sqrt{\pi}\, 2^n 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 (probabilists' or physicists') form an [[orthonormal basis\|orthogonal basis]] of the [[Hilbert space]] of functions satisfying : <math>\int_{-\infty}^\infty \bigl\|f(x)\bigr\|^2\, w(x) \,dx < \infty,</math> in which the inner product is given by the integral : <math>\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 [[Lp space\|{{math\|''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>\int_{-\infty}^\infty f(x) x^n e^{- x^2} \,dx = 0</math> for every {{math\|''n'' ≥ 0}}, then {{math\|''f'' {{=}} 0}}. One possible way to do this is to appreciate that the [[entire function]] : <math>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\|''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 probabilists' Hermite polynomials are solutions of the differential equation : <math>\left(e^{-\frac12 x^2}u'\right)' + \lambda e^{-\frac12 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 He_\lambda(x) </math>, where <math>C_{1}</math> denotes a constant. Rewriting the differential equation as an [[Eigenvalue\|eigenvalue problem]] : <math>L[u] = u'' - x u' = -\lambda u,</math> the Hermite polynomials <math>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>u'' - 2xu' = -2\lambda u.</math> whose solution is uniquely given in terms of physicists' 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 physicists' Hermite equation : <math>u'' - 2xu' + 2\lambda u = 0,</math> the general solution takes the form : <math>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 physicists' Hermite polynomials (of the first kind), and <math>h_\lambda(x)</math> are physicists' 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 probabilists' Hermite polynomials also satisfies the [[recurrence relation]] : <math>\mathit{He}_{n+1}(x) = x \mathit{He}_n(x) - \mathit{He}_n'(x).</math> Individual coefficients are related by the following recursion formula: :<math>a_{n+1,k} = \begin{cases} - n a_{n-1,k} & k = 0, \\ a_{n,k-1} - n a_{n-1,k} & k > 0, \end{cases}</math> and {{math\|''a''<sub>0,0</sub> {{=}} 1}}, {{math\|''a''<sub>1,0</sub> {{=}} 0}}, {{math\|''a''<sub>1,1</sub> {{=}} 1}}. For the physicists' polynomials, assuming : <math>H_n(x) = \sum^n_{k=0} a_{n,k} x^k,</math> we have : <math>H_{n+1}(x) = 2xH_n(x) - H_n'(x).</math> Individual coefficients are related by the following recursion formula: : <math>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\|''a''<sub>0,0</sub> {{=}} 1}}, {{math\|''a''<sub>1,0</sub> {{=}} 0}}, {{math\|''a''<sub>1,1</sub> {{=}} 2}}. The Hermite polynomials constitute an [[Appell sequence]], i.e., they are a polynomial sequence satisfying the identity : <math>\begin{align} \mathit{He}_n'(x) &= n\mathit{He}_{n-1}(x), \\ H_n'(x) &= 2nH_{n-1}(x). \end{align}</math> Equivalently, by [[Taylor series\|Taylor-expanding]], :<math>\begin{align} \mathit{He}_n(x+y) &= \sum_{k=0}^n \binom{n}{k}x^{n-k} \mathit{He}_{k}(y) &&= 2^{-\frac n 2} \sum_{k=0}^n \binom{n}{k} \mathit{He}_{n-k}\left(x\sqrt 2\right) \mathit{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>\begin{align} \mathit{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>\begin{align} \mathit{He}_n^{(m)}(x) &= \frac{n!}{(n-m)!} \mathit{He}_{n-m}(x) &&= m! \binom{n}{m} \mathit{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>\begin{align} \mathit{He}_{n+1}(x) &= x\mathit{He}_n(x) - n\mathit{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>\mathit{He}_n(x)^2 - \mathit{He}_{n-1}(x) \mathit{He}_{n+1}(x) = (n-1)! \sum_{i=0}^{n-1} \frac{2^{n-i}}{i!}\mathit{He}_i(x)^2 > 0.</math> Moreover, the following [[multiplication theorem]] holds: : <math>\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), \\ \mathit{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} \mathit{He}_{n-2i}(x). \end{align}</math> ===Explicit expression=== The physicists' Hermite polynomials can be written explicitly as : <math>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>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 probabilists' 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>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 probabilists’ Hermite polynomials {{mvar\|He}} are : <math>x^n = n! \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{1}{2^m m!(n-2m)!} He_{n-2m}(x).</math> The corresponding expressions for the physicists’ 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 \|accessdate=30 January 2015 \|ref=DLMF}}</ref> : <math>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>\begin{align} e^{xt - \frac12 t^2} &= \sum_{n=0}^\infty \mathit{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 physicists' case). One can also derive the (physicists') generating function by using [[Cauchy's integral formula]] to write the Hermite polynomials as : <math>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>\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>\operatorname{\mathcal E}\left[\mathit{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>\operatorname{\mathcal E}\left[X^{2n}\right] = (-1)^n \mathit{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 probabilists' Hermite polynomials as moments: : <math>\mathit{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>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>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{2 \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>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>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>x = \sqrt{2n + 1}\cos(\varphi), \quad 0 < \varepsilon \leq \varphi \leq \pi - \varepsilon,</math> with which one has the uniform approximation : <math>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>x = \sqrt{2n+1} \cosh(\varphi), \quad 0 < \varepsilon \leq \varphi \leq \omega < \infty,</math> then : <math>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>x = \sqrt{2n + 1} + t</math> with {{mvar\|t}} complex and bounded, the approximation is : <math>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 physicists' Hermite polynomials evaluated at zero argument {{math\|''H<sub>n</sub>''(0)}} are called [[Hermite number]]s. :<math>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\|''H<sub>n</sub>''(0) {{=}} −2(''n'' − 1)''H''<sub>''n'' − 2</sub>(0)}}. In terms of the probabilists' polynomials this translates to :<math>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>\begin{align} H_{2n}(x) &= (-4)^n n! L_n^{\left(-\frac12\right)}(x^2) &&= 4^n n! \sum_{i=0}^n (-1)^{n-i} \binom{n-\frac12}{n-i} \frac{x^{2i}}{i!}, \\ H_{2n+1}(x) &= 2(-4)^n n! x L_n^{\left(\frac12\right)}(x^2) &&= 2\cdot 4^n n!\sum_{i=0}^n (-1)^{n-i} \binom{n+\frac12}{n-i} \frac{x^{2i+1}}{i!}. \end{align}</math> ===Relation to confluent hypergeometric functions=== The physicists' Hermite polynomials can be expressed as a special case of the [[parabolic cylinder function]]s: : <math>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>\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]]. ==Differential-operator representation== The probabilists' Hermite polynomials satisfy the identity : <math>\mathit{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\|''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}} ==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>\begin{align} \mathit{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 probabilists' Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is : <math>\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>\mathit{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>(2\pi\alpha)^{-\frac12} e^{-\frac{x^2}{2\alpha}}.</math> They are given by : <math>\mathit{He}_n^{[\alpha]}(x) = \alpha^{\frac{n}{2}}\mathit{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>\mathit{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>\left(\mathit{He}_n^{[\alpha]} \circ \mathit{He}^{[\beta]}\right)(x) \equiv \sum_{k=0}^n h^{[\alpha]}_{n,k}\,\mathit{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>\left(\mathit{He}_n^{[\alpha]} \circ \mathit{He}^{[\beta]}\right)(x) = \mathit{He}_n^{[\alpha+\beta]}(x)</math> and : <math>\mathit{He}_n^{[\alpha+\beta]}(x + y) = \sum_{k=0}^n \binom{n}{k} \mathit{He}_k^{[\alpha]}(x) \mathit{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\|''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>\mathit{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>\mathit{He}_n^{[-\alpha]}(x)</math> are just the absolute values of the corresponding coefficients of <math>\mathit{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>E[X^n] = \mathit{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>\sum_{k=0}^n \binom{n}{k} \mathit{He}_k^{[\alpha]}(x) \mathit{He}_{n-k}^{[-\alpha]}(y) = \mathit{He}_n^{[0]}(x + y) = (x + y)^n.</math> ==Applications== ===Hermite functions=== One can define the '''Hermite functions''' (often called Hermite-Gaussian functions) from the physicists' polynomials: : <math>\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>\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>\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>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>\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 (black), 1 (red), 2 (blue), 3 (yellow), 4 (green), and 5 (magenta)]] :<math>\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 (black), 2 (blue), 4 (green), and 50 (magenta)]] === Recursion relation === Following recursion relations of Hermite polynomials, the Hermite functions obey : <math>\psi_n'(x) = \sqrt{\frac{n}{2}}\,\psi_{n-1}(x) - \sqrt{\frac{n+1}{2}}\psi_{n+1}(x)</math> and : <math>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>\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) \mathit{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> \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 eigenfunctions of the [[continuous Fourier transform]] {{mathcal\|F}}. To see this, take the physicists' version of the generating function and multiply by {{math\|''e''<sup>−{{sfrac\|1\|2}}''x''<sup>2</sup></sup>}}. This gives : <math>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>\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>\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>\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>L_n(x) := \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{k!}x^k,</math> leading to the oscillator Laguerre functions :<math>l_n (x) := e^{-\frac{x}{2}} L_n(x).</math> For all natural integers {{mvar\|n}}, it is straightforward to see<ref>{{Citation \|authorlink=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>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> 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 \| url = \| 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. ===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}}-member set into {{mvar\|k}} singletons and {{math\|{{sfrac\|''n'' − ''k''\|2}}}} (unordered) pairs. 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>\mathit{He}_n(x) = B_n(x, -1, 0, \ldots, 0),</math> where {{math\|''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}}</ref> : <math>T(n) = \frac{\mathit{He}_n(i)}{i^n}.</math> === Completeness relation === The [[Christoffel–Darboux formula]] for Hermite polynomials reads :<math>\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>\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\|''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>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>\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\|''y'' {{=}} ''x''}}, and very spread out on that line. It follows that : <math>\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>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>\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> 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>\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>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 \|authorlink1=Richard Courant \|last2=Hilbert \|first2=David \|authorlink2=David Hilbert \|title=Methods of Mathematical Physics \|volume=Volume 1 \|publisher=Wiley-Interscience \|origyear=1953 \|year=1989 \|isbn=978-0-471-50447-4}} {{citation \|last=Erdélyi \|first=Arthur \|authorlink1=Arthur Erdélyi \|last2=Magnus \|first2=Wilhelm \|authorlink2=Wilhelm Magnus \|last3=Oberhettinger \|first3=Fritz \|last4=Tricomi \|first4=Francesco G. \|authorlink4=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}} {{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]. {{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 \|volume=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 \|authorlink=Gábor Szegő \|title=Orthogonal Polynomials \|series=Colloquium Publications \|volume=23 \|publisher=American Mathematical Society \|edition=4th \|origyear=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 \|authorlink=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. \|authorlink1=E. T. Whittaker \|last2=Watson \|first2=G. N.\|authorlink2=G. N. Watson \|title=[[A Course of Modern Analysis]] \|origyear=1927 \|year=1996 \|publisher=Cambridge University Press \|location=London \|edition=4th \|isbn=978-0-521-58807-2}} {{refend}} ==External links== {{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]]) {{DEFAULTSORT:Hermite Polynomials}} [[Category:Orthogonal polynomials]] [[Category:Polynomials]] [[Category:Special hypergeometric functions]]
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