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Hermite polynomials

From Wikipedia, the free encyclopedia
Polynomial sequence
This article is about the family of orthogonal polynomials on the real line. For polynomial interpolation on a segment using derivatives, seeHermite interpolation. For integral transform of Hermite polynomials, seeHermite transform.

Inmathematics, theHermite polynomials are a classicalorthogonalpolynomial sequence.

The polynomials arise in:

Hermite polynomials were defined byPierre-Simon Laplace in 1810,[1][2] though in scarcely recognizable form, and studied in detail byPafnuty Chebyshev in 1859.[3] Chebyshev's work was overlooked, and they were named later afterCharles Hermite, who wrote on the polynomials in 1864, describing them as new.[4] They were consequently not new, although Hermite was the first to define the multidimensional polynomials.

Definition

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Like the otherclassical 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:

These equations have the form of aRodrigues' formula and can also be written as,Hen⁡(x)=(x−ddx)n⋅1,Hn(x)=(2x−ddx)n⋅1.{\displaystyle \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.}

The two definitions are not exactly identical; each is a rescaling of the other:Hn(x)=2n2Hen⁡(2x),Hen⁡(x)=2−n2Hn(x2).{\displaystyle 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).}

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

The notationHe andH is that used in the standard references.[5]The polynomialsHen are sometimes denoted byHn, especially in probability theory, because12πe−x22{\displaystyle {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}}is theprobability density function for thenormal distribution withexpected value 0 andstandard deviation 1.

Quick reference table
physicist'sprobabilist's
symbolHn{\displaystyle H_{n}}Hen{\displaystyle \operatorname {He} _{n}}
head coefficient2n{\displaystyle 2^{n}}1{\displaystyle 1}
differential operator(−1)nex2dndxne−x2{\displaystyle (-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}}(−1)nex22dndxne−x22{\displaystyle (-1)^{n}e^{\frac {x^{2}}{2}}{\frac {d^{n}}{dx^{n}}}e^{-{\frac {x^{2}}{2}}}}
orthogonal toe−x2{\displaystyle e^{-x^{2}}}e−12x2{\displaystyle e^{-{\frac {1}{2}}x^{2}}}
inner product∫Hm(x)Hn(x)e−x2πdx=2nn!δmn{\displaystyle \int H_{m}(x)H_{n}(x){\frac {e^{-x^{2}}}{\sqrt {\pi }}}dx=2^{n}n!\delta _{mn}}∫Hem⁡(x)Hen⁡(x)e−x222πdx=n!δnm,{\displaystyle \int \operatorname {He} _{m}(x)\operatorname {He} _{n}(x)\,{\frac {e^{-{\frac {x^{2}}{2}}}}{\sqrt {2\pi }}}\,dx=n!\,\delta _{nm},}
generating functione2xt−t2=∑n=0∞Hn(x)tnn!{\displaystyle e^{2xt-t^{2}}=\sum _{n=0}^{\infty }H_{n}(x){\frac {t^{n}}{n!}}}ext−12t2=∑n=0∞Hen⁡(x)tnn!{\displaystyle e^{xt-{\frac {1}{2}}t^{2}}=\sum _{n=0}^{\infty }\operatorname {He} _{n}(x){\frac {t^{n}}{n!}}}
Rodrigues' formula(2x−ddx)n⋅1{\displaystyle \left(2x-{\frac {d}{dx}}\right)^{n}\cdot 1}(x−ddx)n⋅1{\displaystyle \left(x-{\frac {d}{dx}}\right)^{n}\cdot 1}
recurrence relationHn+1(x)=2xHn(x)−2nHn−1(x){\displaystyle H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x)}Hen+1⁡(x)=xHen⁡(x)−nHen−1⁡(x){\displaystyle \operatorname {He} _{n+1}(x)=x\operatorname {He} _{n}(x)-n\operatorname {He} _{n-1}(x)}

Properties

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Thenth-order Hermite polynomial is a polynomial of degreen. The probabilist's versionHen has leading coefficient 1, while the physicist's versionHn has leading coefficient2n.

Symmetry

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From the Rodrigues formulae given above, we can see thatHn(x) andHen(x) areeven or odd functions depending onn:Hn(−x)=(−1)nHn(x),Hen⁡(−x)=(−1)nHen⁡(x).{\displaystyle H_{n}(-x)=(-1)^{n}H_{n}(x),\quad \operatorname {He} _{n}(-x)=(-1)^{n}\operatorname {He} _{n}(x).}

Orthogonality

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Hn(x) andHen(x) arenth-degree polynomials forn = 0, 1, 2, 3,.... Thesepolynomials are orthogonal with respect to theweight function (measure)w(x)=e−x22(for He){\displaystyle w(x)=e^{-{\frac {x^{2}}{2}}}\quad ({\text{for }}\operatorname {He} )} orw(x)=e−x2(for H),{\displaystyle w(x)=e^{-x^{2}}\quad ({\text{for }}H),}i.e., we have∫−∞∞Hm(x)Hn(x)w(x)dx=0for all m≠n.{\displaystyle \int _{-\infty }^{\infty }H_{m}(x)H_{n}(x)\,w(x)\,dx=0\quad {\text{for all }}m\neq n.}

Furthermore,∫−∞∞Hm(x)Hn(x)e−x2dx=π2nn!δnm,{\displaystyle \int _{-\infty }^{\infty }H_{m}(x)H_{n}(x)\,e^{-x^{2}}\,dx={\sqrt {\pi }}\,2^{n}n!\,\delta _{nm},}and∫−∞∞Hem⁡(x)Hen⁡(x)e−x22dx=2πn!δnm,{\displaystyle \int _{-\infty }^{\infty }\operatorname {He} _{m}(x)\operatorname {He} _{n}(x)\,e^{-{\frac {x^{2}}{2}}}\,dx={\sqrt {2\pi }}\,n!\,\delta _{nm},}whereδnm{\displaystyle \delta _{nm}} is theKronecker delta.

The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.

Completeness

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The Hermite polynomials (probabilist's or physicist's) form anorthogonal basis of theHilbert space of functions satisfying∫−∞∞|f(x)|2w(x)dx<∞,{\displaystyle \int _{-\infty }^{\infty }{\bigl |}f(x){\bigr |}^{2}\,w(x)\,dx<\infty ,}in which the inner product is given by the integral⟨f,g⟩=∫−∞∞f(x)g(x)¯w(x)dx{\displaystyle \langle f,g\rangle =\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,w(x)\,dx}including theGaussian weight functionw(x) defined in the preceding section.

An orthogonal basis forL2(R,w(x)dx) is acomplete orthogonal system. For an orthogonal system,completeness is equivalent to the fact that the 0 function is the only functionf ∈L2(R,w(x)dx) orthogonal toall functions in the system.

Since thelinear span of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that iff satisfies∫−∞∞f(x)xne−x2dx=0{\displaystyle \int _{-\infty }^{\infty }f(x)x^{n}e^{-x^{2}}\,dx=0}for everyn ≥ 0, thenf = 0.

One possible way to do this is to appreciate that theentire functionF(z)=∫−∞∞f(x)ezx−x2dx=∑n=0∞znn!∫f(x)xne−x2dx=0{\displaystyle F(z)=\int _{-\infty }^{\infty }f(x)e^{zx-x^{2}}\,dx=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\int f(x)x^{n}e^{-x^{2}}\,dx=0}vanishes identically. The fact then thatF(it) = 0 for every realt means that theFourier transform off(x)e−x2 is 0, hencef is 0almost everywhere. Variants of the above completeness proof apply to other weights withexponential decay.

In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see section on theCompleteness relation below).

An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis forL2(R,w(x)dx) consists in introducing Hermitefunctions (see below), and in saying that the Hermite functions are an orthonormal basis forL2(R).

Hermite's differential equation

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The probabilist's Hermite polynomials are solutions of thedifferential equation(e−12x2u′)′+λe−12x2u=0,{\displaystyle \left(e^{-{\frac {1}{2}}x^{2}}u'\right)'+\lambda e^{-{\frac {1}{2}}x^{2}}u=0,}whereλ is a constant. Imposing the boundary condition thatu should be polynomially bounded at infinity, the equation has solutions only ifλ is a non-negative integer, and the solution is uniquely given byu(x)=C1Heλ⁡(x){\displaystyle u(x)=C_{1}\operatorname {He} _{\lambda }(x)}, whereC1{\displaystyle C_{1}} denotes a constant.

Rewriting the differential equation as aneigenvalue problemL[u]=u″−xu′=−λu,{\displaystyle L[u]=u''-xu'=-\lambda u,}the Hermite polynomialsHeλ⁡(x){\displaystyle \operatorname {He} _{\lambda }(x)} may be understood aseigenfunctions of the differential operatorL[u]{\displaystyle L[u]} . This eigenvalue problem is called theHermite equation, although the term is also used for the closely related equationu″−2xu′=−2λu.{\displaystyle u''-2xu'=-2\lambda u.} whose solution is uniquely given in terms of physicist's Hermite polynomials in the formu(x)=C1Hλ(x){\displaystyle u(x)=C_{1}H_{\lambda }(x)}, whereC1{\displaystyle C_{1}} denotes a constant, after imposing the boundary condition thatu 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 equationu″−2xu′+2λu=0,{\displaystyle u''-2xu'+2\lambda u=0,}the general solution takes the formu(x)=C1Hλ(x)+C2hλ(x),{\displaystyle u(x)=C_{1}H_{\lambda }(x)+C_{2}h_{\lambda }(x),}whereC1{\displaystyle C_{1}} andC2{\displaystyle C_{2}} are constants,Hλ(x){\displaystyle H_{\lambda }(x)} are physicist's Hermite polynomials (of the first kind), andhλ(x){\displaystyle h_{\lambda }(x)} are physicist's Hermite functions (of the second kind). The latter functions are compactly represented ashλ(x)=1F1(−λ2;12;x2){\displaystyle h_{\lambda }(x)={}_{1}F_{1}(-{\tfrac {\lambda }{2}};{\tfrac {1}{2}};x^{2})} where1F1(a;b;z){\displaystyle {}_{1}F_{1}(a;b;z)} areConfluent hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions, see below.

With more generalboundary conditions, the Hermite polynomials can be generalized to obtain more generalanalytic functions for complex-valuedλ. An explicit formula of Hermite polynomials in terms ofcontour integrals (Courant & Hilbert 1989) is also possible.

Recurrence relation

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The sequence of probabilist's Hermite polynomials also satisfies therecurrence relationHen+1⁡(x)=xHen⁡(x)−Hen′⁡(x).{\displaystyle \operatorname {He} _{n+1}(x)=x\operatorname {He} _{n}(x)-\operatorname {He} _{n}'(x).}Individual coefficients are related by the following recursion formula:an+1,k={−(k+1)an,k+1k=0,an,k−1−(k+1)an,k+1k>0,{\displaystyle 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}}} anda0,0 = 1,a1,0 = 0,a1,1 = 1.

For the physicist's polynomials, assumingHn(x)=∑k=0nan,kxk,{\displaystyle H_{n}(x)=\sum _{k=0}^{n}a_{n,k}x^{k},}we haveHn+1(x)=2xHn(x)−Hn′(x).{\displaystyle H_{n+1}(x)=2xH_{n}(x)-H_{n}'(x).}Individual coefficients are related by the following recursion formula:an+1,k={−an,k+1k=0,2an,k−1−(k+1)an,k+1k>0,{\displaystyle a_{n+1,k}={\begin{cases}-a_{n,k+1}&k=0,\\2a_{n,k-1}-(k+1)a_{n,k+1}&k>0,\end{cases}}}anda0,0 = 1,a1,0 = 0,a1,1 = 2.

The Hermite polynomials constitute anAppell sequence, i.e., they are a polynomial sequence satisfying the identityHen′⁡(x)=nHen−1⁡(x),Hn′(x)=2nHn−1(x).{\displaystyle {\begin{aligned}\operatorname {He} _{n}'(x)&=n\operatorname {He} _{n-1}(x),\\H_{n}'(x)&=2nH_{n-1}(x).\end{aligned}}}

An integral recurrence that is deduced and demonstrated in[6] is as follows:Hen+1⁡(x)=(n+1)∫0xHen⁡(t)dt−Hen′(0),{\displaystyle \operatorname {He} _{n+1}(x)=(n+1)\int _{0}^{x}\operatorname {He} _{n}(t)dt-He'_{n}(0),}

Hn+1(x)=2(n+1)∫0xHn(t)dt−Hn′(0).{\displaystyle H_{n+1}(x)=2(n+1)\int _{0}^{x}H_{n}(t)dt-H'_{n}(0).}

Equivalently, byTaylor-expanding,Hen⁡(x+y)=∑k=0n(nk)xn−kHek⁡(y)=2−n2∑k=0n(nk)Hen−k⁡(x2)Hek⁡(y2),Hn(x+y)=∑k=0n(nk)Hk(x)(2y)n−k=2−n2⋅∑k=0n(nk)Hn−k(x2)Hk(y2).{\displaystyle {\begin{aligned}\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{aligned}}}Theseumbral identities are self-evident andincluded in thedifferential operator representation detailed below,Hen⁡(x)=e−D22xn,Hn(x)=2ne−D24xn.{\displaystyle {\begin{aligned}\operatorname {He} _{n}(x)&=e^{-{\frac {D^{2}}{2}}}x^{n},\\H_{n}(x)&=2^{n}e^{-{\frac {D^{2}}{4}}}x^{n}.\end{aligned}}}

In consequence, for themth derivatives the following relations hold:Hen(m)⁡(x)=n!(n−m)!Hen−m⁡(x)=m!(nm)Hen−m⁡(x),Hn(m)(x)=2mn!(n−m)!Hn−m(x)=2mm!(nm)Hn−m(x).{\displaystyle {\begin{aligned}\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{aligned}}}

It follows that the Hermite polynomials also satisfy therecurrence relationHen+1⁡(x)=xHen⁡(x)−nHen−1⁡(x),Hn+1(x)=2xHn(x)−2nHn−1(x).{\displaystyle {\begin{aligned}\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{aligned}}}

These last relations, together with the initial polynomialsH0(x) andH1(x), can be used in practice to compute the polynomials quickly.

Turán's inequalities areHn(x)2−Hn−1(x)Hn+1(x)=(n−1)!∑i=0n−12n−ii!Hi(x)2>0.{\displaystyle {\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.}

Moreover, the followingmultiplication theorem holds:Hn(γx)=∑i=0⌊n2⌋γn−2i(γ2−1)i(n2i)(2i)!i!Hn−2i(x),Hen⁡(γx)=∑i=0⌊n2⌋γn−2i(γ2−1)i(n2i)(2i)!i!2−iHen−2i⁡(x).{\displaystyle {\begin{aligned}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{aligned}}}

Explicit expression

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The physicist's Hermite polynomials can be written explicitly asHn(x)={n!∑l=0n2(−1)n2−l(2l)!(n2−l)!(2x)2lfor even n,n!∑l=0n−12(−1)n−12−l(2l+1)!(n−12−l)!(2x)2l+1for odd n.{\displaystyle 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}}}

These two equations may be combined into one using thefloor function:Hn(x)=n!∑m=0⌊n2⌋(−1)mm!(n−2m)!(2x)n−2m.{\displaystyle H_{n}(x)=n!\sum _{m=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }{\frac {(-1)^{m}}{m!(n-2m)!}}(2x)^{n-2m}.}

The probabilist's Hermite polynomialsHe have similar formulas, which may be obtained from these by replacing the power of2x with the corresponding power of√2 x and multiplying the entire sum by2−⁠n/2⁠:Hen⁡(x)=n!∑m=0⌊n2⌋(−1)mm!(n−2m)!xn−2m2m.{\displaystyle \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}}}.}

Inverse explicit expression

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The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomialsHe arexn=n!∑m=0⌊n2⌋12mm!(n−2m)!Hen−2m⁡(x).{\displaystyle 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).}

The corresponding expressions for the physicist's Hermite polynomialsH follow directly by properly scaling this:[7]xn=n!2n∑m=0⌊n2⌋1m!(n−2m)!Hn−2m(x).{\displaystyle 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).}

Generating function

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The Hermite polynomials are given by theexponential generating functionext−12t2=∑n=0∞Hen⁡(x)tnn!,e2xt−t2=∑n=0∞Hn(x)tnn!.{\displaystyle {\begin{aligned}e^{xt-{\frac {1}{2}}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{aligned}}}

This equality is valid for allcomplex values ofx andt, and can be obtained by writing the Taylor expansion atx of the entire functionz →e−z2 (in the physicist's case). One can also derive the (physicist's) generating function by usingCauchy's integral formula to write the Hermite polynomials asHn(x)=(−1)nex2dndxne−x2=(−1)nex2n!2πi∮γe−z2(z−x)n+1dz.{\displaystyle 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.}

Using this in the sum∑n=0∞Hn(x)tnn!,{\displaystyle \sum _{n=0}^{\infty }H_{n}(x){\frac {t^{n}}{n!}},} one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.

A slight generalization states[8]e2xt−t2Hk(x−t)=∑n=0∞Hn+k(x)tnn!{\displaystyle e^{2xt-t^{2}}H_{k}(x-t)=\sum _{n=0}^{\infty }{\frac {H_{n+k}(x)t^{n}}{n!}}}

Expected values

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IfX is arandom variable with anormal distribution with standard deviation 1 and expected valueμ, thenE⁡[Hen⁡(X)]=μn.{\displaystyle \operatorname {\mathbb {E} } \left[\operatorname {He} _{n}(X)\right]=\mu ^{n}.}

The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices:E⁡[X2n]=(−1)nHe2n⁡(0)=(2n−1)!!,{\displaystyle \operatorname {\mathbb {E} } \left[X^{2n}\right]=(-1)^{n}\operatorname {He} _{2n}(0)=(2n-1)!!,}where(2n − 1)!! is thedouble factorial. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments:Hen⁡(x)=12π∫−∞∞(x+iy)ne−y22dy.{\displaystyle \operatorname {He} _{n}(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }(x+iy)^{n}e^{-{\frac {y^{2}}{2}}}\,dy.}

Integral representations

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From the generating-function representation above, we see that the Hermite polynomials have a representation in terms of acontour integral, asHen⁡(x)=n!2πi∮Cetx−t22tn+1dt,Hn(x)=n!2πi∮Ce2tx−t2tn+1dt,{\displaystyle {\begin{aligned}\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{aligned}}}with the contour encircling the origin.

Using the Fourier transform of the gaussiane−x2=1π∫e−t2+2ixtdt{\displaystyle e^{-x^{2}}={\frac {1}{\sqrt {\pi }}}\int e^{-t^{2}+2ixt}dt}, we haveHn(x)=(−1)nex2dndxne−x2=(−2i)nex2π∫tne−t2+2ixtdtHen⁡(x)=(−i)nex2/22π∫tne−t2/2+ixtdt.{\displaystyle {\begin{aligned}H_{n}(x)&=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}={\frac {(-2i)^{n}e^{x^{2}}}{\sqrt {\pi }}}\int t^{n}e^{-t^{2}+2ixt}dt\\\operatorname {He} _{n}(x)&={\frac {(-i)^{n}e^{x^{2}/2}}{\sqrt {2\pi }}}\int t^{n}\,e^{-t^{2}/2+ixt}\,dt.\end{aligned}}}

Other properties

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The addition theorem, or the summation theorem, states that[9][10]: 8.958 (∑k=1rak2)n2n!Hn(∑k=1rakxk∑k=1rak2)=∑m1+m2+…+mr=n,mi≥0∏k=1r{akmkmk!Hmk(xk)}{\displaystyle {\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\}}for any nonzero vectora1:r{\displaystyle a_{1:r}}.

The multiplication theorem states that[9]Hn(λx)=λn∑ℓ=0⌊n/2⌋(−n)2ℓℓ!(1−λ−2)ℓHn−2ℓ(x){\displaystyle 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)}for any nonzeroλ{\displaystyle \lambda }.

Feldheim formula[11]: Eq 46 1aπ∫−∞+∞e−x2aHm(x+yλ)Hn(x+zμ)dx=(1−aλ2)m2(1−aμ2)n2∑r=0min(m,n)r!(mr)(nr)(2a(λ2−a)(μ2−a))rHm−r(yλ2−a)Hn−r(zμ2−a){\displaystyle {\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)dx\\&=\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 {2a}{\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}}}wherea∈C{\displaystyle a\in \mathbb {C} } has a positive real part. As a special case,[11]: Eq 52 1π∫−∞+∞e−t2Hm(tsin⁡θ+vcos⁡θ)Hn(tcos⁡θ−vsin⁡θ)dt=(−1)ncosm⁡θsinn⁡θHm+n(v){\displaystyle {\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 )dt=(-1)^{n}\cos ^{m}\theta \sin ^{n}\theta H_{m+n}(v)}

Asymptotic expansion

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Asymptotically, asn → ∞, the expansion[12]e−x22⋅Hn(x)∼2nπΓ(n+12)cos⁡(x2n−nπ2){\displaystyle 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 {2n}}-{\frac {n\pi }{2}}\right)} holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude:e−x22⋅Hn(x)∼2nπΓ(n+12)cos⁡(x2n−nπ2)(1−x22n+1)−14=Γ(n)Γ(n2)cos⁡(x2n−nπ2)(1−x22n+1)−14,{\displaystyle 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 {2n}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}}={\frac {\Gamma (n)}{\Gamma \left({\frac {n}{2}}\right)}}\cos \left(x{\sqrt {2n}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}},}which, usingStirling's approximation, can be further simplified, in the limit, toe−x22⋅Hn(x)∼(2ne)n22cos⁡(x2n−nπ2)(1−x22n+1)−14.{\displaystyle 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)^{-{\frac {1}{4}}}.}

This expansion is needed to resolve thewavefunction of aquantum harmonic oscillator such that it agrees with the classical approximation in the limit of thecorrespondence principle.

A better approximation, which accounts for the variation in frequency, is given bye−x22⋅Hn(x)∼(2ne)n22cos⁡(x2n+1−x23−nπ2)(1−x22n+1)−14.{\displaystyle 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)^{-{\frac {1}{4}}}.}

A finer approximation,[13] which takes into account the uneven spacing of the zeros near the edges, makes use of the substitutionx=2n+1cos⁡(φ),0<ε≤φ≤π−ε,{\displaystyle x={\sqrt {2n+1}}\cos(\varphi ),\quad 0<\varepsilon \leq \varphi \leq \pi -\varepsilon ,} with which one has the uniform approximatione−x22⋅Hn(x)=2n2+14n!(πn)−14(sin⁡φ)−12⋅(sin⁡(3π4+(n2+14)(sin⁡2φ−2φ))+O(n−1)).{\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=2^{{\frac {n}{2}}+{\frac {1}{4}}}{\sqrt {n!}}(\pi n)^{-{\frac {1}{4}}}(\sin \varphi )^{-{\frac {1}{2}}}\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).}

Similar approximations hold for the monotonic and transition regions. Specifically, ifx=2n+1cosh⁡(φ),0<ε≤φ≤ω<∞,{\displaystyle x={\sqrt {2n+1}}\cosh(\varphi ),\quad 0<\varepsilon \leq \varphi \leq \omega <\infty ,} thene−x22⋅Hn(x)=2n2−34n!(πn)−14(sinh⁡φ)−12⋅e(n2+14)(2φ−sinh⁡2φ)(1+O(n−1)),{\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=2^{{\frac {n}{2}}-{\frac {3}{4}}}{\sqrt {n!}}(\pi n)^{-{\frac {1}{4}}}(\sinh \varphi )^{-{\frac {1}{2}}}\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),}while forx=2n+1+t{\displaystyle x={\sqrt {2n+1}}+t} witht complex and bounded, the approximation ise−x22⋅Hn(x)=π142n2+14n!n−112(Ai⁡(212n16t)+O(n−23)),{\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=\pi ^{\frac {1}{4}}2^{{\frac {n}{2}}+{\frac {1}{4}}}{\sqrt {n!}}\,n^{-{\frac {1}{12}}}\left(\operatorname {Ai} \left(2^{\frac {1}{2}}n^{\frac {1}{6}}t\right)+O\left(n^{-{\frac {2}{3}}}\right)\right),}whereAi is theAiry function of the first kind.

Special values

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The physicist's Hermite polynomials evaluated at zero argumentHn(0) are calledHermite numbers.

Hn(0)={0for odd n,(−2)n2(n−1)!!for even n,{\displaystyle H_{n}(0)={\begin{cases}0&{\text{for odd }}n,\\(-2)^{\frac {n}{2}}(n-1)!!&{\text{for even }}n,\end{cases}}}which satisfy the recursion relationHn(0) = −2(n − 1)Hn − 2(0). Equivalently,H2n(0)=(−2)n(2n−1)!!{\displaystyle H_{2n}(0)=(-2)^{n}(2n-1)!!}.

In terms of the probabilist's polynomials this translates toHen⁡(0)={0for odd n,(−1)n2(n−1)!!for even n.{\displaystyle \operatorname {He} _{n}(0)={\begin{cases}0&{\text{for odd }}n,\\(-1)^{\frac {n}{2}}(n-1)!!&{\text{for even }}n.\end{cases}}}

Kibble–Slepian formula

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LetM{\textstyle M} be a realn×n{\textstyle n\times n} symmetric matrix, then theKibble–Slepian formula states thatdet(I+M)−12exTM(I+M)−1x=∑K[∏1≤i≤j≤n(Mij/2)kijkij!]2−tr(K)Hk1(x1)⋯Hkn(xn){\displaystyle \det(I+M)^{-{\frac {1}{2}}}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})} where∑K{\textstyle \sum _{K}} is then(n+1)2{\displaystyle {\frac {n(n+1)}{2}}}-fold summation over alln×n{\textstyle n\times n} symmetric matrices with non-negative integer entries,tr(K){\displaystyle tr(K)} is thetrace ofK{\displaystyle K}, andki{\textstyle k_{i}} is defined askii+∑j=1nkij{\textstyle k_{ii}+\sum _{j=1}^{n}k_{ij}}. This givesMehler's formula whenM=[0uu0]{\displaystyle M={\begin{bmatrix}0&u\\u&0\end{bmatrix}}}.

Equivalently stated, ifT{\textstyle T} is apositive semidefinite matrix, then setM=−T(I+T)−1{\textstyle M=-T(I+T)^{-1}}, we haveM(I+M)−1=−T{\textstyle M(I+M)^{-1}=-T}, soe−xTTx=det(I+T)−12∑K[∏1≤i≤j≤n(Mij/2)kijkij!]2−tr(K)Hk1(x1)…Hkn(xn){\displaystyle e^{-x^{T}Tx}=\det(I+T)^{-{\frac {1}{2}}}\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})}Equivalently stated in a form closer to thebosonquantum mechanics of theharmonic oscillator:[14]π−n/4det(I+M)−12e−12xT(I−M)(I+M)−1x=∑K[∏1≤i≤j≤nMijkij/kij!][∏1≤i≤nki!]1/22−tr⁡Kψk1(x1)⋯ψkn(xn).{\displaystyle \pi ^{-n/4}\det(I+M)^{-{\frac {1}{2}}}e^{-{\frac {1}{2}}x^{T}(I-M)(I+M)^{-1}x}=\sum _{K}\left[\prod _{1\leq i\leq j\leq n}M_{ij}^{k_{ij}}/k_{ij}!\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).} where eachψn(x){\textstyle \psi _{n}(x)} is then{\textstyle n}-th eigenfunction of the harmonic oscillator, defined asψn(x):=12nn!(1π)14e−12x2Hn(x){\displaystyle \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)}The Kibble–Slepian formula was proposed by Kibble in 1945[15] and proven by Slepian in 1972 using Fourier analysis.[16] Foata gave a combinatorial proof[17] while Louck gave a proof via boson quantum mechanics.[14] It has a generalization for complex-argument Hermite polynomials.[18][19]

Relations to other functions

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Laguerre polynomials

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The Hermite polynomials can be expressed as a special case of theLaguerre polynomials:H2n(x)=(−4)nn!Ln(−12)(x2)=4nn!∑k=0n(−1)n−k(n−12n−k)x2kk!,H2n+1(x)=2(−4)nn!xLn(12)(x2)=2⋅4nn!∑k=0n(−1)n−k(n+12n−k)x2k+1k!.{\displaystyle {\begin{aligned}H_{2n}(x)&=(-4)^{n}n!L_{n}^{\left(-{\frac {1}{2}}\right)}(x^{2})&&=4^{n}n!\sum _{k=0}^{n}(-1)^{n-k}{\binom {n-{\frac {1}{2}}}{n-k}}{\frac {x^{2k}}{k!}},\\H_{2n+1}(x)&=2(-4)^{n}n!xL_{n}^{\left({\frac {1}{2}}\right)}(x^{2})&&=2\cdot 4^{n}n!\sum _{k=0}^{n}(-1)^{n-k}{\binom {n+{\frac {1}{2}}}{n-k}}{\frac {x^{2k+1}}{k!}}.\end{aligned}}}

Hypergeometric functions

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The physicist's Hermite polynomials can be expressed as a special case of theparabolic cylinder functions:Hn(x)=2nU(−12n,12,x2){\displaystyle H_{n}(x)=2^{n}U\left(-{\tfrac {1}{2}}n,{\tfrac {1}{2}},x^{2}\right)}in theright half-plane, whereU(a,b,z) isTricomi's confluent hypergeometric function. Similarly,H2n(x)=(−1)n(2n)!n!1F1(−n,12;x2),H2n+1(x)=(−1)n(2n+1)!n!2x1F1(−n,32;x2),{\displaystyle {\begin{aligned}H_{2n}(x)&=(-1)^{n}{\frac {(2n)!}{n!}}\,_{1}F_{1}{\big (}-n,{\tfrac {1}{2}};x^{2}{\big )},\\H_{2n+1}(x)&=(-1)^{n}{\frac {(2n+1)!}{n!}}\,2x\,_{1}F_{1}{\big (}-n,{\tfrac {3}{2}};x^{2}{\big )},\end{aligned}}}where1F1(a,b;z) =M(a,b;z) isKummer's confluent hypergeometric function.

There is also[20]Hn(x)=(2x)n2F0(−12n,−12n+12−;−1x2).{\displaystyle 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).}

Limit relations

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The Hermite polynomials can be obtained as the limit of various other polynomials.[21]

As a limit of Jacobi polynomials:limα→∞α−12nPn(α,α)(α−12x)=Hn(x)2nn!.{\displaystyle \lim _{\alpha \to \infty }\alpha ^{-{\frac {1}{2}}n}P_{n}^{(\alpha ,\alpha )}\left(\alpha ^{-{\frac {1}{2}}}x\right)={\frac {H_{n}\left(x\right)}{2^{n}n!}}.}As a limit of ultraspherical polynomials:limλ→∞λ−12nCn(λ)(λ−12x)=Hn(x)n!.{\displaystyle \lim _{\lambda \to \infty }\lambda ^{-{\frac {1}{2}}n}C_{n}^{(\lambda )}\left(\lambda ^{-{\frac {1}{2}}}x\right)={\frac {H_{n}\left(x\right)}{n!}}.}As a limit of associated Laguerre polynomials:limα→∞(2α)12nLn(α)((2α)12x+α)=(−1)nn!Hn(x).{\displaystyle \lim _{\alpha \to \infty }\left({\frac {2}{\alpha }}\right)^{{\frac {1}{2}}n}L_{n}^{(\alpha )}\left((2\alpha )^{\frac {1}{2}}x+\alpha \right)={\frac {(-1)^{n}}{n!}}H_{n}\left(x\right).}

Hermite polynomial expansion

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Similar to Taylor expansion, some functions are expressible as an infinite sum of Hermite polynomials. Specifically, if∫e−x2f(x)2dx<∞{\displaystyle \int e^{-x^{2}}f(x)^{2}dx<\infty }, then it has an expansion in the physicist's Hermite polynomials.[22]

Given suchf{\displaystyle f}, the partial sums of the Hermite expansion off{\displaystyle f} converges to in theLp{\displaystyle L^{p}} norm if and only if4/3<p<4{\displaystyle 4/3<p<4}.[23]xn=n!2n∑k=0⌊n/2⌋1k!(n−2k)!Hn−2k(x)=n!∑k=0⌊n/2⌋1k!2k(n−2k)!Hen−2k⁡(x),n∈Z+.{\displaystyle 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} _{+}.}eax=ea2/4∑n≥0ann!2nHn(x),a∈C,x∈R.{\displaystyle e^{ax}=e^{a^{2}/4}\sum _{n\geq 0}{\frac {a^{n}}{n!\,2^{n}}}\,H_{n}(x),\qquad a\in \mathbb {C} ,\quad x\in \mathbb {R} .}e−a2x2=∑n≥0(−1)na2nn!(1+a2)n+1/222nH2n(x).{\displaystyle e^{-a^{2}x^{2}}=\sum _{n\geq 0}{\frac {(-1)^{n}a^{2n}}{n!\left(1+a^{2}\right)^{n+1/2}2^{2n}}}\,H_{2n}(x).}erf⁡(x)=2π∫0xe−t2 dt=12π∑k≥0(−1)kk!(2k+1)23kH2k(x).{\displaystyle \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!(2k+1)2^{3k}}}H_{2k}(x).}cosh⁡(2x)=e∑k≥01(2k)!H2k(x),sinh⁡(2x)=e∑k≥01(2k+1)!H2k+1(x).{\displaystyle \cosh(2x)=e\sum _{k\geq 0}{\frac {1}{(2k)!}}\,H_{2k}(x),\qquad \sinh(2x)=e\sum _{k\geq 0}{\frac {1}{(2k+1)!}}\,H_{2k+1}(x).}cos⁡(x)=e−1/4∑k≥0(−1)k22k(2k)!H2k(x)sin⁡(x)=e−1/4∑k≥0(−1)k22k+1(2k+1)!H2k+1(x){\displaystyle \cos(x)=e^{-1/4}\,\sum _{k\geq 0}{\frac {(-1)^{k}}{2^{2k}\,(2k)!}}\,H_{2k}(x)\quad \sin(x)=e^{-1/4}\,\sum _{k\geq 0}{\frac {(-1)^{k}}{2^{2k+1}\,(2k+1)!}}\,H_{2k+1}(x)}

Differential-operator representation

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The probabilist's Hermite polynomials satisfy the identity[24]Hen⁡(x)=e−D22xn,{\displaystyle \operatorname {He} _{n}(x)=e^{-{\frac {D^{2}}{2}}}x^{n},} whereD represents differentiation with respect tox, and theexponential is interpreted by expanding it as apower 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 monomialxn can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients ofHn that can be used to quickly compute these polynomials.

Since the formal expression for theWeierstrass transformW iseD2, we see that the Weierstrass transform of(√2)nHen(⁠x/√2⁠) isxn. Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a correspondingMaclaurin series.

The existence of some formal power seriesg(D) with nonzero constant coefficient, such thatHen(x) =g(D)xn, is another equivalent to the statement that these polynomials form anAppell sequence. Since they are an Appell sequence, they area fortiori aSheffer sequence.

Further information:Weierstrass transform § The inverse transform

Generalizations

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The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is12πe−x22,{\displaystyle {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}},}which has expected value 0 and variance 1.

Scaling, one may analogously speak ofgeneralized Hermite polynomials[25]Hen[α]⁡(x){\displaystyle \operatorname {He} _{n}^{[\alpha ]}(x)}of varianceα, whereα is any positive number. These are then orthogonal with respect to the normal probability distribution whose density function is12παe−x22α.{\displaystyle {\frac {1}{\sqrt {2\pi \alpha }}}e^{-{\frac {x^{2}}{2\alpha }}}.}They are given byHen[α]⁡(x)=αn2Hen⁡(xα)=(α2)n2Hn(x2α)=e−αD22(xn).{\displaystyle \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).}

Now, ifHen[α]⁡(x)=∑k=0nhn,k[α]xk,{\displaystyle \operatorname {He} _{n}^{[\alpha ]}(x)=\sum _{k=0}^{n}h_{n,k}^{[\alpha ]}x^{k},}then the polynomial sequence whosenth term is(Hen[α]∘He[β])(x)≡∑k=0nhn,k[α]Hek[β]⁡(x){\displaystyle \left(\operatorname {He} _{n}^{[\alpha ]}\circ \operatorname {He} ^{[\beta ]}\right)(x)\equiv \sum _{k=0}^{n}h_{n,k}^{[\alpha ]}\,\operatorname {He} _{k}^{[\beta ]}(x)}is called theumbral composition of the two polynomial sequences. It can be shown to satisfy the identities(Hen[α]∘He[β])(x)=Hen[α+β]⁡(x){\displaystyle \left(\operatorname {He} _{n}^{[\alpha ]}\circ \operatorname {He} ^{[\beta ]}\right)(x)=\operatorname {He} _{n}^{[\alpha +\beta ]}(x)}andHen[α+β]⁡(x+y)=∑k=0n(nk)Hek[α]⁡(x)Hen−k[β]⁡(y).{\displaystyle \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).}The last identity is expressed by saying that thisparameterized family of polynomial sequences is known as a cross-sequence. (See the above section on Appell sequences and on thedifferential-operator representation, which leads to a ready derivation of it. Thisbinomial type identity, forα =β =⁠1/2⁠, has already been encountered in the above section on#Recursion relations.)

"Negative variance"

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Since polynomial sequences form agroup under the operation ofumbral composition, one may denote byHen[−α]⁡(x){\displaystyle \operatorname {He} _{n}^{[-\alpha ]}(x)}the sequence that is inverse to the one similarly denoted, but without the minus sign, and thus speak of Hermite polynomials of negative variance. Forα > 0, the coefficients ofHen[−α]⁡(x){\displaystyle \operatorname {He} _{n}^{[-\alpha ]}(x)} are just the absolute values of the corresponding coefficients ofHen[α]⁡(x){\displaystyle \operatorname {He} _{n}^{[\alpha ]}(x)}.

These arise as moments of normal probability distributions: Thenth moment of the normal distribution with expected valueμ and varianceσ2 isE[Xn]=Hen[−σ2]⁡(μ),{\displaystyle E[X^{n}]=\operatorname {He} _{n}^{[-\sigma ^{2}]}(\mu ),}whereX is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that∑k=0n(nk)Hek[α]⁡(x)Hen−k[−α]⁡(y)=Hen[0]⁡(x+y)=(x+y)n.{\displaystyle \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}.}

Hermite functions

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Definition

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One can define theHermite functions (often called Hermite-Gaussian functions) from the physicist's polynomials:ψn(x)=(2nn!π)−12e−x22Hn(x)=(−1)n(2nn!π)−12ex22dndxne−x2.{\displaystyle \psi _{n}(x)=\left(2^{n}n!{\sqrt {\pi }}\right)^{-{\frac {1}{2}}}e^{-{\frac {x^{2}}{2}}}H_{n}(x)=(-1)^{n}\left(2^{n}n!{\sqrt {\pi }}\right)^{-{\frac {1}{2}}}e^{\frac {x^{2}}{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}.}Thus,2(n+1)  ψn+1(x)=(x−ddx)ψn(x).{\displaystyle {\sqrt {2(n+1)}}~~\psi _{n+1}(x)=\left(x-{d \over dx}\right)\psi _{n}(x).}

Since these functions contain the square root of theweight function and have been scaled appropriately, they areorthonormal:∫−∞∞ψn(x)ψm(x)dx=δnm,{\displaystyle \int _{-\infty }^{\infty }\psi _{n}(x)\psi _{m}(x)\,dx=\delta _{nm},}and they form an orthonormal basis ofL2(R). This fact is equivalent to the corresponding statement for Hermite polynomials (see above).

The Hermite functions are closely related to theWhittaker function (Whittaker & Watson 1996)Dn(z):Dn(z)=(n!π)12ψn(z2)=(−1)nez24dndzne−z22{\displaystyle D_{n}(z)=\left(n!{\sqrt {\pi }}\right)^{\frac {1}{2}}\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}}}and thereby to otherparabolic cylinder functions.

The Hermite functions satisfy the differential equationψn″(x)+(2n+1−x2)ψn(x)=0.{\displaystyle \psi _{n}''(x)+\left(2n+1-x^{2}\right)\psi _{n}(x)=0.}This equation is equivalent to theSchrödinger equation for a harmonic oscillator in quantum mechanics, so these functions are theeigenfunctions.

Hermite functions: 0 (blue, solid), 1 (orange, dashed), 2 (green, dot-dashed), 3 (red, dotted), 4 (purple, solid), and 5 (brown, dashed)

ψ0(x)=π−14e−12x2,ψ1(x)=2π−14xe−12x2,ψ2(x)=(2π14)−1(2x2−1)e−12x2,ψ3(x)=(3π14)−1(2x3−3x)e−12x2,ψ4(x)=(26π14)−1(4x4−12x2+3)e−12x2,ψ5(x)=(215π14)−1(4x5−20x3+15x)e−12x2.{\displaystyle {\begin{aligned}\psi _{0}(x)&=\pi ^{-{\frac {1}{4}}}\,e^{-{\frac {1}{2}}x^{2}},\\\psi _{1}(x)&={\sqrt {2}}\,\pi ^{-{\frac {1}{4}}}\,x\,e^{-{\frac {1}{2}}x^{2}},\\\psi _{2}(x)&=\left({\sqrt {2}}\,\pi ^{\frac {1}{4}}\right)^{-1}\,\left(2x^{2}-1\right)\,e^{-{\frac {1}{2}}x^{2}},\\\psi _{3}(x)&=\left({\sqrt {3}}\,\pi ^{\frac {1}{4}}\right)^{-1}\,\left(2x^{3}-3x\right)\,e^{-{\frac {1}{2}}x^{2}},\\\psi _{4}(x)&=\left(2{\sqrt {6}}\,\pi ^{\frac {1}{4}}\right)^{-1}\,\left(4x^{4}-12x^{2}+3\right)\,e^{-{\frac {1}{2}}x^{2}},\\\psi _{5}(x)&=\left(2{\sqrt {15}}\,\pi ^{\frac {1}{4}}\right)^{-1}\,\left(4x^{5}-20x^{3}+15x\right)\,e^{-{\frac {1}{2}}x^{2}}.\end{aligned}}}

Hermite functions: 0 (blue, solid), 2 (orange, dashed), 4 (green, dot-dashed), and 50 (red, solid)

Recursion relation

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Following recursion relations of Hermite polynomials, the Hermite functions obeyψn′(x)=n2ψn−1(x)−n+12ψn+1(x){\displaystyle \psi _{n}'(x)={\sqrt {\frac {n}{2}}}\,\psi _{n-1}(x)-{\sqrt {\frac {n+1}{2}}}\psi _{n+1}(x)}andxψn(x)=n2ψn−1(x)+n+12ψn+1(x).{\displaystyle x\psi _{n}(x)={\sqrt {\frac {n}{2}}}\,\psi _{n-1}(x)+{\sqrt {\frac {n+1}{2}}}\psi _{n+1}(x).}

Extending the first relation to the arbitrarymth derivatives for any positive integerm leads toψn(m)(x)=∑k=0m(mk)(−1)k2m−k2n!(n−m+k)!ψn−m+k(x)Hek⁡(x).{\displaystyle \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).}

This formula can be used in connection with the recurrence relations forHen andψn to calculate any derivative of the Hermite functions efficiently.

Cramér's inequality

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For realx, the Hermite functions satisfy the following bound due toHarald Cramér[26][27] and Jack Indritz:[28]|ψn(x)|≤π−14.{\displaystyle {\bigl |}\psi _{n}(x){\bigr |}\leq \pi ^{-{\frac {1}{4}}}.}

Hermite functions as eigenfunctions of the Fourier transform

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The Hermite functionsψn(x) are a set ofeigenfunctions of thecontinuous Fourier transformF. To see this, take the physicist's version of the generating function and multiply bye−⁠1/2⁠x2. This givese−12x2+2xt−t2=∑n=0∞e−12x2Hn(x)tnn!.{\displaystyle e^{-{\frac {1}{2}}x^{2}+2xt-t^{2}}=\sum _{n=0}^{\infty }e^{-{\frac {1}{2}}x^{2}}H_{n}(x){\frac {t^{n}}{n!}}.}

The Fourier transform of the left side is given byF{e−12x2+2xt−t2}(k)=12π∫−∞∞e−ixke−12x2+2xt−t2dx=e−12k2−2kit+t2=∑n=0∞e−12k2Hn(k)(−it)nn!.{\displaystyle {\begin{aligned}{\mathcal {F}}\left\{e^{-{\frac {1}{2}}x^{2}+2xt-t^{2}}\right\}(k)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }e^{-ixk}e^{-{\frac {1}{2}}x^{2}+2xt-t^{2}}\,dx\\&=e^{-{\frac {1}{2}}k^{2}-2kit+t^{2}}\\&=\sum _{n=0}^{\infty }e^{-{\frac {1}{2}}k^{2}}H_{n}(k){\frac {(-it)^{n}}{n!}}.\end{aligned}}}

The Fourier transform of the right side is given byF{∑n=0∞e−12x2Hn(x)tnn!}=∑n=0∞F{e−12x2Hn(x)}tnn!.{\displaystyle {\mathcal {F}}\left\{\sum _{n=0}^{\infty }e^{-{\frac {1}{2}}x^{2}}H_{n}(x){\frac {t^{n}}{n!}}\right\}=\sum _{n=0}^{\infty }{\mathcal {F}}\left\{e^{-{\frac {1}{2}}x^{2}}H_{n}(x)\right\}{\frac {t^{n}}{n!}}.}

Equating like powers oft in the transformed versions of the left and right sides finally yieldsF{e−12x2Hn(x)}=(−i)ne−12k2Hn(k).{\displaystyle {\mathcal {F}}\left\{e^{-{\frac {1}{2}}x^{2}}H_{n}(x)\right\}=(-i)^{n}e^{-{\frac {1}{2}}k^{2}}H_{n}(k).}

The Hermite functionsψn(x) are thus an orthonormal basis ofL2(R), whichdiagonalizes the Fourier transform operator.[29] In short, we have:12π∫e−ikxψn(x)dx=(−i)nψn(k),12π∫e+ikxψn(k)dk=inψn(x){\displaystyle {\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)}

Wigner distributions of Hermite functions

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TheWigner distribution function of thenth-order Hermite function is related to thenth-orderLaguerre polynomial. The Laguerre polynomials areLn(x):=∑k=0n(nk)(−1)kk!xk,{\displaystyle L_{n}(x):=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{k!}}x^{k},}leading to the oscillator Laguerre functionsln(x):=e−x2Ln(x).{\displaystyle l_{n}(x):=e^{-{\frac {x}{2}}}L_{n}(x).}For all natural integersn, it is straightforward to see[30] thatWψn(t,f)=(−1)nln(4π(t2+f2)),{\displaystyle W_{\psi _{n}}(t,f)=(-1)^{n}l_{n}{\big (}4\pi (t^{2}+f^{2}){\big )},}where the Wigner distribution of a functionx ∈L2(R,C) is defined asWx(t,f)=∫−∞∞x(t+τ2)x(t−τ2)∗e−2πiτfdτ.{\displaystyle 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 .}This is a fundamental result for thequantum harmonic oscillator discovered byHip Groenewold in 1946 in his PhD thesis.[31] It is the standard paradigm ofquantum mechanics in phase space.

There arefurther relations between the two families of polynomials.

Partial Overlap Integrals

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It can be shown[32][33] that the overlap between two different Hermite functions (k≠ℓ{\displaystyle k\neq \ell }) over a given interval has the exact result:∫x1x2ψk(x)ψℓ(x)dx=12(ℓ−k)(ψk′(x2)ψℓ(x2)−ψℓ′(x2)ψk(x2)−ψk′(x1)ψℓ(x1)+ψℓ′(x1)ψk(x1)).{\displaystyle \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).}

Combinatorial interpretation of coefficients

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In the Hermite polynomialHen(x) of variance 1, the absolute value of the coefficient ofxk is the number of (unordered) partitions of ann-element set intok singletons and⁠n −k/2⁠ (unordered) pairs. Equivalently, it is the number of involutions of ann-element set with preciselyk fixed points, or in other words, the number of matchings in thecomplete graph onn vertices that leavek vertices uncovered (indeed, the Hermite polynomials are thematching 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-calledtelephone numbers

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496,... (sequenceA000085 in theOEIS).

This combinatorial interpretation can be related to complete exponentialBell polynomials asHen⁡(x)=Bn(x,−1,0,…,0),{\displaystyle \operatorname {He} _{n}(x)=B_{n}(x,-1,0,\ldots ,0),}wherexi = 0 for alli > 2.

These numbers may also be expressed as a special value of the Hermite polynomials:[34]T(n)=Hen⁡(i)in.{\displaystyle T(n)={\frac {\operatorname {He} _{n}(i)}{i^{n}}}.}

Completeness relation

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TheChristoffel–Darboux formula for Hermite polynomials reads∑k=0nHk(x)Hk(y)k!2k=1n!2n+1Hn(y)Hn+1(x)−Hn(x)Hn+1(y)x−y.{\displaystyle \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}}.}

Moreover, the followingcompleteness identity for the above Hermite functions holds in the sense ofdistributions:∑n=0∞ψn(x)ψn(y)=δ(x−y),{\displaystyle \sum _{n=0}^{\infty }\psi _{n}(x)\psi _{n}(y)=\delta (x-y),}whereδ is theDirac delta function,ψn the Hermite functions, andδ(x −y) represents theLebesgue measure on the liney =x inR2, normalized so that its projection on the horizontal axis is the usual Lebesgue measure.

This distributional identity followsWiener (1958) by takingu → 1 inMehler's formula, valid when−1 <u < 1:E(x,y;u):=∑n=0∞unψn(x)ψn(y)=1π(1−u2)exp⁡(−1−u1+u(x+y)24−1+u1−u(x−y)24),{\displaystyle 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),}which is often stated equivalently as a separable kernel,[35][36]∑n=0∞Hn(x)Hn(y)n!(u2)n=11−u2e2u1+uxy−u21−u2(x−y)2.{\displaystyle \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}}.}

The function(x,y) →E(x,y;u) is the bivariate Gaussian probability density onR2, which is, whenu is close to 1, very concentrated around the liney =x, and very spread out on that line. It follows that∑n=0∞un⟨f,ψn⟩⟨ψn,g⟩=∬E(x,y;u)f(x)g(y)¯dxdy→∫f(x)g(x)¯dx=⟨f,g⟩{\displaystyle \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 }whenf andg are continuous and compactly supported.

This yields thatf can be expressed in Hermite functions as the sum of a series of vectors inL2(R), namely,f=∑n=0∞⟨f,ψn⟩ψn.{\displaystyle f=\sum _{n=0}^{\infty }\langle f,\psi _{n}\rangle \psi _{n}.}

In order to prove the above equality forE(x,y;u), theFourier transform ofGaussian functions is used repeatedly:ρπe−ρ2x24=∫eisx−s2ρ2dsfor ρ>0.{\displaystyle \rho {\sqrt {\pi }}e^{-{\frac {\rho ^{2}x^{2}}{4}}}=\int e^{isx-{\frac {s^{2}}{\rho ^{2}}}}\,ds\quad {\text{for }}\rho >0.}

The Hermite polynomial is then represented asHn(x)=(−1)nex2dndxn(12π∫eisx−s24ds)=(−1)nex212π∫(is)neisx−s24ds.{\displaystyle 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.}

With this representation forHn(x) andHn(y), it is evident thatE(x,y;u)=∑n=0∞un2nn!πHn(x)Hn(y)e−x2+y22=ex2+y224ππ∬(∑n=0∞12nn!(−ust)n)eisx+ity−s24−t24dsdt=ex2+y224ππ∬e−ust2eisx+ity−s24−t24dsdt,{\displaystyle {\begin{aligned}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{aligned}}}and this yields the desired resolution of the identity result, using again the Fourier transform of Gaussian kernels under the substitutions=σ+τ2,t=σ−τ2.{\displaystyle s={\frac {\sigma +\tau }{\sqrt {2}}},\quad t={\frac {\sigma -\tau }{\sqrt {2}}}.}

See also

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Notes

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  1. ^Laplace (1811)."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" [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].Mémoires de la Classe des Sciences Mathématiques et Physiques de l'Institut Impérial de France (in French).11:297–347.
  2. ^Laplace, P.-S. (1812),Théorie analytique des probabilités [Analytic Probability Theory], vol. 2, pp. 194–203 Collected inŒuvres complètesVII.
  3. ^Tchébychef, P. (1860)."Sur le développement des fonctions à une seule variable" [On the development of single-variable functions].Bulletin de l'Académie impériale des sciences de St.-Pétersbourg (in French).1:193–200. Collected inŒuvresI, 501–508.
  4. ^Hermite, C. (1864)."Sur un nouveau développement en série de fonctions" [On a new development in function series].C. R. Acad. Sci. Paris (in French).58:93–100,266–273. Collected inŒuvresII, 293–308.
  5. ^Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al. (2010) andAbramowitz & Stegun.
  6. ^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.
  7. ^"18. Orthogonal Polynomials, Classical Orthogonal Polynomials, Sums".Digital Library of Mathematical Functions. National Institute of Standards and Technology. Retrieved30 January 2015.
  8. ^(Rainville 1971), p. 198
  9. ^ab"DLMF: §18.18 Sums ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials".dlmf.nist.gov. Retrieved2025-03-18.
  10. ^Gradshteĭn, I. S.; Zwillinger, Daniel (2015).Table of integrals, series, and products (8 ed.). Amsterdam ; Boston: Elsevier, Academic Press is an imprint of Elsevier.ISBN 978-0-12-384933-5.
  11. ^abFeldheim, 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). PartI,II,III
  12. ^Abramowitz & Stegun 1983, p. 508–510,13.6.38 and 13.5.16.
  13. ^Szegő 1955, p. 201
  14. ^abLouck, J. D (1981-09-01)."Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods".Advances in Applied Mathematics.2 (3):239–249.doi:10.1016/0196-8858(81)90005-1.ISSN 0196-8858.
  15. ^Kibble, W. F. (June 1945)."An extension of a theorem of Mehler's on Hermite polynomials".Mathematical Proceedings of the Cambridge Philosophical Society.41 (1):12–15.Bibcode:1945PCPS...41...12K.doi:10.1017/S0305004100022313.ISSN 1469-8064.
  16. ^Slepian, David (November 1972)."On the Symmetrized Kronecker Power of a Matrix and Extensions of Mehler's Formula for Hermite Polynomials".SIAM Journal on Mathematical Analysis.3 (4):606–616.doi:10.1137/0503060.ISSN 0036-1410.
  17. ^Foata, Dominique (1981-09-01)."Some Hermite polynomial identities and their combinatorics".Advances in Applied Mathematics.2 (3):250–259.doi:10.1016/0196-8858(81)90006-3.ISSN 0196-8858.
  18. ^Ismail, Mourad E.H.; Zhang, Ruiming (September 2016)."Kibble–Slepian formula and generating functions for 2D polynomials".Advances in Applied Mathematics.80:70–92.arXiv:1508.01816.doi:10.1016/j.aam.2016.05.003.ISSN 0196-8858.
  19. ^Ismail, Mourad E. H.; Zhang, Ruiming (2017-04-01)."A review of multivariate orthogonal polynomials".Journal of the Egyptian Mathematical Society.25 (2):91–110.doi:10.1016/j.joems.2016.11.001.ISSN 1110-256X.
  20. ^DLMF Equation 18.5.13
  21. ^DLMF §18.7(iii) Limit Relations
  22. ^"MATHEMATICA tutorial, part 2.5: Hermite expansion".www.cfm.brown.edu. Retrieved2023-12-24.
  23. ^Askey, Richard; Wainger, Stephen (1965)."Mean Convergence of Expansions in Laguerre and Hermite Series".American Journal of Mathematics.87 (3):695–708.doi:10.2307/2373069.ISSN 0002-9327.JSTOR 2373069.
  24. ^Rota, Gian-Carlo; Doubilet, P. (1975).Finite operator calculus. New York: Academic Press. p. 44.ISBN 9780125966504.
  25. ^Roman, Steven (1984),The Umbral Calculus, Pure and Applied Mathematics, vol. 111 (1st ed.), Academic Press, pp. 87–93,ISBN 978-0-12-594380-2
  26. ^Erdélyi et al. 1955, p. 207.
  27. ^Szegő 1955.
  28. ^Indritz, Jack (1961), "An inequality for Hermite polynomials",Proceedings of the American Mathematical Society,12 (6):981–983,doi:10.1090/S0002-9939-1961-0132852-2,MR 0132852
  29. ^In this case, we used the unitary version of the Fourier transform, so theeigenvalues are(−i)n. The ensuing resolution of the identity then serves to define powers, including fractional ones, of the Fourier transform, to wit aFractional Fourier transform generalization, in effect aMehler kernel.
  30. ^Folland, G. B. (1989),Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122, Princeton University Press,ISBN 978-0-691-08528-9
  31. ^Groenewold, H. J. (1946). "On the Principles of elementary quantum mechanics".Physica.12 (7):405–460.Bibcode:1946Phy....12..405G.doi:10.1016/S0031-8914(46)80059-4.
  32. ^Mawby, Clement (2024). "Tests of Macrorealism in Discrete and Continuous Variable Systems".arXiv:2402.16537 [quant-ph].
  33. ^Moriconi, Marco (2007). "Nodes of Wavefunctions".arXiv:quant-ph/0702260.
  34. ^Banderier, Cyril;Bousquet-Mélou, Mireille; Denise, Alain;Flajolet, Philippe; Gardy, Danièle; Gouyou-Beauchamps, Dominique (2002), "Generating functions for generating trees",Discrete Mathematics,246 (1–3):29–55,arXiv:math/0411250,doi:10.1016/S0012-365X(01)00250-3,MR 1884885,S2CID 14804110
  35. ^Mehler, F. G. (1866),"Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung" [On the development of a function of arbitrarily many variables according to higher-order Laplace functions],Journal für die Reine und Angewandte Mathematik (in German) (66):161–176,ISSN 0075-4102,ERAM 066.1720cj. See p. 174, eq. (18) and p. 173, eq. (13).
  36. ^Erdélyi et al. 1955, p. 194, 10.13 (22).

References

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