Editing Hermite polynomials (section)

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

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

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

Furthermore,
<math display="block">\int_{-\infty}^\infty H_m(x) H_n(x)\, e^{-x^2} \,dx = \sqrt{\pi}\, 2^n n!\, \delta_{nm},</math>
and
<math display="block">\int_{-\infty}^\infty \operatorname{He}_m(x) \operatorname{He}_n(x)\, e^{-\frac{x^2}{2}} \,dx = \sqrt{2 \pi}\, n!\, \delta_{nm},</math>
where <math>\delta_{nm}</math> 
is the [[Kronecker delta]].

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

=== Completeness ===
The Hermite polynomials (probabilist's or physicist's) form an [[orthonormal basis|orthogonal basis]] of the [[Hilbert space]] of functions satisfying
<math display="block">\int_{-\infty}^\infty \bigl|f(x)\bigr|^2\, w(x) \,dx < \infty,</math>
in which the inner product is given by the integral
<math display="block">\langle f,g\rangle = \int_{-\infty}^\infty f(x) \overline{g(x)}\, w(x) \,dx</math>
including the [[gaussian function|Gaussian]] weight function {{math|''w''(''x'')}} defined in the preceding section.

An orthogonal basis for {{math|[[Lp space|''L''<sup>2</sup>('''R''', ''w''(''x'') ''dx'')]]}} is a [[Hilbert space#Orthonormal bases|''complete'' orthogonal system]]. For an orthogonal system, ''completeness'' is equivalent to the fact that the 0 function is the only function {{math|''f'' ∈ ''L''<sup>2</sup>('''R''', ''w''(''x'') ''dx'')}} orthogonal to ''all'' functions in the system.

Since the [[linear span]] of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that if {{mvar|f}} satisfies
<math display="block">\int_{-\infty}^\infty f(x) x^n e^{- x^2} \,dx = 0</math>
for every {{math|''n'' ≥ 0}}, then {{math|1=''f'' = 0}}.

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

In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see section on the [[#Completeness_relation|Completeness relation]] below).

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

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

Rewriting the differential equation as an [[Eigenvalue|eigenvalue problem]]
<math display="block">L[u] = u'' - x u' = -\lambda u,</math>
the Hermite polynomials <math>\operatorname{He}_\lambda(x) </math> may be understood as [[eigenfunction]]s of the differential operator <math>L[u]</math> . This eigenvalue problem is called the '''Hermite equation''', although the term is also used for the closely related equation
<math display="block">u'' - 2xu' = -2\lambda u.</math> 
whose solution is uniquely given in terms of physicist's Hermite polynomials in the form <math>u(x) = C_1 H_\lambda(x) </math>, where <math>C_{1}</math> denotes a constant, after imposing the boundary condition that {{mvar|u}} should be polynomially bounded at infinity.

The general solutions to the above second-order differential equations are in fact linear combinations of both Hermite polynomials and confluent hypergeometric functions of the first kind. For example, for the physicist's Hermite equation
<math display="block">u'' - 2xu' + 2\lambda u = 0,</math>
the general solution takes the form
<math display="block">u(x) = C_1   H_\lambda(x) + C_2  h_\lambda(x),</math>
where <math>C_{1}</math> and <math>C_{2}</math> are constants, <math>H_\lambda(x)</math> are physicist's Hermite polynomials (of the first kind), and  <math>h_\lambda(x)</math> are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as <math> h_\lambda(x) = {}_1F_1(-\tfrac{\lambda}{2};\tfrac{1}{2};x^2)</math> where  <math>{}_1F_1(a;b;z)</math> are [[Confluent hypergeometric function|Confluent hypergeometric functions of the first kind]]. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions, see below.

With more general [[boundary conditions]], the Hermite polynomials can be generalized to obtain more general [[analytic function]]s for complex-valued {{mvar|λ}}. An explicit formula of Hermite polynomials in terms of [[contour integral]]s {{harv|Courant|Hilbert|1989}} is also possible.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This equality is valid for all [[complex number|complex]] values of {{mvar|x}} and {{mvar|t}}, and can be obtained by writing the Taylor expansion at {{mvar|x}} of the entire function {{math|''z'' → ''e''<sup>−''z''<sup>2</sup></sup>}} (in the physicist's case). One can also derive the (physicist's) generating function by using [[Cauchy's integral formula]] to write the Hermite polynomials as
<math display="block">H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} = (-1)^n e^{x^2} \frac{n!}{2\pi i} \oint_\gamma \frac{e^{-z^2}}{(z-x)^{n+1}} \,dz.</math>

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

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

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

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

=== Integral representations ===
From the generating-function representation above, we see that the Hermite polynomials have a representation in terms of a [[contour integral]], as
<math display="block">\begin{align}
 \operatorname{He}_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{tx-\frac{t^2}{2}}}{t^{n+1}}\,dt, \\
 H_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{2tx-t^2}}{t^{n+1}}\,dt,
\end{align}</math>
with the contour encircling the origin.

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

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

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

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

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

This expansion is needed to resolve the [[wavefunction]] of a [[quantum harmonic oscillator]] such that it agrees with the classical approximation in the limit of the [[correspondence principle]].

A better approximation, which accounts for the variation in frequency, is given by
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \left(\frac{2n}{e}\right)^{\frac{n}{2}} \sqrt{2} \cos \left(x \sqrt{2n+1-\frac{x^2}{3}}- \frac {n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}.</math>

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

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

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

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

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

=== Kibble–Slepian formula ===

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

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