Hermite polynomials#Cramér's inequality

Like the other classical orthogonal polynomials, the Hermite polynomials can be defined from several different starting points. Noting from the outset that there are two different standardizations in common use, one convenient method is as follows:

• The "probabilist's Hermite polynomials" are given by $$\backslash operatorname\{He\}\_n(x)\; =\; (-1)^n\; e^\{\backslash frac\{x^2\}\{2\}\}\backslash frac\{d^n\}\{dx^n\}e^\{-\backslash frac\{x^2\}\{2\}\},$$
• while the "physicist's Hermite polynomials" are given by $$H\_n(x)\; =\; (-1)^n\; e^\{x^2\}\backslash frac\{d^n\}\{dx^n\}e^\{-x^2\}.$$

These equations have the form of a Rodrigues' formula and can also be written as,

$$\backslash operatorname\{He\}\_n(x)\; =\; \backslash left(x\; -\; \backslash frac\{d\}\{dx\}\; \backslash right)^n\; \backslash cdot\; 1,\; \backslash quad\; H\_n(x)\; =\; \backslash left(2x\; -\; \backslash frac\{d\}\{dx\}\; \backslash right)^n\; \backslash cdot\; 1.$$

The two definitions are not exactly identical; each is a rescaling of the other:

$$H\_n(x)=2^\backslash frac\{n\}\{2\}\; \backslash operatorname\{He\}\_n\backslash left(\backslash sqrt\{2\}\; \backslash ,x\backslash right),\; \backslash quad\; \backslash operatorname\{He\}\_n(x)=2^\{-\backslash frac\{n\}\{2\}\}\; H\_n\backslash left(\backslash frac\; \{x\}\{\backslash sqrt\; 2\}\; \backslash right).$$

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.{{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.

The polynomials {{mvar|He_n}} are sometimes denoted by {{mvar|H_n}}, especially in probability theory, because

$$\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}e^\{-\backslash frac\{x^2\}\{2\}\}$$

is the probability density function for the normal distribution with expected value 0 and standard deviation 1.

• The first eleven probabilist's Hermite polynomials are: $$\backslash begin\{align\}$$

\operatorname{He}_0(x) &= 1, \\

\operatorname{He}_1(x) &= x, \\

\operatorname{He}_2(x) &= x^2 - 1, \\

\operatorname{He}_3(x) &= x^3 - 3x, \\

\operatorname{He}_4(x) &= x^4 - 6x^2 + 3, \\

\operatorname{He}_5(x) &= x^5 - 10x^3 + 15x, \\

\operatorname{He}_6(x) &= x^6 - 15x^4 + 45x^2 - 15, \\

\operatorname{He}_7(x) &= x^7 - 21x^5 + 105x^3 - 105x, \\

\operatorname{He}_8(x) &= x^8 - 28x^6 + 210x^4 - 420x^2 + 105, \\

\operatorname{He}_9(x) &= x^9 - 36x^7 + 378x^5 - 1260x^3 + 945x, \\

\operatorname{He}_{10}(x) &= x^{10} - 45x^8 + 630x^6 - 3150x^4 + 4725x^2 - 945.

\end{align}

• The first eleven physicist's Hermite polynomials are: $$\backslash begin\{align\}$$

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}

File:Hermite poly.svg|The first six probabilist's Hermite polynomials $\backslash operatorname\{He\}\_n(x)$

File:Hermite poly phys.svg|The first six physicist's Hermite polynomials $H\_n(x)$

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

From the Rodrigues formulae given above, we can see that {{math|H_n(x)}} and {{math|He_n(x)}} are even or odd functions depending on {{mvar|n}}:

$$H\_n(-x)=(-1)^nH\_n(x),\backslash quad\; \backslash operatorname\{He\}\_n(-x)=(-1)^n\backslash operatorname\{He\}\_n(x).$$

{{math|H_n(x)}} and {{math|He_n(x)}} are {{mvar|n}}th-degree polynomials for {{math|n {{=}} 0, 1, 2, 3,...}}. These polynomials are orthogonal with respect to the weight function (measure)

$$w(x)\; =\; e^\{-\backslash frac\{x^2\}\{2\}\}\; \backslash quad\; (\backslash text\{for\; \}\backslash operatorname\{He\})$$ or $$w(x)\; =\; e^\{-x^2\}\; \backslash quad\; (\backslash text\{for\; \}\; H),$$

i.e., we have

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; H\_m(x)\; H\_n(x)\backslash ,\; w(x)\; \backslash ,dx\; =\; 0\; \backslash quad\; \backslash text\{for\; all\; \}m\; \backslash neq\; n.$$

Furthermore,

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; H\_m(x)\; H\_n(x)\backslash ,\; e^\{-x^2\}\; \backslash ,dx\; =\; \backslash sqrt\{\backslash pi\}\backslash ,\; 2^n\; n!\backslash ,\; \backslash delta\_\{nm\},$$

and

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash operatorname\{He\}\_m(x)\; \backslash operatorname\{He\}\_n(x)\backslash ,\; e^\{-\backslash frac\{x^2\}\{2\}\}\; \backslash ,dx\; =\; \backslash sqrt\{2\; \backslash pi\}\backslash ,\; n!\backslash ,\; \backslash delta\_\{nm\},$$

where $\backslash delta\_\{nm\}$

is the Kronecker delta.

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

The Hermite polynomials (probabilist's or physicist's) form an orthogonal basis of the Hilbert space of functions satisfying

in which the inner product is given by the integral

$$\backslash langle\; f,g\backslash rangle\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)\; \backslash overline\{g(x)\}\backslash ,\; w(x)\; \backslash ,dx$$

including the Gaussian weight function {{math|w(x)}} defined in the preceding section.

An orthogonal basis for {{math|L²(R, w(x) dx)}} is a complete orthogonal system. For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function {{math|f ∈ L²(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

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)\; x^n\; e^\{-\; x^2\}\; \backslash ,dx\; =\; 0$$

for every {{math|n ≥ 0}}, then {{math|1=f = 0}}.

One possible way to do this is to appreciate that the entire function

$$F(z)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)\; e^\{z\; x\; -\; x^2\}\; \backslash ,dx\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{z^n\}\{n!\}\; \backslash int\; f(x)\; x^n\; e^\{-\; x^2\}\; \backslash ,dx\; =\; 0$$

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^−x2}} is 0, hence {{mvar|f}} is 0 almost everywhere. Variants of the above completeness proof apply to other weights with exponential decay.

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

An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for {{math|L²(R, w(x) dx)}} consists in introducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis for {{math|L²(R)}}.

The probabilist's Hermite polynomials are solutions of the differential equation

$$\backslash left(e^\{-\backslash frac12\; x^2\}u\text{'}\backslash right)\text{'}\; +\; \backslash lambda\; e^\{-\backslash frac\; 1\; 2\; x^2\}u\; =\; 0,$$

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 $u(x)\; =\; C\_1\; \backslash operatorname\{He\}\_\backslash lambda(x)$, where $C\_\{1\}$ denotes a constant.

Rewriting the differential equation as an eigenvalue problem

$$L[u]\; =\; u\text{'}\text{'}\; -\; x\; u\text{'}\; =\; -\backslash lambda\; u,$$

the Hermite polynomials $\backslash operatorname\{He\}\_\backslash lambda(x)$ may be understood as eigenfunctions of the differential operator $L[u]$ . This eigenvalue problem is called the Hermite equation, although the term is also used for the closely related equation

$$u\text{'}\text{'}\; -\; 2xu\text{'}\; =\; -2\backslash lambda\; u.$$

whose solution is uniquely given in terms of physicist's Hermite polynomials in the form $u(x)\; =\; C\_1\; H\_\backslash lambda(x)$, where $C\_\{1\}$ 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

$$u\text{'}\text{'}\; -\; 2xu\text{'}\; +\; 2\backslash lambda\; u\; =\; 0,$$

the general solution takes the form

$$u(x)\; =\; C\_1\; H\_\backslash lambda(x)\; +\; C\_2\; h\_\backslash lambda(x),$$

where $C\_\{1\}$ and $C\_\{2\}$ are constants, $H\_\backslash lambda(x)$ are physicist's Hermite polynomials (of the first kind), and $h\_\backslash lambda(x)$ are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as $h\_\backslash lambda(x)\; =\; \{\}\_1F\_1(-\backslash tfrac\{\backslash lambda\}\{2\};\backslash tfrac\{1\}\{2\};x^2)$ where $\{\}\_1F\_1(a;b;z)$ are Confluent hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions, see below.

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

The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation

$$\backslash operatorname\{He\}\_\{n+1\}(x)\; =\; x\; \backslash operatorname\{He\}\_n(x)\; -\; \backslash operatorname\{He\}\_n\text{'}(x).$$

Individual coefficients are related by the following recursion formula:

$$a\_\{n+1,k\}\; =\; \backslash begin\{cases\}$$

- (k+1) a_{n,k+1} & k = 0, \\

a_{n,k-1} - (k+1) a_{n,k+1} & k > 0,

\end{cases}

and {{math|1=a_0,0 = 1}}, {{math|1=a_1,0 = 0}}, {{math|1=a_1,1 = 1}}.

For the physicist's polynomials, assuming

$$H\_n(x)\; =\; \backslash sum^n\_\{k=0\}\; a\_\{n,k\}\; x^k,$$

we have

$$H\_\{n+1\}(x)\; =\; 2xH\_n(x)\; -\; H\_n\text{'}(x).$$

Individual coefficients are related by the following recursion formula:

$$a\_\{n+1,k\}\; =\; \backslash begin\{cases\}$$

- a_{n,k+1} & k = 0, \\

2 a_{n,k-1} - (k+1)a_{n,k+1} & k > 0,

\end{cases}

and {{math|1=a_0,0 = 1}}, {{math|1=a_1,0 = 0}}, {{math|1=a_1,1 = 2}}.

The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity

$$\backslash begin\{align\}$$

\operatorname{He}_n'(x) &= n\operatorname{He}_{n-1}(x), \\

H_n'(x) &= 2nH_{n-1}(x).

\end{align}

An integral recurrence that is deduced and demonstrated in 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. is as follows:

$$\backslash operatorname\{He\}\_\{n+1\}(x)\; =\; (n+1)\backslash int\_0^x\; \backslash operatorname\{He\}\_n(t)dt\; -\; He\text{'}\_n(0),$$

$$H\_\{n+1\}(x)\; =\; 2(n+1)\backslash int\_0^x\; H\_n(t)dt\; -\; H\text{'}\_n(0).$$

Equivalently, by Taylor-expanding,

$$\backslash begin\{align\}$$

\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}

These umbral identities are self-evident and included in the differential operator representation detailed below,

$$\backslash begin\{align\}$$

\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}

In consequence, for the {{mvar|m}}th derivatives the following relations hold:

$$\backslash begin\{align\}$$

\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}

It follows that the Hermite polynomials also satisfy the recurrence relation

$$\backslash begin\{align\}$$

\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}

These last relations, together with the initial polynomials {{math|H₀(x)}} and {{math|H₁(x)}}, can be used in practice to compute the polynomials quickly.

Turán's inequalities are

$$\backslash mathit\{H\}\_n(x)^2\; -\; \backslash mathit\{H\}\_\{n-1\}(x)\; \backslash mathit\{H\}\_\{n+1\}(x)\; =\; (n-1)!\; \backslash sum\_\{i=0\}^\{n-1\}\; \backslash frac\{2^\{n-i\}\}\{i!\}\backslash mathit\{H\}\_i(x)^2\; >\; 0.$$

Moreover, the following multiplication theorem holds:

$$\backslash begin\{align\}$$

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}

The physicist's Hermite polynomials can be written explicitly as

$$H\_n(x)\; =\; \backslash begin\{cases\}$$

\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 the floor function:

$$H\_n(x)\; =\; n!\; \backslash sum\_\{m=0\}^\{\backslash left\backslash lfloor\; \backslash tfrac\{n\}\{2\}\; \backslash right\backslash rfloor\}\; \backslash frac\{(-1)^m\}\{m!(n\; -\; 2m)!\}\; (2x)^\{n\; -\; 2m\}.$$

The probabilist's Hermite polynomials {{mvar|He}} have similar formulas, which may be obtained from these by replacing the power of {{math|2x}} with the corresponding power of {{math|{{sqrt|2}} x}} and multiplying the entire sum by {{math|2^{−{{sfrac|n|2}}}}}:

$$\backslash operatorname\{He\}\_n(x)\; =\; n!\; \backslash sum\_\{m=0\}^\{\backslash left\backslash lfloor\; \backslash tfrac\{n\}\{2\}\; \backslash right\backslash rfloor\}\; \backslash frac\{(-1)^m\}\{m!(n\; -\; 2m)!\}\; \backslash frac\{x^\{n\; -\; 2m\}\}\{2^m\}.$$

The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials {{mvar|He}} are

$$x^n\; =\; n!\; \backslash sum\_\{m=0\}^\{\backslash left\backslash lfloor\; \backslash tfrac\{n\}\{2\}\; \backslash right\backslash rfloor\}\; \backslash frac\{1\}\{2^m\; m!(n-2m)!\}\; \backslash operatorname\{He\}\_\{n-2m\}(x).$$

The corresponding expressions for the physicist's Hermite polynomials {{mvar|H}} follow directly by properly scaling this:{{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}}

$$x^n\; =\; \backslash frac\{n!\}\{2^n\}\; \backslash sum\_\{m=0\}^\{\backslash left\backslash lfloor\; \backslash tfrac\{n\}\{2\}\; \backslash right\backslash rfloor\}\; \backslash frac\{1\}\{m!(n-2m)!\; \}\; H\_\{n-2m\}(x).$$

The Hermite polynomials are given by the exponential generating function

$$\backslash begin\{align\}$$

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}

This equality is valid for all complex values of {{mvar|x}} and {{mvar|t}}, and can be obtained by writing the Taylor expansion at {{mvar|x}} of the entire function {{math|z → e^−z2}} (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

$$H\_n(x)\; =\; (-1)^n\; e^\{x^2\}\; \backslash frac\{d^n\}\{dx^n\}\; e^\{-x^2\}\; =\; (-1)^n\; e^\{x^2\}\; \backslash frac\{n!\}\{2\backslash pi\; i\}\; \backslash oint\_\backslash gamma\; \backslash frac\{e^\{-z^2\}\}\{(z-x)^\{n+1\}\}\; \backslash ,dz.$$

Using this in the sum

$$\backslash sum\_\{n=0\}^\backslash infty\; H\_n(x)\; \backslash frac\; \{t^n\}\{n!\},$$

one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.

A slight generalization states(Rainville 1971), p. 198$$e^\{2\; x\; t-t^2\}\; H\_k(x-t)\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{H\_\{n+k\}(x)\; t^n\}\{n!\}$$

If {{mvar|X}} is a random variable with a normal distribution with standard deviation 1 and expected value {{mvar|μ}}, then

$$\backslash operatorname\{\backslash mathbb\; E\}\backslash left[\backslash operatorname\{He\}\_n(X)\backslash right]\; =\; \backslash mu^n.$$

The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices:

$$\backslash operatorname\{\backslash mathbb\; E\}\backslash left[X^\{2n\}\backslash right]\; =\; (-1)^n\; \backslash operatorname\{He\}\_\{2n\}(0)\; =\; (2n-1)!!,$$

where {{math|(2n − 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:

$$\backslash operatorname\{He\}\_n(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\backslash infty\; (x\; +\; iy)^n\; e^\{-\backslash frac\{y^2\}\{2\}\}\; \backslash ,dy.$$

From the generating-function representation above, we see that the Hermite polynomials have a representation in terms of a contour integral, as

$$\backslash begin\{align\}$$

\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}

with the contour encircling the origin.

Using the Fourier transform of the gaussian $e^\{-x^2\}=\backslash frac\{1\}\{\backslash sqrt\{\; \backslash pi\}\}\; \backslash int\; e^\{-t^2+2\; i\; x\; t\}\; dt$, we have$$\backslash begin\{align\}$$

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}

The addition theorem, or the summation theorem, states that{{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}}{{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}}{{Pg|location=8.958}}$$\backslash frac\{\backslash left(\backslash sum\_\{k=1\}^r\; a\_k^2\backslash right)^\{\backslash frac\{n\}\{2\}\}\}\{n!\}\; H\_n\backslash left(\backslash frac\{\backslash sum\_\{k=1\}^r\; a\_k\; x\_k\}\{\backslash sqrt\{\backslash sum\_\{k=1\}^r\; a\_k^2\}\}\backslash right)=\backslash sum\_\{m\_1+m\_2+\backslash ldots+m\_r=n,\; m\_i\; \backslash geq\; 0\}\; \backslash prod\_\{k=1\}^r\backslash left\backslash \{\backslash frac\{a\_k^\{m\_k\}\}\{m\_\{k\}!\}\; H\_\{m\_k\}\backslash left(x\_k\backslash right)\backslash right\backslash \}$$for any nonzero vector $a\_\{1:r\}$.

The multiplication theorem states that$$H\_\{n\}\backslash left(\backslash lambda\; x\backslash right)=\backslash lambda^\{n\}\backslash sum\_\{\backslash ell=0\}^\{\backslash left\backslash lfloor\; n/2\backslash right\backslash rfloor\}\backslash frac\{\{\backslash left(-n\backslash right)\_\{2\backslash ell\}\}\}\{\backslash ell!\}(1-\backslash lambda^\{-2\})^\{\backslash ell\}H\_\{n-2\backslash ell\}\backslash left(x\backslash right)$$for any nonzero $\backslash lambda$.

Feldheim formulaFeldheim, 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]{{Pg|location=Eq 46}}$$\backslash begin\{aligned\}$$

\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}where $a\; \backslash in\; \backslash mathbb\; C$ has a positive real part. As a special case,{{Pg|location=Eq 52}}$$\backslash frac\{1\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; e^\{-t^2\}\; H\_m(t\; \backslash sin\; \backslash theta+v\; \backslash cos\; \backslash theta)\; H\_n(t\; \backslash cos\; \backslash theta-v\; \backslash sin\; \backslash theta)\; d\; t\; =(-1)^n\; \backslash cos\; ^m\; \backslash theta\; \backslash sin\; ^n\; \backslash theta\; H\_\{m+n\}(v)$$

Asymptotically, as {{math|n → ∞}}, the expansion{{harvnb|Abramowitz|Stegun|1983|page=508–510}}, [http://www.math.sfu.ca/~cbm/aands/page_508.htm 13.6.38 and 13.5.16].

$$e^\{-\backslash frac\{x^2\}\{2\}\}\backslash cdot\; H\_n(x)\; \backslash sim\; \backslash frac\{2^n\}\{\backslash sqrt\; \backslash pi\}\backslash Gamma\backslash left(\backslash frac\{n+1\}2\backslash right)\; \backslash cos\; \backslash left(x\; \backslash sqrt\{2\; n\}-\; \backslash frac\{n\backslash pi\}\{2\}\; \backslash right)$$

holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude:

$$e^\{-\backslash frac\{x^2\}\{2\}\}\backslash cdot\; H\_n(x)\; \backslash sim\; \backslash frac\{2^n\}\{\backslash sqrt\; \backslash pi\}\backslash Gamma\backslash left(\backslash frac\{n+1\}2\backslash right)\; \backslash cos\; \backslash left(x\; \backslash sqrt\{2\; n\}-\; \backslash frac\{n\backslash pi\}\{2\}\; \backslash right)\backslash left(1-\backslash frac\{x^2\}\{2n+1\}\backslash right)^\{-\backslash frac14\}=\backslash frac\{\backslash Gamma(n)\}\{\backslash Gamma\backslash left(\backslash frac\{n\}2\backslash right)\}\; \backslash cos\; \backslash left(x\; \backslash sqrt\{2\; n\}-\; \backslash frac\{n\backslash pi\}\{2\}\; \backslash right)\backslash left(1-\backslash frac\{x^2\}\{2n+1\}\backslash right)^\{-\backslash frac14\},$$

which, using Stirling's approximation, can be further simplified, in the limit, to

$$e^\{-\backslash frac\{x^2\}\{2\}\}\backslash cdot\; H\_n(x)\; \backslash sim\; \backslash left(\backslash frac\{2n\}\{e\}\backslash right)^\{\backslash frac\{n\}\{2\}\}\; \backslash sqrt\{2\}\; \backslash cos\; \backslash left(x\; \backslash sqrt\{2n\}-\; \backslash frac\{n\backslash pi\}\{2\}\; \backslash right)\backslash left(1-\backslash frac\{x^2\}\{2n+1\}\backslash right)^\{-\backslash frac14\}.$$

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

$$e^\{-\backslash frac\{x^2\}\{2\}\}\backslash cdot\; H\_n(x)\; \backslash sim\; \backslash left(\backslash frac\{2n\}\{e\}\backslash right)^\{\backslash frac\{n\}\{2\}\}\; \backslash sqrt\{2\}\; \backslash cos\; \backslash left(x\; \backslash sqrt\{2n+1-\backslash frac\{x^2\}\{3\}\}-\; \backslash frac\; \{n\backslash pi\}\{2\}\; \backslash right)\backslash left(1-\backslash frac\{x^2\}\{2n+1\}\backslash right)^\{-\backslash frac14\}.$$

A finer approximation,{{harvnb|Szegő|1955|p=201}} which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution

with which one has the uniform approximation

$$e^\{-\backslash frac\{x^2\}\{2\}\}\backslash cdot\; H\_n(x)\; =\; 2^\{\backslash frac\{n\}\{2\}+\backslash frac14\}\backslash sqrt\{n!\}(\backslash pi\; n)^\{-\backslash frac14\}(\backslash sin\; \backslash varphi)^\{-\backslash frac12\}\; \backslash cdot\; \backslash left(\backslash sin\backslash left(\backslash frac\{3\backslash pi\}\{4\}\; +\; \backslash left(\backslash frac\{n\}\{2\}\; +\; \backslash frac\{1\}\{4\}\backslash right)\backslash left(\backslash sin\; 2\backslash varphi-2\backslash varphi\backslash right)\; \backslash right)+O\backslash left(n^\{-1\}\backslash right)\; \backslash right).$$

Similar approximations hold for the monotonic and transition regions. Specifically, if

then

$$e^\{-\backslash frac\{x^2\}\{2\}\}\backslash cdot\; H\_n(x)\; =\; 2^\{\backslash frac\{n\}\{2\}-\backslash frac34\}\backslash sqrt\{n!\}(\backslash pi\; n)^\{-\backslash frac14\}(\backslash sinh\; \backslash varphi)^\{-\backslash frac12\}\; \backslash cdot\; e^\{\backslash left(\backslash frac\{n\}\{2\}+\backslash frac\{1\}\{4\}\backslash right)\backslash left(2\backslash varphi-\backslash sinh\; 2\backslash varphi\backslash right)\}\backslash left(1+O\backslash left(n^\{-1\}\backslash right)\; \backslash right),$$

while for $$x\; =\; \backslash sqrt\{2n\; +\; 1\}\; +\; t$$ with {{mvar|t}} complex and bounded, the approximation is

$$e^\{-\backslash frac\{x^2\}\{2\}\}\backslash cdot\; H\_n(x)\; =\backslash pi^\{\backslash frac14\}2^\{\backslash frac\{n\}\{2\}+\backslash frac14\}\backslash sqrt\{n!\}\backslash ,\; n^\{-\backslash frac\{1\}\{12\}\}\backslash left(\; \backslash operatorname\{Ai\}\backslash left(2^\{\backslash frac12\}n^\{\backslash frac16\}t\backslash right)+\; O\backslash left(n^\{-\backslash frac23\}\backslash right)\; \backslash right),$$

where {{math|Ai}} is the Airy function of the first kind.

The physicist's Hermite polynomials evaluated at zero argument {{math|H_n(0)}} are called Hermite numbers.

$$H\_n(0)\; =\; \backslash begin\{cases\}$$

0 & \text{for odd }n, \\

(-2)^\frac{n}{2} (n-1)!! & \text{for even }n,

\end{cases}

which satisfy the recursion relation {{math|1=H_n(0) = −2(n − 1)H_{n − 2}(0)}}. Equivalently, $H\_\{2n\}(0)\; =\; (-2)^n\; (2n-1)!!$.

In terms of the probabilist's polynomials this translates to

$$\backslash operatorname\{He\}\_n(0)\; =\; \backslash begin\{cases\}$$

0 & \text{for odd }n, \\

(-1)^\frac{n}{2} (n-1)!! & \text{for even }n.

\end{cases}

Let $M$ be a real $n\backslash times\; n$ symmetric matrix, then the Kibble–Slepian formula states that$$\backslash det(I+M)^\{-\backslash frac\; 12\; \}\; e^\{x^T\; M\; (I+M)^\{-1\}x\}\; =\; \backslash sum\_K\; \backslash left[\backslash prod\_\{1\backslash leq\; i\; \backslash leq\; j\; \backslash leq\; n\}\; \backslash frac\{(M\_\{ij\}/2)^\{k\_\{ij\}\}\}\{k\_\{ij\}!\}\backslash right]\; 2^\{-tr(K)\}\; H\_\{k\_1\}(x\_1)\; \backslash cdots\; H\_\{k\_n\}(x\_n)$$ where $\backslash sum\_K$ is the $\backslash frac\{n(n+1)\}\{2\}$-fold summation over all $n\; \backslash times\; n$ symmetric matrices with non-negative integer entries, $tr(K)$ is the trace of $K$, and $k\_i$ is defined as $k\_\{ii\}\; +\; \backslash sum\_\{j=1\}^n\; k\_\{ij\}$. This gives Mehler's formula when .

Equivalently stated, if $T$ is a positive semidefinite matrix, then set $M\; =\; -T(I+T)^\{-1\}$, we have $M(I+M)^\{-1\}\; =\; -T$, so $$$$

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)

Equivalently stated in a form closer to the boson quantum mechanics of the harmonic oscillator:{{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}}$$$$

\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) .

where each $\backslash psi\_n(x)$ is the $n$-th eigenfunction of the harmonic oscillator, defined as $$\backslash psi\_n(x)\; :=\; \backslash frac\{1\}\{\backslash sqrt\{2^n\; n!\}\}\backslash left(\backslash frac\{1\}\{\backslash pi\}\backslash right)^\{\backslash frac\{1\}\{4\}\}\; e^\{-\backslash frac\{1\}\{2\}\; x^2\}\; H\_n(x)$$The Kibble–Slepian formula was proposed by Kibble in 1945{{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}} and proven by Slepian in 1972 using Fourier analysis.{{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}} Foata gave a combinatorial proof{{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}} while Louck gave a proof via boson quantum mechanics. It has a generalization for complex-argument Hermite polynomials.{{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 }}{{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 }}

The Hermite polynomials can be expressed as a special case of the Laguerre polynomials:

$$\backslash begin\{align\}$$

H_{2n}(x) &= (-4)^n n! L_n^{\left(-\frac12\right)}(x^2)

&&= 4^n n! \sum_{k=0}^n (-1)^{n-k} \binom{n-\frac12}{n-k} \frac{x^{2k}}{k!}, \\

H_{2n+1}(x) &= 2(-4)^n n! x L_n^{\left(\frac12\right)}(x^2)

&&= 2\cdot 4^n n!\sum_{k=0}^n (-1)^{n-k} \binom{n+\frac12}{n-k} \frac{x^{2k+1}}{k!}.

\end{align}

The physicist's Hermite polynomials can be expressed as a special case of the parabolic cylinder functions:

$$H\_n(x)\; =\; 2^n\; U\backslash left(-\backslash tfrac12\; n,\; \backslash tfrac12,\; x^2\backslash right)$$

in the right half-plane, where {{math|U(a, b, z)}} is Tricomi's confluent hypergeometric function. Similarly,

$$\backslash begin\{align\}$$

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}

where {{math|₁F₁(a, b; z) {{=}} M(a, b; z)}} is Kummer's confluent hypergeometric function.

There is also[https://dlmf.nist.gov/18.5#E13 DLMF Equation 18.5.13]$$H\_\{n\}\backslash left(x\backslash right)=(2x)^\{n\}\{\{\}\_\{2\}F\_\{0\}\}\backslash left(\{-\backslash tfrac\{1\}\{2\}n,-\backslash tfrac\{1\}\{2\}n+\backslash tfrac\{1\}\{2\}\backslash atop-\};-\backslash frac\{1\}\{x^\{2\}\}\backslash right).$$

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

As a limit of Jacobi polynomials:$$\backslash lim\_\{\backslash alpha\backslash to\backslash infty\}\backslash alpha^\{-\backslash frac\{1\}\{2\}n\}P^\{(\backslash alpha,\backslash alpha)\}\_\{n\}\backslash left(\backslash alpha^\{-\backslash frac\{1\}\{2\}\}x\backslash right)=\backslash frac\{H\_\{n\}\backslash left(x\backslash right)\}\{2^\{n\}n!\}.$$As a limit of ultraspherical polynomials:$$\backslash lim\_\{\backslash lambda\backslash to\backslash infty\}\backslash lambda^\{-\backslash frac\{1\}\{2\}n\}C^\{(\backslash lambda)\}\_\{n\}\backslash left(\backslash lambda^\{-\backslash frac\{1\}\{2\}\}x\backslash right)=\backslash frac\{H\_\{n\}\backslash left(x\backslash right)\}\{n!\}.$$As a limit of associated Laguerre polynomials:$$\backslash lim\_\{\backslash alpha\backslash to\backslash infty\}\backslash left(\backslash frac\{2\}\{\backslash alpha\}\backslash right)^\{\backslash frac\{1\}\{2\}n\}L^\{(\backslash alpha)\}\_\{n\}\backslash left((2\backslash alpha)^\{\backslash frac\{1\}\{2\}\}x+\backslash alpha\backslash right)=\backslash frac\{(-1)^\{n\}\}\{n!\}H\_\{n\}\backslash left(x\backslash right).$$

Similar to Taylor expansion, some functions are expressible as an infinite sum of Hermite polynomials. Specifically, if , then it has an expansion in the physicist's Hermite polynomials.{{Cite web |title=MATHEMATICA tutorial, part 2.5: Hermite expansion |url=https://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch5/hermite.html |access-date=2023-12-24 |website=www.cfm.brown.edu}}

Given such $f$, the partial sums of the Hermite expansion of $f$ converges to in the $L^p$ norm if and only if

$$e^\{ax\}\; =\; e^\{a^2\; /4\}\; \backslash sum\_\{n\backslash ge\; 0\}\; \backslash frac\{a^n\}\{n!\; \backslash ,2^n\}\; \backslash ,\; H\_n\; (x)\; ,\; \backslash qquad\; a\backslash in\; \backslash mathbb\{C\},\; \backslash quad\; x\backslash in\; \backslash mathbb\{R\}\; .$$$$e^\{-a^2\; x^2\}\; =\; \backslash sum\_\{n\backslash ge\; 0\}\; \backslash frac\{(-1)^n\; a^\{2n\}\}\{n!\; \backslash left(\; 1\; +\; a^2\; \backslash right)^\{n\; +\; 1/2\}\; 2^\{2n\}\}\backslash ,\; H\_\{2n\}\; (x)\; .$$$$\backslash operatorname\{erf\}(x)=\backslash frac\{2\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash int\_0^x\; e^\{-t^2\}\; ~dt=\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash sum\_\{k\; \backslash geq\; 0\}\; \backslash frac\{(-1)^k\}\{k\; !(2\; k+1)\; 2^\{3\; k\}\}\; H\_\{2\; k\}(x)\; .$$$$\backslash cosh\; (2x)\; =\; e\; \backslash sum\_\{k\backslash ge\; 0\}\; \backslash frac\{1\}\{(2k)!\}\backslash ,\; H\_\{2k\}\; (x)\; ,\; \backslash qquad\; \backslash sinh\; (2x)\; =\; e\; \backslash sum\_\{k\backslash ge\; 0\}\; \backslash frac\{1\}\{(2k+1)!\}\; \backslash ,\; H\_\{2k+1\}\; (x)\; .$$$$\backslash cos\; (x)\; =\; e^\{-1/4\}\; \backslash ,\backslash sum\_\{k\backslash ge\; 0\}\; \backslash frac\{(-1)^k\}\{2^\{2k\}\; \backslash ,\; (2k)!\}\; \backslash ,\; H\_\{2k\}\; (x)$$

\quad \sin (x) = e^{-1/4} \,\sum_{k\ge 0} \frac{(-1)^k}{2^{2k+1} \, (2k+1)!} \, H_{2k+1} (x)

The probabilist's Hermite polynomials satisfy the identity{{cite book |last1=Rota |first1=Gian-Carlo |last2=Doubilet |first2=P. |title=Finite operator calculus |date=1975 |publisher=Academic Press |location=New York |isbn=9780125966504 |page=44}} $$\backslash operatorname\{He\}\_n(x)\; =\; e^\{-\backslash frac\{D^2\}\{2\}\}x^n,$$ where {{mvar|D}} represents differentiation with respect to {{mvar|x}}, and the exponential is interpreted by expanding it as a power series. There are no delicate questions of convergence of this series when it operates on polynomials, since all but finitely many terms vanish.

Since the power-series coefficients of the exponential are well known, and higher-order derivatives of the monomial {{math|xⁿ}} can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients of {{math|H_n}} that can be used to quickly compute these polynomials.

Since the formal expression for the Weierstrass transform {{mvar|W}} is {{math|e^D2}}, we see that the Weierstrass transform of {{math|({{sqrt|2}})ⁿHe_n({{sfrac|x|{{sqrt|2}}}})}} is {{math|xⁿ}}. Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series.

The existence of some formal power series {{math|g(D)}} with nonzero constant coefficient, such that {{math|1=He_n(x) = g(D)xⁿ}}, is another equivalent to the statement that these polynomials form an Appell sequence. Since they are an Appell sequence, they are a fortiori a Sheffer sequence.

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

The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is

$$\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; e^\{-\backslash frac\{x^2\}\{2\}\},$$

which has expected value 0 and variance 1.

Scaling, one may analogously speak of generalized Hermite polynomials{{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}}

$$\backslash operatorname\{He\}\_n^\{[\backslash alpha]\}(x)$$

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

$$\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\; \backslash alpha\}\}\; e^\{-\backslash frac\{x^2\}\{2\backslash alpha\}\}.$$

They are given by

$$\backslash operatorname\{He\}\_n^\{[\backslash alpha]\}(x)\; =\; \backslash alpha^\{\backslash frac\{n\}\{2\}\}\backslash operatorname\{He\}\_n\backslash left(\backslash frac\{x\}\{\backslash sqrt\{\backslash alpha\}\}\backslash right)\; =\; \backslash left(\backslash frac\{\backslash alpha\}\{2\}\backslash right)^\{\backslash frac\{n\}\{2\}\}\; H\_n\backslash left(\; \backslash frac\{x\}\{\backslash sqrt\{2\; \backslash alpha\}\}\backslash right)\; =\; e^\{-\backslash frac\{\backslash alpha\; D^2\}\{2\}\}\; \backslash left(x^n\backslash right).$$

Now, if

$$\backslash operatorname\{He\}\_n^\{[\backslash alpha]\}(x)\; =\; \backslash sum\_\{k=0\}^n\; h^\{[\backslash alpha]\}\_\{n,k\}\; x^k,$$

then the polynomial sequence whose {{mvar|n}}th term is

$$\backslash left(\backslash operatorname\{He\}\_n^\{[\backslash alpha]\}\; \backslash circ\; \backslash operatorname\{He\}^\{[\backslash beta]\}\backslash right)(x)\; \backslash equiv\; \backslash sum\_\{k=0\}^n\; h^\{[\backslash alpha]\}\_\{n,k\}\backslash ,\backslash operatorname\{He\}\_k^\{[\backslash beta]\}(x)$$

is called the umbral composition of the two polynomial sequences. It can be shown to satisfy the identities

$$\backslash left(\backslash operatorname\{He\}\_n^\{[\backslash alpha]\}\; \backslash circ\; \backslash operatorname\{He\}^\{[\backslash beta]\}\backslash right)(x)\; =\; \backslash operatorname\{He\}\_n^\{[\backslash alpha+\backslash beta]\}(x)$$

and

$$\backslash operatorname\{He\}\_n^\{[\backslash alpha+\backslash beta]\}(x\; +\; y)\; =\; \backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; \backslash operatorname\{He\}\_k^\{[\backslash alpha]\}(x)\; \backslash operatorname\{He\}\_\{n-k\}^\{[\backslash beta]\}(y).$$

The last identity is expressed by saying that this parameterized family of polynomial sequences is known as a cross-sequence. (See the above section on Appell sequences and on the differential-operator representation, which leads to a ready derivation of it. This binomial type identity, for {{math|1=α = β = {{sfrac|1|2}}}}, has already been encountered in the above section on #Recursion relations.)

Since polynomial sequences form a group under the operation of umbral composition, one may denote by

$$\backslash operatorname\{He\}\_n^\{[-\backslash alpha]\}(x)$$

the sequence that is inverse to the one similarly denoted, but without the minus sign, and thus speak of Hermite polynomials of negative variance. For {{math|α > 0}}, the coefficients of $\backslash operatorname\{He\}\_n^\{[-\backslash alpha]\}(x)$ are just the absolute values of the corresponding coefficients of $\backslash operatorname\{He\}\_n^\{[\backslash alpha]\}(x)$.

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|σ²}} is

$$E[X^n]\; =\; \backslash operatorname\{He\}\_n^\{[-\backslash sigma^2]\}(\backslash mu),$$

where {{mvar|X}} is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that

$$\backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; \backslash operatorname\{He\}\_k^\{[\backslash alpha]\}(x)\; \backslash operatorname\{He\}\_\{n-k\}^\{[-\backslash alpha]\}(y)\; =\; \backslash operatorname\{He\}\_n^\{[0]\}(x\; +\; y)\; =\; (x\; +\; y)^n.$$

One can define the Hermite functions (often called Hermite-Gaussian functions) from the physicist's polynomials:

$$\backslash psi\_n(x)\; =\; \backslash left\; (2^n\; n!\; \backslash sqrt\{\backslash pi\}\; \backslash right\; )^\{-\backslash frac12\}\; e^\{-\backslash frac\{x^2\}\{2\}\}\; H\_n(x)\; =\; (-1)^n\; \backslash left\; (2^n\; n!\; \backslash sqrt\{\backslash pi\}\; \backslash right)^\{-\backslash frac12\}\; e^\{\backslash frac\{x^2\}\{2\}\}\backslash frac\{d^n\}\{dx^n\}\; e^\{-x^2\}.$$

Thus,

$$\backslash sqrt\{2(n+1)\}~~\backslash psi\_\{n+1\}(x)=\; \backslash left\; (\; x-\; \{d\backslash over\; dx\}\backslash right\; )\; \backslash psi\_n(x).$$

Since these functions contain the square root of the weight function and have been scaled appropriately, they are orthonormal:

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash psi\_n(x)\; \backslash psi\_m(x)\; \backslash ,dx\; =\; \backslash delta\_\{nm\},$$

and they form an orthonormal basis of {{math|L²(R)}}. This fact is equivalent to the corresponding statement for Hermite polynomials (see above).

The Hermite functions are closely related to the Whittaker function {{Harv|Whittaker|Watson|1996}} {{math|D_n(z)}}:

$$D\_n(z)\; =\; \backslash left(n!\; \backslash sqrt\{\backslash pi\}\backslash right)^\{\backslash frac12\}\; \backslash psi\_n\backslash left(\backslash frac\{z\}\{\backslash sqrt\; 2\}\backslash right)\; =\; (-1)^n\; e^\backslash frac\{z^2\}\{4\}\; \backslash frac\{d^n\}\{dz^n\}\; e^\backslash frac\{-z^2\}\{2\}$$

and thereby to other parabolic cylinder functions.

The Hermite functions satisfy the differential equation

$$\backslash psi\_n\text{'}\text{'}(x)\; +\; \backslash left(2n\; +\; 1\; -\; x^2\backslash right)\; \backslash psi\_n(x)\; =\; 0.$$

This equation is equivalent to the Schrödinger equation for a harmonic oscillator in quantum mechanics, so these functions are the eigenfunctions.

Image:Herm5.svg

$$\backslash begin\{align\}$$

\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}

Image:Herm50.svg

Following recursion relations of Hermite polynomials, the Hermite functions obey

$$\backslash psi\_n\text{'}(x)\; =\; \backslash sqrt\{\backslash frac\{n\}\{2\}\}\backslash ,\backslash psi\_\{n-1\}(x)\; -\; \backslash sqrt\{\backslash frac\{n+1\}\{2\}\}\backslash psi\_\{n+1\}(x)$$

and

$$x\backslash psi\_n(x)\; =\; \backslash sqrt\{\backslash frac\{n\}\{2\}\}\backslash ,\backslash psi\_\{n-1\}(x)\; +\; \backslash sqrt\{\backslash frac\{n+1\}\{2\}\}\backslash psi\_\{n+1\}(x).$$

Extending the first relation to the arbitrary {{mvar|m}}th derivatives for any positive integer {{mvar|m}} leads to

$$\backslash psi\_n^\{(m)\}(x)\; =\; \backslash sum\_\{k=0\}^m\; \backslash binom\{m\}\{k\}\; (-1)^k\; 2^\backslash frac\{m-k\}\{2\}\; \backslash sqrt\{\backslash frac\{n!\}\{(n-m+k)!\}\}\; \backslash psi\_\{n-m+k\}(x)\; \backslash operatorname\{He\}\_k(x).$$

This formula can be used in connection with the recurrence relations for {{math|He_n}} and {{math|ψ_n}} to calculate any derivative of the Hermite functions efficiently.

For real {{mvar|x}}, the Hermite functions satisfy the following bound due to Harald Cramér{{harvnb|Erdélyi|Magnus|Oberhettinger|Tricomi|1955|p=207}}.{{harvnb|Szegő|1955}}. and Jack 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

}}

$$\backslash bigl|\backslash psi\_n(x)\backslash bigr|\; \backslash le\; \backslash pi^\{-\backslash frac14\}.$$

The Hermite functions {{math|ψ_n(x)}} are a set of eigenfunctions of the continuous Fourier transform {{mathcal|F}}. To see this, take the physicist's version of the generating function and multiply by {{math|e^{−{{sfrac|1|2}}x2}}}. This gives

$$e^\{-\backslash frac12\; x^2\; +\; 2xt\; -\; t^2\}\; =\; \backslash sum\_\{n=0\}^\backslash infty\; e^\{-\backslash frac12\; x^2\}\; H\_n(x)\; \backslash frac\{t^n\}\{n!\}.$$

The Fourier transform of the left side is given by

$$\backslash begin\{align\}$$

\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}

The Fourier transform of the right side is given by

$$\backslash mathcal\{F\}\; \backslash left\backslash \{\; \backslash sum\_\{n=0\}^\backslash infty\; e^\{-\backslash frac12\; x^2\}\; H\_n(x)\; \backslash frac\; \{t^n\}\{n!\}\; \backslash right\backslash \}\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash mathcal\{F\}\; \backslash left\; \backslash \{\; e^\{-\backslash frac12\; x^2\}\; H\_n(x)\; \backslash right\backslash \}\; \backslash frac\{t^n\}\{n!\}.$$

Equating like powers of {{mvar|t}} in the transformed versions of the left and right sides finally yields

$$\backslash mathcal\{F\}\; \backslash left\backslash \{\; e^\{-\backslash frac12\; x^2\}\; H\_n(x)\; \backslash right\backslash \}\; =\; (-i)^n\; e^\{-\backslash frac12\; k^2\}\; H\_n(k).$$

The Hermite functions {{math|ψ_n(x)}} are thus an orthonormal basis of {{math|L²(R)}}, which diagonalizes the Fourier transform operator.In this case, we used the unitary version of the Fourier transform, so the eigenvalues are {{math|(−i)ⁿ}}. The ensuing resolution of the identity then serves to define powers, including fractional ones, of the Fourier transform, to wit a Fractional Fourier transform generalization, in effect a Mehler kernel. In short, we have:$$\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\; e^\{-ikx\}\; \backslash psi\_n(x)\; dx\; =\; (-i)^n\; \backslash psi\_n(k),\; \backslash quad\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\; e^\{+ikx\}\; \backslash psi\_n(k)\; dk\; =\; i^n\; \backslash psi\_n(x)$$

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

$$L\_n(x)\; :=\; \backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; \backslash frac\{(-1)^k\}\{k!\}x^k,$$

leading to the oscillator Laguerre functions

$$l\_n\; (x)\; :=\; e^\{-\backslash frac\{x\}\{2\}\}\; L\_n(x).$$

For all natural integers {{mvar|n}}, it is straightforward to see{{Citation |author-link=Gerald Folland |first=G. B. |last=Folland |title=Harmonic Analysis in Phase Space | series=Annals of Mathematics Studies |volume=122 |publisher=Princeton University Press |date=1989 |isbn=978-0-691-08528-9}} that

$$W\_\{\backslash psi\_n\}(t,f)\; =\; (-1)^n\; l\_n\; \backslash big(4\backslash pi\; (t^2\; +\; f^2)\; \backslash big),$$

where the Wigner distribution of a function {{math|x ∈ L²(R, C)}} is defined as

$$W\_x(t,f)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; x\backslash left(t\; +\; \backslash frac\{\backslash tau\}\{2\}\backslash right)\; \backslash ,\; x\backslash left(t\; -\; \backslash frac\{\backslash tau\}\{2\}\backslash right)^*\; \backslash ,\; e^\{-2\backslash pi\; i\backslash tau\; f\}\; \backslash ,d\backslash tau.$$

This is a fundamental result for the quantum harmonic oscillator discovered by Hip Groenewold in 1946 in his PhD thesis.{{cite journal | last1 = Groenewold | first1 = H. J. | year = 1946 | title = On the Principles of elementary quantum mechanics | journal = Physica | volume = 12 | issue = 7| pages = 405–460 | doi = 10.1016/S0031-8914(46)80059-4 | bibcode=1946Phy....12..405G}} It is the standard paradigm of quantum mechanics in phase space.

There are further relations between the two families of polynomials.

It can be shown{{cite arXiv |last=Mawby|first=Clement|title=Tests of Macrorealism in Discrete and Continuous Variable Systems |date=2024 |class=quant-ph |eprint=2402.16537}}{{cite arXiv |last=Moriconi|first=Marco|title=Nodes of Wavefunctions |date=2007 |eprint=quant-ph/0702260}} that the overlap between two different Hermite functions ($k\backslash neq\; \backslash ell$) over a given interval has the exact result:

$$\backslash int\_\{x\_1\}^\{x\_2\}\backslash psi\_\{k\}(x)\; \backslash psi\_\{\backslash ell\}(x)\backslash ,dx\; =\backslash frac\{1\}\{2(\backslash ell\; -\; k)\}\backslash left(\backslash psi\_k\text{'}(x\_2)\backslash psi\_\backslash ell(x\_2)-\backslash psi\_\backslash ell\text{'}(x\_2)\backslash psi\_k(x\_2)-\backslash psi\_k\text{'}(x\_1)\backslash psi\_\backslash ell(x\_1)+\backslash psi\_\backslash ell\text{'}(x\_1)\backslash psi\_k(x\_1)\backslash right).$$

In the Hermite polynomial {{math|He_n(x)}} of variance 1, the absolute value of the coefficient of {{math|x^k}} is the number of (unordered) partitions of an {{mvar|n}}-element set into {{mvar|k}} singletons and {{math|{{sfrac|n − k|2}}}} (unordered) pairs. Equivalently, it is the number of involutions of an {{mvar|n}}-element set with precisely {{mvar|k}} fixed points, or in other words, the number of matchings in the complete graph on {{mvar|n}} vertices that leave {{mvar|k}} vertices uncovered (indeed, the Hermite polynomials are the matching polynomials of these graphs). The sum of the absolute values of the coefficients gives the total number of partitions into singletons and pairs, the so-called telephone numbers

: 1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496,... {{OEIS|A000085}}.

This combinatorial interpretation can be related to complete exponential Bell polynomials as

$$\backslash operatorname\{He\}\_n(x)\; =\; B\_n(x,\; -1,\; 0,\; \backslash ldots,\; 0),$$

where {{math|1=x_i = 0}} for all {{math|i > 2}}.

These numbers may also be expressed as a special value of the Hermite polynomials:{{citation

| last1 = Banderier | first1 = Cyril

| last2 = Bousquet-Mélou | first2 = Mireille | author2-link = Mireille Bousquet-Mélou

| last3 = Denise | first3 = Alain

| last4 = Flajolet | first4 = Philippe | author4-link = Philippe Flajolet

| last5 = Gardy | first5 = Danièle

| last6 = Gouyou-Beauchamps | first6 = Dominique

| arxiv = math/0411250 | doi = 10.1016/S0012-365X(01)00250-3 | issue = 1–3 | journal = Discrete Mathematics | mr = 1884885 | pages = 29–55 | title = Generating functions for generating trees | volume = 246 | year = 2002| s2cid = 14804110

}}

$$T(n)\; =\; \backslash frac\{\backslash operatorname\{He\}\_n(i)\}\{i^n\}.$$

The Christoffel–Darboux formula for Hermite polynomials reads

$$\backslash sum\_\{k=0\}^n\; \backslash frac\{H\_k(x)\; H\_k(y)\}\{k!2^k\}\; =\; \backslash frac\{1\}\{n!2^\{n+1\}\}\backslash ,\backslash frac\{H\_n(y)\; H\_\{n+1\}(x)\; -\; H\_n(x)\; H\_\{n+1\}(y)\}\{x\; -\; y\}.$$

Moreover, the following completeness identity for the above Hermite functions holds in the sense of distributions:

$$\backslash sum\_\{n=0\}^\backslash infty\; \backslash psi\_n(x)\; \backslash psi\_n(y)\; =\; \backslash delta(x\; -\; y),$$

where {{mvar|δ}} is the Dirac delta function, {{math|ψ_n}} the Hermite functions, and {{math|δ(x − y)}} represents the Lebesgue measure on the line {{math|1=y = x}} in {{math|R²}}, 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}}:

$$E(x,\; y;\; u)\; :=\; \backslash sum\_\{n=0\}^\backslash infty\; u^n\; \backslash ,\; \backslash psi\_n\; (x)\; \backslash ,\; \backslash psi\_n\; (y)\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash pi\; (1\; -\; u^2)\}\}\; \backslash ,\; \backslash exp\backslash left(-\backslash frac\{1\; -\; u\}\{1\; +\; u\}\; \backslash ,\; \backslash frac\{(x\; +\; y)^2\}\{4\}\; -\; \backslash frac\{1\; +\; u\}\{1\; -\; u\}\; \backslash ,\; \backslash frac\{(x\; -\; y)^2\}\{4\}\backslash right),$$

which is often stated equivalently as a separable kernel,{{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).{{harvnb|Erdélyi|Magnus|Oberhettinger|Tricomi|1955|page=194}}, 10.13 (22).

$$\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{H\_n(x)\; H\_n(y)\}\{n!\}\; \backslash left(\backslash frac\; u\; 2\backslash right)^n\; =\; \backslash frac\{1\}\{\backslash sqrt\{1\; -\; u^2\}\}\; e^\{\backslash frac\{2u\}\{1\; +\; u\}xy\; -\; \backslash frac\{u^2\}\{1\; -\; u^2\}(x\; -\; y)^2\}.$$

The function {{math|(x, y) → E(x, y; u)}} is the bivariate Gaussian probability density on {{math|R²}}, which is, when {{mvar|u}} is close to 1, very concentrated around the line {{math|1=y = x}}, and very spread out on that line. It follows that

$$\backslash sum\_\{n=0\}^\backslash infty\; u^n\; \backslash langle\; f,\; \backslash psi\_n\; \backslash rangle\; \backslash langle\; \backslash psi\_n,\; g\; \backslash rangle\; =\; \backslash iint\; E(x,\; y;\; u)\; f(x)\; \backslash overline\{g(y)\}\; \backslash ,dx\; \backslash ,dy\; \backslash to\; \backslash int\; f(x)\; \backslash overline\{g(x)\}\; \backslash ,dx\; =\; \backslash langle\; f,\; g\; \backslash rangle$$

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|L²(R)}}, namely,

$$f\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash langle\; f,\; \backslash psi\_n\; \backslash rangle\; \backslash psi\_n.$$

In order to prove the above equality for {{math|E(x,y;u)}}, the Fourier transform of Gaussian functions is used repeatedly:

$$\backslash rho\; \backslash sqrt\{\backslash pi\}\; e^\{-\backslash frac\{\backslash rho^2\; x^2\}\{4\}\}\; =\; \backslash int\; e^\{isx\; -\; \backslash frac\{s^2\}\{\backslash rho^2\}\}\; \backslash ,ds\; \backslash quad\; \backslash text\{for\; \}\backslash rho\; >\; 0.$$

The Hermite polynomial is then represented as

$$H\_n(x)\; =\; (-1)^n\; e^\{x^2\}\; \backslash frac\; \{d^n\}\{dx^n\}\; \backslash left(\; \backslash frac\; \{1\}\{2\backslash sqrt\{\backslash pi\}\}\; \backslash int\; e^\{isx\; -\; \backslash frac\{s^2\}\{4\}\}\; \backslash ,ds\; \backslash right)\; =\; (-1)^n\; e^\{x^2\}\backslash frac\{1\}\{2\backslash sqrt\{\backslash pi\}\}\; \backslash int\; (is)^n\; e^\{isx\; -\; \backslash frac\{s^2\}\{4\}\}\; \backslash ,ds.$$

With this representation for {{math|H_n(x)}} and {{math|H_n(y)}}, it is evident that

$$\backslash begin\{align\}$$

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}

and this yields the desired resolution of the identity result, using again the Fourier transform of Gaussian kernels under the substitution

$$s\; =\; \backslash frac\{\backslash sigma\; +\; \backslash tau\}\{\backslash sqrt\; 2\},\; \backslash quad\; t\; =\; \backslash frac\{\backslash sigma\; -\; \backslash tau\}\{\backslash sqrt\; 2\}.$$

{{Div col}}

• Hermite transform
• Legendre polynomials
• Mehler kernel
• Parabolic cylinder function
• Romanovski polynomials
• Turán's inequalities

{{Div col end}}

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{{refend}}

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

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{{DEFAULTSORT:Hermite Polynomials}}

Category:Orthogonal polynomials

Category:Polynomials

Category:Special hypergeometric functions