Hermite polynomials

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.

The polynomials arise in:

• signal processing as Hermitian wavelets for wavelet transform analysis
• probability, such as the Edgeworth series, as well as in connection with Brownian motion;
• combinatorics, as an example of an Appell sequence, obeying the umbral calculus;
• numerical analysis as Gaussian quadrature;
• physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term ${\displaystyle {\begin{aligned}xu_{x}\end{aligned}}}$ is present);
• systems theory in connection with nonlinear operations on Gaussian noise.
• random matrix theory in Gaussian ensembles.

Hermite polynomials were defined by Pierre-Simon Laplace in 1810,^[1][2] though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859.^[3] Chebyshev's work was overlooked, and they were named later after Charles 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.

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 $${\displaystyle \operatorname {He} _{n}(x)=(-1)^{n}e^{\frac {x^{2}}{2}}{\frac {d^{n}}{dx^{n}}}e^{-{\frac {x^{2}}{2}}},}$$
• while the "physicist's Hermite polynomials" are given by $${\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}.}$$

These equations have the form of a Rodrigues' formula and can also be written as, $${\displaystyle \operatorname {He} _{n}(x)=\left(x-{\frac {d}{dx}}\right)^{n}\cdot 1,\quad H_{n}(x)=\left(2x-{\frac {d}{dx}}\right)^{n}\cdot 1.}$$

The two definitions are not exactly identical; each is a rescaling of the other: $${\displaystyle H_{n}(x)=2^{\frac {n}{2}}\operatorname {He} _{n}\left({\sqrt {2}}\,x\right),\quad \operatorname {He} _{n}(x)=2^{-{\frac {n}{2}}}H_{n}\left({\frac {x}{\sqrt {2}}}\right).}$$

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

The notation ${\displaystyle \operatorname {He} }$ and ${\displaystyle H}$ is that used in the standard references.^[5] The polynomials ${\displaystyle \operatorname {He} _{n}}$ are sometimes denoted by ${\displaystyle H_{n}}$, especially in probability theory, because $${\displaystyle {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}}$$ is the probability density function for the normal distribution with expected value 0 and standard deviation 1. The probabilist's Hermite polynomials are also called the monic Hermite polynomials, because they are monic.

• The first eleven probabilist's Hermite polynomials are: $${\displaystyle {\begin{aligned}\operatorname {He} _{0}(x)&=1,\\\operatorname {He} _{1}(x)&=x,\\\operatorname {He} _{2}(x)&=x^{2}-1,\\\operatorname {He} _{3}(x)&=x^{3}-3x,\\\operatorname {He} _{4}(x)&=x^{4}-6x^{2}+3,\\\operatorname {He} _{5}(x)&=x^{5}-10x^{3}+15x,\\\operatorname {He} _{6}(x)&=x^{6}-15x^{4}+45x^{2}-15,\\\operatorname {He} _{7}(x)&=x^{7}-21x^{5}+105x^{3}-105x,\\\operatorname {He} _{8}(x)&=x^{8}-28x^{6}+210x^{4}-420x^{2}+105,\\\operatorname {He} _{9}(x)&=x^{9}-36x^{7}+378x^{5}-1260x^{3}+945x,\\\operatorname {He} _{10}(x)&=x^{10}-45x^{8}+630x^{6}-3150x^{4}+4725x^{2}-945.\end{aligned}}}$$
• The first eleven physicist's Hermite polynomials are: $${\displaystyle {\begin{aligned}H_{0}(x)&=1,\\H_{1}(x)&=2x,\\H_{2}(x)&=4x^{2}-2,\\H_{3}(x)&=8x^{3}-12x,\\H_{4}(x)&=16x^{4}-48x^{2}+12,\\H_{5}(x)&=32x^{5}-160x^{3}+120x,\\H_{6}(x)&=64x^{6}-480x^{4}+720x^{2}-120,\\H_{7}(x)&=128x^{7}-1344x^{5}+3360x^{3}-1680x,\\H_{8}(x)&=256x^{8}-3584x^{6}+13440x^{4}-13440x^{2}+1680,\\H_{9}(x)&=512x^{9}-9216x^{7}+48384x^{5}-80640x^{3}+30240x,\\H_{10}(x)&=1024x^{10}-23040x^{8}+161280x^{6}-403200x^{4}+302400x^{2}-30240.\end{aligned}}}$$

Quick reference table

	physicist's	probabilist's
symbol	${\displaystyle H_{n}}$	${\displaystyle \operatorname {He} _{n}}$
head coefficient	${\displaystyle 2^{n}}$	${\displaystyle 1}$
differential operator	${\displaystyle (-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}}$	${\displaystyle (-1)^{n}e^{\frac {x^{2}}{2}}{\frac {d^{n}}{dx^{n}}}e^{-{\frac {x^{2}}{2}}}}$
orthogonal to	${\displaystyle e^{-x^{2}}}$	${\displaystyle e^{-{\frac {1}{2}}x^{2}}}$
inner product	${\displaystyle \int H_{m}(x)H_{n}(x){\frac {e^{-x^{2}}}{\sqrt {\pi }}}dx=2^{n}n!\delta _{mn}}$	${\displaystyle \int \operatorname {He} _{m}(x)\operatorname {He} _{n}(x)\,{\frac {e^{-{\frac {x^{2}}{2}}}}{\sqrt {2\pi }}}\,dx=n!\,\delta _{nm},}$
generating function	${\displaystyle e^{2xt-t^{2}}=\sum _{n=0}^{\infty }H_{n}(x){\frac {t^{n}}{n!}}}$	${\displaystyle e^{xt-{\frac {1}{2}}t^{2}}=\sum _{n=0}^{\infty }\operatorname {He} _{n}(x){\frac {t^{n}}{n!}}}$
Rodrigues' formula	${\displaystyle \left(2x-{\frac {d}{dx}}\right)^{n}\cdot 1}$	${\displaystyle \left(x-{\frac {d}{dx}}\right)^{n}\cdot 1}$
recurrence relation	${\displaystyle H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x)}$	${\displaystyle \operatorname {He} _{n+1}(x)=x\operatorname {He} _{n}(x)-n\operatorname {He} _{n-1}(x)}$

• 
  The first six probabilist's Hermite polynomials ${\displaystyle \operatorname {He} _{n}(x)}$
• 
  The first six physicist's Hermite polynomials ${\displaystyle H_{n}(x)}$

The nth-order Hermite polynomial is a polynomial of degree n. The probabilist's version He_n has leading coefficient 1, while the physicist's version H_n has leading coefficient 2ⁿ.

From the Rodrigues formulae given above, we can see that H_n(x) and He_n(x) are even or odd functions depending on n: $${\displaystyle H_{n}(-x)=(-1)^{n}H_{n}(x),\quad \operatorname {He} _{n}(-x)=(-1)^{n}\operatorname {He} _{n}(x).}$$

H_n(x) and He_n(x) are nth-degree polynomials for n = 0, 1, 2, 3,.... These polynomials are orthogonal with respect to the weight function (measure) $${\displaystyle w(x)=e^{-{\frac {x^{2}}{2}}}\quad ({\text{for }}\operatorname {He} )}$$ or $${\displaystyle w(x)=e^{-x^{2}}\quad ({\text{for }}H),}$$ i.e., we have $${\displaystyle \int _{-\infty }^{\infty }H_{m}(x)H_{n}(x)\,w(x)\,dx=0\quad {\text{for all }}m\neq n.}$$

Furthermore, $${\displaystyle \int _{-\infty }^{\infty }H_{m}(x)H_{n}(x)\,e^{-x^{2}}\,dx={\sqrt {\pi }}\,2^{n}n!\,\delta _{nm},}$$ and $${\displaystyle \int _{-\infty }^{\infty }\operatorname {He} _{m}(x)\operatorname {He} _{n}(x)\,e^{-{\frac {x^{2}}{2}}}\,dx={\sqrt {2\pi }}\,n!\,\delta _{nm},}$$ where ${\displaystyle \delta _{nm}}$ 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 $${\displaystyle \int _{-\infty }^{\infty }{\bigl |}f(x){\bigr |}^{2}\,w(x)\,dx<\infty ,}$$ in which the inner product is given by the integral $${\displaystyle \langle f,g\rangle =\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,w(x)\,dx}$$ including the Gaussian weight function w(x) defined in the preceding section.

An orthogonal basis for 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 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 f satisfies $${\displaystyle \int _{-\infty }^{\infty }f(x)x^{n}e^{-x^{2}}\,dx=0}$$ for every n ≥ 0, then f = 0.

One possible way to do this is to appreciate that the entire function $${\displaystyle F(z)=\int _{-\infty }^{\infty }f(x)e^{zx-x^{2}}\,dx=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\int f(x)x^{n}e^{-x^{2}}\,dx=0}$$ vanishes identically. The fact then that F(it) = 0 for every real t means that the Fourier transform of f(x)e^−x2 is 0, hence 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 L²(R, w(x) dx) consists in introducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis for L²(R).

The probabilist's Hermite polynomials are solutions of the Sturm–Liouville differential equation $${\displaystyle \left(e^{-{\frac {1}{2}}x^{2}}u'\right)'+\lambda e^{-{\frac {1}{2}}x^{2}}u=0,}$$ where λ is a constant. Imposing the boundary condition that u should be polynomially bounded at infinity, the equation has solutions only if λ is a non-negative integer, and the solution is uniquely given by ${\displaystyle u(x)=C_{1}\operatorname {He} _{\lambda }(x)}$, where ${\displaystyle C_{1}}$ denotes a constant.

Rewriting the differential equation as an eigenvalue problem $${\displaystyle L[u]=u''-xu'=-\lambda u,}$$ the Hermite polynomials ${\displaystyle \operatorname {He} _{\lambda }(x)}$ may be understood as eigenfunctions of the differential operator ${\displaystyle L[u]}$ . This eigenvalue problem is called the Hermite equation, although the term is also used for the closely related equation $${\displaystyle u''-2xu'=-2\lambda u.}$$ whose solution is uniquely given in terms of physicist's Hermite polynomials in the form ${\displaystyle u(x)=C_{1}H_{\lambda }(x)}$, where ${\displaystyle C_{1}}$ denotes a constant, after imposing the boundary condition that 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 $${\displaystyle u''-2xu'+2\lambda u=0,}$$ the general solution takes the form $${\displaystyle u(x)=C_{1}H_{\lambda }(x)+C_{2}h_{\lambda }(x),}$$ where ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ are constants, ${\displaystyle H_{\lambda }(x)}$ are physicist's Hermite polynomials (of the first kind), and ${\displaystyle h_{\lambda }(x)}$ are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as ${\displaystyle h_{\lambda }(x)={}_{1}F_{1}(-{\tfrac {\lambda }{2}};{\tfrac {1}{2}};x^{2})}$ where ${\displaystyle {}_{1}F_{1}(a;b;z)}$ 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 λ. An explicit formula of Hermite polynomials in terms of contour integrals (Courant & Hilbert 1989) is also possible.

The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation $${\displaystyle \operatorname {He} _{n+1}(x)=x\operatorname {He} _{n}(x)-\operatorname {He} _{n}'(x).}$$ Individual coefficients are related by the following recursion formula: $${\displaystyle a_{n+1,k}={\begin{cases}-(k+1)a_{n,k+1}&k=0,\\a_{n,k-1}-(k+1)a_{n,k+1}&k>0,\end{cases}}}$$ and a_0,0 = 1, a_1,0 = 0, a_1,1 = 1.

For the physicist's polynomials, assuming $${\displaystyle H_{n}(x)=\sum _{k=0}^{n}a_{n,k}x^{k},}$$ we have $${\displaystyle H_{n+1}(x)=2xH_{n}(x)-H_{n}'(x).}$$ Individual coefficients are related by the following recursion formula: $${\displaystyle a_{n+1,k}={\begin{cases}-a_{n,k+1}&k=0,\\2a_{n,k-1}-(k+1)a_{n,k+1}&k>0,\end{cases}}}$$ and a_0,0 = 1, a_1,0 = 0, a_1,1 = 2.

The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity $${\displaystyle {\begin{aligned}\operatorname {He} _{n}'(x)&=n\operatorname {He} _{n-1}(x),\\H_{n}'(x)&=2nH_{n-1}(x).\end{aligned}}}$$

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

$${\displaystyle H_{n+1}(x)=2(n+1)\int _{0}^{x}H_{n}(t)dt-H'_{n}(0).}$$

Equivalently, by Taylor-expanding, $${\displaystyle {\begin{aligned}\operatorname {He} _{n}(x+y)&=\sum _{k=0}^{n}{\binom {n}{k}}x^{n-k}\operatorname {He} _{k}(y)&&=2^{-{\frac {n}{2}}}\sum _{k=0}^{n}{\binom {n}{k}}\operatorname {He} _{n-k}\left(x{\sqrt {2}}\right)\operatorname {He} _{k}\left(y{\sqrt {2}}\right),\\H_{n}(x+y)&=\sum _{k=0}^{n}{\binom {n}{k}}H_{k}(x)(2y)^{n-k}&&=2^{-{\frac {n}{2}}}\cdot \sum _{k=0}^{n}{\binom {n}{k}}H_{n-k}\left(x{\sqrt {2}}\right)H_{k}\left(y{\sqrt {2}}\right).\end{aligned}}}$$ These umbral identities are self-evident and included in the differential operator representation detailed below, $${\displaystyle {\begin{aligned}\operatorname {He} _{n}(x)&=e^{-{\frac {D^{2}}{2}}}x^{n},\\H_{n}(x)&=2^{n}e^{-{\frac {D^{2}}{4}}}x^{n}.\end{aligned}}}$$

In consequence, for the mth derivatives the following relations hold: $${\displaystyle {\begin{aligned}\operatorname {He} _{n}^{(m)}(x)&={\frac {n!}{(n-m)!}}\operatorname {He} _{n-m}(x)&&=m!{\binom {n}{m}}\operatorname {He} _{n-m}(x),\\H_{n}^{(m)}(x)&=2^{m}{\frac {n!}{(n-m)!}}H_{n-m}(x)&&=2^{m}m!{\binom {n}{m}}H_{n-m}(x).\end{aligned}}}$$

It follows that the Hermite polynomials also satisfy the recurrence relation $${\displaystyle {\begin{aligned}\operatorname {He} _{n+1}(x)&=x\operatorname {He} _{n}(x)-n\operatorname {He} _{n-1}(x),\\H_{n+1}(x)&=2xH_{n}(x)-2nH_{n-1}(x).\end{aligned}}}$$

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

Turán's inequalities are $${\displaystyle {\mathit {H}}_{n}(x)^{2}-{\mathit {H}}_{n-1}(x){\mathit {H}}_{n+1}(x)=(n-1)!\sum _{i=0}^{n-1}{\frac {2^{n-i}}{i!}}{\mathit {H}}_{i}(x)^{2}>0.}$$

Moreover, the following multiplication theorem holds: $${\displaystyle {\begin{aligned}H_{n}(\gamma x)&=\sum _{i=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }\gamma ^{n-2i}(\gamma ^{2}-1)^{i}{\binom {n}{2i}}{\frac {(2i)!}{i!}}H_{n-2i}(x),\\\operatorname {He} _{n}(\gamma x)&=\sum _{i=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }\gamma ^{n-2i}(\gamma ^{2}-1)^{i}{\binom {n}{2i}}{\frac {(2i)!}{i!}}2^{-i}\operatorname {He} _{n-2i}(x).\end{aligned}}}$$

The physicist's Hermite polynomials can be written explicitly as $${\displaystyle H_{n}(x)={\begin{cases}\displaystyle n!\sum _{l=0}^{\frac {n}{2}}{\frac {(-1)^{{\tfrac {n}{2}}-l}}{(2l)!\left({\tfrac {n}{2}}-l\right)!}}(2x)^{2l}&{\text{for even }}n,\\\displaystyle n!\sum _{l=0}^{\frac {n-1}{2}}{\frac {(-1)^{{\frac {n-1}{2}}-l}}{(2l+1)!\left({\frac {n-1}{2}}-l\right)!}}(2x)^{2l+1}&{\text{for odd }}n.\end{cases}}}$$

These two equations may be combined into one using the floor function: $${\displaystyle H_{n}(x)=n!\sum _{m=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }{\frac {(-1)^{m}}{m!(n-2m)!}}(2x)^{n-2m}.}$$

The probabilist's Hermite polynomials He have similar formulas, which may be obtained from these by replacing the power of 2x with the corresponding power of √2 x and multiplying the entire sum by 2^−⁠n/2⁠: $${\displaystyle \operatorname {He} _{n}(x)=n!\sum _{m=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }{\frac {(-1)^{m}}{m!(n-2m)!}}{\frac {x^{n-2m}}{2^{m}}}.}$$

The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials He are $${\displaystyle x^{n}=n!\sum _{m=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }{\frac {1}{2^{m}m!(n-2m)!}}\operatorname {He} _{n-2m}(x).}$$

The corresponding expressions for the physicist's Hermite polynomials H follow directly by properly scaling this:^[7] $${\displaystyle x^{n}={\frac {n!}{2^{n}}}\sum _{m=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }{\frac {1}{m!(n-2m)!}}H_{n-2m}(x).}$$

The Hermite polynomials are given by the exponential generating function $${\displaystyle {\begin{aligned}e^{xt-{\frac {1}{2}}t^{2}}&=\sum _{n=0}^{\infty }\operatorname {He} _{n}(x){\frac {t^{n}}{n!}},\\e^{2xt-t^{2}}&=\sum _{n=0}^{\infty }H_{n}(x){\frac {t^{n}}{n!}}.\end{aligned}}}$$

This equality is valid for all complex values of x and t, and can be obtained by writing the Taylor expansion at x of the entire function 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 $${\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}=(-1)^{n}e^{x^{2}}{\frac {n!}{2\pi i}}\oint _{\gamma }{\frac {e^{-z^{2}}}{(z-x)^{n+1}}}\,dz.}$$

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

A slight generalization states^[8]$${\displaystyle e^{2xt-t^{2}}H_{k}(x-t)=\sum _{n=0}^{\infty }{\frac {H_{n+k}(x)t^{n}}{n!}}}$$

If X is a random variable with a normal distribution with standard deviation 1 and expected value μ, then $${\displaystyle \operatorname {\mathbb {E} } \left[\operatorname {He} _{n}(X)\right]=\mu ^{n}.}$$

The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices: $${\displaystyle \operatorname {\mathbb {E} } \left[X^{2n}\right]=(-1)^{n}\operatorname {He} _{2n}(0)=(2n-1)!!,}$$ where (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: $${\displaystyle \operatorname {He} _{n}(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }(x+iy)^{n}e^{-{\frac {y^{2}}{2}}}\,dy.}$$

From the generating-function representation above, we see that the Hermite polynomials have a representation in terms of a contour integral, as $${\displaystyle {\begin{aligned}\operatorname {He} _{n}(x)&={\frac {n!}{2\pi i}}\oint _{C}{\frac {e^{tx-{\frac {t^{2}}{2}}}}{t^{n+1}}}\,dt,\\H_{n}(x)&={\frac {n!}{2\pi i}}\oint _{C}{\frac {e^{2tx-t^{2}}}{t^{n+1}}}\,dt,\end{aligned}}}$$ with the contour encircling the origin.

Using the Fourier transform of the gaussian ${\displaystyle e^{-x^{2}}={\frac {1}{\sqrt {\pi }}}\int e^{-t^{2}+2ixt}dt}$, we have$${\displaystyle {\begin{aligned}H_{n}(x)&=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}={\frac {(-2i)^{n}e^{x^{2}}}{\sqrt {\pi }}}\int t^{n}e^{-t^{2}+2ixt}dt\\\operatorname {He} _{n}(x)&={\frac {(-i)^{n}e^{x^{2}/2}}{\sqrt {2\pi }}}\int t^{n}\,e^{-t^{2}/2+ixt}\,dt.\end{aligned}}}$$

The discriminant is expressed as a hyperfactorial:^[9]

$${\displaystyle {\begin{aligned}\operatorname {Disc} (H_{n})&=2^{{\frac {3}{2}}n(n-1)}\prod _{j=1}^{n}j^{j}\\\operatorname {Disc} (\operatorname {He} _{n})&=\prod _{j=1}^{n}j^{j}\end{aligned}}}$$

The addition theorem, or the summation theorem, states that^{[10][11]: 8.958}$${\displaystyle {\frac {\left(\sum _{k=1}^{r}a_{k}^{2}\right)^{\frac {n}{2}}}{n!}}H_{n}\left({\frac {\sum _{k=1}^{r}a_{k}x_{k}}{\sqrt {\sum _{k=1}^{r}a_{k}^{2}}}}\right)=\sum _{m_{1}+m_{2}+\ldots +m_{r}=n,m_{i}\geq 0}\prod _{k=1}^{r}\left\{{\frac {a_{k}^{m_{k}}}{m_{k}!}}H_{m_{k}}\left(x_{k}\right)\right\}}$$for any nonzero vector ${\displaystyle a_{1:r}}$.

The multiplication theorem states that^[10]$${\displaystyle H_{n}\left(\lambda x\right)=\lambda ^{n}\sum _{\ell =0}^{\left\lfloor n/2\right\rfloor }{\frac {\left(-n\right)_{2\ell }}{\ell !}}(1-\lambda ^{-2})^{\ell }H_{n-2\ell }\left(x\right)}$$for any nonzero ${\displaystyle \lambda }$.

Feldheim formula^{[12]: Eq 46}$${\displaystyle {\begin{aligned}{\frac {1}{\sqrt {a\pi }}}&\int _{-\infty }^{+\infty }e^{-{\frac {x^{2}}{a}}}H_{m}\left({\frac {x+y}{\lambda }}\right)H_{n}\left({\frac {x+z}{\mu }}\right)dx\\&=\left(1-{\frac {a}{\lambda ^{2}}}\right)^{\frac {m}{2}}\left(1-{\frac {a}{\mu ^{2}}}\right)^{\frac {n}{2}}\sum _{r=0}^{\min(m,n)}r!{\binom {m}{r}}{\binom {n}{r}}\left({\frac {2a}{\sqrt {\left(\lambda ^{2}-a\right)\left(\mu ^{2}-a\right)}}}\right)^{r}H_{m-r}\left({\frac {y}{\sqrt {\lambda ^{2}-a}}}\right)H_{n-r}\left({\frac {z}{\sqrt {\mu ^{2}-a}}}\right)\end{aligned}}}$$where ${\displaystyle a\in \mathbb {C} }$ has a positive real part. As a special case,^{[12]: Eq 52}$${\displaystyle {\frac {1}{\sqrt {\pi }}}\int _{-\infty }^{+\infty }e^{-t^{2}}H_{m}(t\sin \theta +v\cos \theta )H_{n}(t\cos \theta -v\sin \theta )dt=(-1)^{n}\cos ^{m}\theta \sin ^{n}\theta H_{m+n}(v)}$$

Asymptotically, as n → ∞, the expansion^[13] $${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim {\frac {2^{n}}{\sqrt {\pi }}}\Gamma \left({\frac {n+1}{2}}\right)\cos \left(x{\sqrt {2n}}-{\frac {n\pi }{2}}\right)}$$ holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude: $${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim {\frac {2^{n}}{\sqrt {\pi }}}\Gamma \left({\frac {n+1}{2}}\right)\cos \left(x{\sqrt {2n}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}}={\frac {\Gamma (n+1)}{\Gamma \left({\frac {n}{2}}+1\right)}}\cos \left(x{\sqrt {2n}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}},}$$ which, using Stirling's approximation, can be further simplified, in the limit, to $${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim \left({\frac {2n}{e}}\right)^{\frac {n}{2}}{\sqrt {2}}\cos \left(x{\sqrt {2n}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}}.}$$This expansion is needed to resolve the wavefunction of a quantum harmonic oscillator such that it agrees with the classical approximation in the limit of the correspondence principle. The term ${\displaystyle \left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{2}}}}$ corresponds to the probability of finding a classical particle in a potential well of shape ${\displaystyle V(x)={\frac {1}{2}}x^{2}}$ at location ${\displaystyle x}$, if its total energy is ${\displaystyle n+{\frac {1}{2}}}$. This is a general method in semiclassical analysis. The semiclassical approximation breaks down near ${\displaystyle \pm {\sqrt {2n+1}}}$, the location where the classical particle would be turned back. This is a fold catastrophe, at which point the Airy function is needed.^[14]

A better approximation, which accounts for the variation in frequency, is given by $${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim \left({\frac {2n}{e}}\right)^{\frac {n}{2}}{\sqrt {2}}\cos \left(x{\sqrt {2n+1-{\frac {x^{2}}{3}}}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}}.}$$

The Plancherel–Rotach asymptotics method, applied to Hermite polynomials, takes into account the uneven spacing of the zeros near the edges.^[15] It makes use of the substitution $${\displaystyle x={\sqrt {2n+1}}\cos(\varphi ),\quad 0<\varepsilon \leq \varphi \leq \pi -\varepsilon ,}$$ with which one has the uniform approximation $${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=2^{{\frac {n}{2}}+{\frac {1}{4}}}{\sqrt {n!}}(\pi n)^{-{\frac {1}{4}}}(\sin \varphi )^{-{\frac {1}{2}}}\cdot \left(\sin \left({\frac {3\pi }{4}}+\left({\frac {n}{2}}+{\frac {1}{4}}\right)\left(\sin 2\varphi -2\varphi \right)\right)+O\left(n^{-1}\right)\right).}$$

Similar approximations hold for the monotonic and transition regions. Specifically, if $${\displaystyle x={\sqrt {2n+1}}\cosh(\varphi ),\quad 0<\varepsilon \leq \varphi \leq \omega <\infty ,}$$ then $${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=2^{{\frac {n}{2}}-{\frac {3}{4}}}{\sqrt {n!}}(\pi n)^{-{\frac {1}{4}}}(\sinh \varphi )^{-{\frac {1}{2}}}\cdot e^{\left({\frac {n}{2}}+{\frac {1}{4}}\right)\left(2\varphi -\sinh 2\varphi \right)}\left(1+O\left(n^{-1}\right)\right),}$$ while for $${\displaystyle x={\sqrt {2n+1}}+t}$$ with t complex and bounded, the approximation is $${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=\pi ^{\frac {1}{4}}2^{{\frac {n}{2}}+{\frac {1}{4}}}{\sqrt {n!}}\,n^{-{\frac {1}{12}}}\left(\operatorname {Ai} \left(2^{\frac {1}{2}}n^{\frac {1}{6}}t\right)+O\left(n^{-{\frac {2}{3}}}\right)\right),}$$ where Ai is the Airy function of the first kind.

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

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

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

Let ${\textstyle M}$ be a real ${\textstyle n\times n}$ symmetric matrix, then the Kibble–Slepian formula states that$${\displaystyle \det(I+M)^{-{\frac {1}{2}}}e^{x^{T}M(I+M)^{-1}x}=\sum _{K}\left[\prod _{1\leq i\leq j\leq n}{\frac {(M_{ij}/2)^{k_{ij}}}{k_{ij}!}}\right]2^{-tr(K)}H_{k_{1}}(x_{1})\cdots H_{k_{n}}(x_{n})}$$ where ${\textstyle \sum _{K}}$ is the ${\displaystyle {\frac {n(n+1)}{2}}}$-fold summation over all ${\textstyle n\times n}$ symmetric matrices with non-negative integer entries, ${\displaystyle tr(K)}$ is the trace of ${\displaystyle K}$, and ${\textstyle k_{i}}$ is defined as ${\textstyle k_{ii}+\sum _{j=1}^{n}k_{ij}}$. This gives Mehler's formula when ${\displaystyle M={\begin{bmatrix}0&u\\u&0\end{bmatrix}}}$.

Equivalently stated, if ${\textstyle T}$ is a positive semidefinite matrix, then set ${\textstyle M=-T(I+T)^{-1}}$, we have ${\textstyle M(I+M)^{-1}=-T}$, so $${\displaystyle e^{-x^{T}Tx}=\det(I+T)^{-{\frac {1}{2}}}\sum _{K}\left[\prod _{1\leq i\leq j\leq n}{\frac {(M_{ij}/2)^{k_{ij}}}{k_{ij}!}}\right]2^{-tr(K)}H_{k_{1}}(x_{1})\dots H_{k_{n}}(x_{n})}$$Equivalently stated in a form closer to the boson quantum mechanics of the harmonic oscillator:^[16]$${\displaystyle \pi ^{-n/4}\det(I+M)^{-{\frac {1}{2}}}e^{-{\frac {1}{2}}x^{T}(I-M)(I+M)^{-1}x}=\sum _{K}\left[\prod _{1\leq i\leq j\leq n}M_{ij}^{k_{ij}}/k_{ij}!\right]\left[\prod _{1\leq i\leq n}k_{i}!\right]^{1/2}2^{-\operatorname {tr} K}\psi _{k_{1}}\left(x_{1}\right)\cdots \psi _{k_{n}}\left(x_{n}\right).}$$ where each ${\textstyle \psi _{n}(x)}$ is the ${\textstyle n}$-th eigenfunction of the harmonic oscillator, defined as $${\displaystyle \psi _{n}(x):={\frac {1}{\sqrt {2^{n}n!}}}\left({\frac {1}{\pi }}\right)^{\frac {1}{4}}e^{-{\frac {1}{2}}x^{2}}H_{n}(x)}$$The Kibble–Slepian formula was proposed by Kibble in 1945^[17] and proven by Slepian in 1972 using Fourier analysis.^[18] Foata gave a combinatorial proof^[19] while Louck gave a proof via boson quantum mechanics.^[16] It has a generalization for complex-argument Hermite polynomials.^[20][21]

Let ${\displaystyle x_{n,1}>\dots >x_{n,n}}$ be the roots of ${\displaystyle H_{n}}$ in descending order. Let ${\displaystyle a_{m}}$ be the ${\displaystyle m}$-th zero of the Airy function ${\displaystyle \operatorname {Ai} (x)}$ in descending order: ${\displaystyle 0>a_{1}>a_{2}>\cdots }$. By the symmetry of ${\displaystyle H_{n}}$, we need only consider the positive half of its roots.

We have^[9]$${\displaystyle (2n+1)^{\frac {1}{2}}>x_{n,1}>x_{n,2}>\cdots >x_{n,\lfloor n/2\rfloor }>0.}$$For each ${\displaystyle m}$, asymptotically at ${\displaystyle n\to \infty }$,^[9]$${\displaystyle x_{n,m}=(2n+1)^{\frac {1}{2}}+2^{-{\frac {1}{3}}}(2n+1)^{-{\frac {1}{6}}}a_{m}+\epsilon _{n,m},}$$where ${\displaystyle \epsilon _{n,m}=O\left(n^{-{\frac {5}{6}}}\right)}$, and ${\displaystyle \epsilon _{n,m}<0}$.

See also ^[22], and the formulas involving the zeroes of Laguerre polynomials.

Let ${\displaystyle F_{n}(t):={\frac {1}{n}}\#\{i:x_{n,i}\leq t\}}$ be the cumulative distribution function for the roots of ${\displaystyle H_{n}}$, then we have the semicircle law^[23]$${\displaystyle \lim _{n\to \infty }F_{n}({\sqrt {2n}}t)={\frac {2}{\pi }}\int _{-1}^{t}{\sqrt {1-s^{2}}}ds\quad t\in (-1,+1)}$$The Stieltjes relation states that^[24][25]$${\displaystyle -x_{n,i}+\sum _{1\leq j\leq n,i\neq j}{\frac {1}{x_{n,i}-x_{n,j}}}=0}$$and can be physically interpreted as the equilibrium position of ${\displaystyle n}$ particles on a line, such that each particle ${\displaystyle i}$ is attracted to the origin by a linear force ${\displaystyle -x_{n,i}}$, and repelled by each other particle ${\displaystyle j}$ by a reciprocal force ${\displaystyle {\frac {1}{x_{n,i}-x_{n,j}}}}$. This can be constructed by confining ${\displaystyle n}$ positively charged particles in ${\displaystyle \mathbb {R} ^{2}}$ to the real line, and connecting each particle to the origin by a spring. This is also called the electrostatic model, and relates to the Coulomb gas interpretation of the eigenvalues of gaussian ensembles.