The th-order Hermite polynomial is a polynomial of degree . The probabilist's version has leading coefficient 1, while the physicist's version has leading coefficient .
Symmetry From the Rodrigues formulae given above, we can see that and are
even or odd functions, with the same
parity as : H_n(-x)=(-1)^nH_n(x),\quad \operatorname{He}_n(-x)=(-1)^n\operatorname{He}_n(x).
Orthogonality and are th-degree polynomials for . These
polynomials are orthogonal with respect to the
weight function (
measure) w(x) = e^{-\frac{x^2}{2}} \quad (\text{for }\operatorname{He}) or w(x) = e^{-x^2} \quad (\text{for } H), i.e., we have \int_{-\infty}^\infty H_m(x) H_n(x)\, w(x) \,dx = 0 \quad \text{for all }m \neq n. Furthermore, \int_{-\infty}^\infty H_m(x) H_n(x)\, e^{-x^2} \,dx = \sqrt{\pi}\, 2^n n!\, \delta_{nm}, and \int_{-\infty}^\infty \operatorname{He}_m(x) \operatorname{He}_n(x)\, e^{-\frac{x^2}{2}} \,dx = \sqrt{2 \pi}\, n!\, \delta_{nm}, where \delta_{nm} 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
orthogonal basis of the
Hilbert space of functions satisfying \int_{-\infty}^\infty \bigl|f(x)\bigr|^2\, w(x) \,dx in which the inner product is given by the integral \langle f,g\rangle = \int_{-\infty}^\infty f(x) \overline{g(x)}\, w(x) \,dx including the
Gaussian weight function defined in the preceding section. An orthogonal basis for is a
complete orthogonal system. For an orthogonal system,
completeness is equivalent to the fact that the 0 function is the only function 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 satisfies \int_{-\infty}^\infty f(x) x^n e^{- x^2} \,dx = 0 for every , then . One possible way to do this is to appreciate that the
entire function 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 vanishes identically. The fact then that for every real means that the
Fourier transform of is 0, hence 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 consists in introducing Hermite
functions (see below), and in saying that the Hermite functions are an orthonormal basis for .
Hermite's differential equation The probabilist's Hermite polynomials are solutions of the
Sturm–Liouville differential equation \left(e^{-\frac12 x^2}u'\right)' + \lambda e^{-\frac 1 2 x^2}u = 0, where is a constant. Imposing the boundary condition that should be polynomially bounded at infinity, the equation has solutions only if is a non-negative integer, and the solution is uniquely given by u(x) = C_1 \operatorname{He}_\lambda(x) , where C_{1} denotes a constant. Rewriting the differential equation as an
eigenvalue problem L[u] = u'' - x u' = -\lambda u, the Hermite polynomials \operatorname{He}_\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'' - 2xu' = -2\lambda u. whose solution is uniquely given in terms of physicist's Hermite polynomials in the form u(x) = C_1 H_\lambda(x) , where C_{1} denotes a constant, after imposing the boundary condition that 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'' - 2xu' + 2\lambda u = 0, the general solution takes the form u(x) = C_1 H_\lambda(x) + C_2 h_\lambda(x), where C_{1} and C_{2} are constants, H_\lambda(x) are physicist's Hermite polynomials (of the first kind), and h_\lambda(x) are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as h_\lambda(x) = {}_1F_1(-\tfrac{\lambda}{2};\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 . An explicit formula of Hermite polynomials in terms of
contour integrals is also possible.
Recurrence relation The sequence of probabilist's Hermite polynomials also satisfies the
recurrence relation \operatorname{He}_{n+1}(x) = x \operatorname{He}_n(x) - \operatorname{He}_n'(x). Individual coefficients are related by the following recursion formula: 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 , , . For the physicist's polynomials, assuming H_n(x) = \sum^n_{k=0} a_{n,k} x^k, we have H_{n+1}(x) = 2xH_n(x) - H_n'(x). Individual coefficients are related by the following recursion formula: 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} and , , . The Hermite polynomials constitute an
Appell sequence, i.e., they are a polynomial sequence satisfying the identity \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 is as follows: \operatorname{He}_{n+1}(x) = (n+1)\int_0^x \operatorname{He}_n(t)dt - He'_n(0), H_{n+1}(x) = 2(n+1)\int_0^x H_n(t)dt - H'_n(0). Equivalently, by
Taylor-expanding, \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, \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 th derivatives the following relations hold: \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 \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 and , can be used in practice to compute the polynomials quickly.
Turán's inequalities are \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: \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}
Explicit expression The physicist's Hermite polynomials can be written explicitly as 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: 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 have similar formulas, which may be obtained from these by replacing the power of with the corresponding power of and multiplying the entire sum by : \operatorname{He}_n(x) = n! \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{(-1)^m}{m!(n - 2m)!} \frac{x^{n - 2m}}{2^m}.
Inverse explicit expression The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials are 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 follow directly by properly scaling this: x^n = \frac{n!}{2^n} \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{1}{m!(n-2m)! } H_{n-2m}(x).
Generating function The Hermite polynomials are given by the
exponential generating function \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 and , and can be obtained by writing the Taylor expansion at of the entire function (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} \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 \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 statese^{2 x t-t^2} H_k(x-t) = \sum_{n=0}^{\infty} \frac{H_{n+k}(x) t^n}{n!}
Expected values If is a
random variable with a
normal distribution with standard deviation 1 and expected value , then \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: \operatorname{\mathbb E}\left[X^{2n}\right] = (-1)^n \operatorname{He}_{2n}(0) = (2n-1)!!, where is the
double factorial. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments: \operatorname{He}_n(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty (x + iy)^n e^{-\frac{y^2}{2}} \,dy.
Integral representations From the generating-function representation above, we see that the Hermite polynomials have a representation in terms of a
contour integral, as \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}=\frac{1}{\sqrt{ \pi}} \int e^{-t^2+2 i x t} dt , we have\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}
Other properties The
discriminant is expressed as a
hyperfactorial: \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\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 a_{1:r}. The multiplication theorem states that\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 \in \mathbb C has a positive real part. As a special case, 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)For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude: 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+1)}{\Gamma\left(\frac{n}{2} +1\right)} \cos \left(x \sqrt{2 n}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}, which, using
Stirling's approximation, can be further simplified, in the limit, to 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}.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 \left(1-\frac{x^2}{2n+1}\right)^{-\frac12} corresponds to the probability of finding a classical particle in a
potential well of shape V(x) = \frac 12 x^2 at location x, if its total energy is n + \frac 12. This is a general method in
semiclassical analysis. The semiclassical approximation breaks down near \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. A better approximation, which accounts for the variation in frequency, is given by 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}. The
Plancherel–Rotach asymptotics method, applied to Hermite polynomials, takes into account the uneven spacing of the zeros near the edges. It makes use of the substitution x = \sqrt{2n + 1}\cos(\varphi), \quad 0 with which one has the uniform approximation 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). Similar approximations hold for the monotonic and transition regions. Specifically, if x = \sqrt{2n+1} \cosh(\varphi), \quad 0 then 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), while for x = \sqrt{2n + 1} + t with complex and bounded, the approximation is 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), where is the
Airy function of the first kind.
Special values The physicist's Hermite polynomials evaluated at zero argument are called
Hermite numbers. 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 . Equivalently, H_{2n}(0) = (-2)^n (2n-1)!!. In terms of the probabilist's polynomials this translates to \operatorname{He}_n(0) = \begin{cases} 0 & \text{for odd }n, \\ (-1)^\frac{n}{2} (n-1)!! & \text{for even }n. \end{cases}
Kibble–Slepian formula Let M be a real n\times n
symmetric matrix, then the
Kibble–Slepian formula states that\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) where \sum_K is the \frac{n(n+1)}{2}-fold summation over all n \times n symmetric matrices with non-negative integer entries, tr(K) is the
trace of K, and k_i is defined as k_{ii} + \sum_{j=1}^n k_{ij}. This gives
Mehler's formula when M = \begin{bmatrix} 0 & u \\ u & 0\end{bmatrix}. 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: \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 \psi_n(x) is the n-th eigenfunction of the harmonic oscillator, defined as \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 and proven by Slepian in 1972 using Fourier analysis. Foata gave a
combinatorial proof while Louck gave a proof via boson quantum mechanics.
Zeroes Let x_{n,1} > \dots > x_{n,n} be the roots of H_n in descending order. Let a_m be the m-th zero of the
Airy function \operatorname{Ai}(x) in descending order: 0 > a_1 > a_2 > \cdots. By the symmetry of H_n, we need only consider the positive half of its roots. We have and the formulas involving the zeroes of
Laguerre polynomials. Let F_n(t) := \frac 1n \#\{i : x_{n, i} \leq t\} be the
cumulative distribution function for the roots of H_n, then we have the
semicircle law\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-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 n particles on a line, such that each particle i is attracted to the origin by a linear force -x_{n,i}, and repelled by each other particle j by a reciprocal force \frac{1}{x_{n,i} - x_{n,j}}. This can be constructed by confining n positively charged particles in \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. As the zeroes specify the polynomial up to scaling, the Stieltjes relation provides an alternative way to uniquely characterize the Hermite polynomials. Similarly, we have\begin{aligned} \sum_i x_{n,i}^2 &= \sum_{1 \leq i \leq n}^n \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{n,i} - x_{n,j})^2}\\ x_{n,i} &= \sum_{1 \leq j \leq n, i \neq j} \frac{1}{x_{n,i}-x_{n,j}}\\ \frac{2n - 2 - x_{n,i}^2}{3} &= \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{n,i}-x_{n,j})^2}\\ \frac 12 x_{n,i} &= \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{n,i}-x_{n,j})^3} \end{aligned} ==Relations to other functions==