Hermite polynomials

This article is about the family of orthogonal polynomials on the real line. For polynomial interpolation on a segment using derivatives, see Hermite interpolation. For integral transform of Hermite polynomials, see Hermite transform.

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 He and H is that used in the standard references.^[5] The polynomials He_n are sometimes denoted by 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 first six probabilist's Hermite polynomials He_n(x)

The first six (physicist's) Hermite polynomials H_n(x)

• 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: $${\displaystyle {\begin{aligned}H_{0}(x)&=1,\\H_{1}(x)&=2x,\\H_{2}(x)&=4x^{2}-2,\\H_{3}(x)&=8x^{3}-12x,\\H_{4}(x)&=16x^{4}-48x^{2}+12,\\H_{5}(x)&=32x^{5}-160x^{3}+120x,\\H_{6}(x)&=64x^{6}-480x^{4}+720x^{2}-120,\\H_{7}(x)&=128x^{7}-1344x^{5}+3360x^{3}-1680x,\\H_{8}(x)&=256x^{8}-3584x^{6}+13440x^{4}-13440x^{2}+1680,\\H_{9}(x)&=512x^{9}-9216x^{7}+48384x^{5}-80640x^{3}+30240x,\\H_{10}(x)&=1024x^{10}-23040x^{8}+161280x^{6}-403200x^{4}+302400x^{2}-30240.\end{aligned}}}$$

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.

Symmetry

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: $${\displaystyle H_{n}(-x)=(-1)^{n}H_{n}(x),\quad \operatorname {He} _{n}(-x)=(-1)^{n}\operatorname {He} _{n}(x).}$$

Orthogonality

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.

Completeness

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).

Hermite's differential equation

The probabilist's Hermite polynomials are solutions of the 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.

Recurrence relation

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