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 ${\begin{aligned}xu_{x}\end{aligned}}$ 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 in his later 1865 publications.

Definition

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

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

{\mathit {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:

H_{n}(x)=2^{\frac {n}{2}}{\mathit {He}}_{n}\left({\sqrt {2}}\,x\right),\quad {\mathit {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

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

Properties

The $n$ th-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 $2 n$ .

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 $n$ :

H_{n}(-x)=(-1)^{n}H_{n}(x),\quad {\mathit {He}}_{n}(-x)=(-1)^{n}{\mathit {He}}_{n}(x).

Orthogonality

$H n (x)$ and $He n (x)$ are $n$ th-degree polynomials for $n = 0, 1, 2, 3,...$ . These polynomials are orthogonal with respect to the weight function (measure)

w(x)=e^{-{\frac {x^{2}}{2}}}\quad ({\text{for }}{\mathit {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 }{\mathit {He}}_{m}(x){\mathit {He}}_{n}(x)\,e^{-{\frac {x^{2}}{2}}}\,dx={\sqrt {2\pi }}\,n!\,\delta _{nm},

or

\int _{-\infty }^{\infty }H_{m}(x)H_{n}(x)\,e^{-x^{2}}\,dx={\sqrt {\pi }}\,2^{n}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<\infty ,

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 $w (x)$ defined in the preceding section

An orthogonal basis for $L 2 (R, w (x) dx)$ is a complete orthogonal system. For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function $f \in L 2 (R, w (x) dx)$ 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

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

for every $n \geq 0$ , then $f = 0$ .

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

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 $f (x) e - x 2$ 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 $L 2 (R, w (x) dx)$ consists in introducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis for $L 2 (R)$ .

Hermite's differential equation

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

\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 $u(x)=C_{1}He_{\lambda }(x)$ , where $C_{1}$ denotes a constant.

Rewriting the differential equation as an eigenvalue problem

L[u]=u''-xu'=-\lambda u,

the Hermite polynomials $He_{\lambda }(x)$ 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 $u$ 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''-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 $h_{\lambda }(x)={}_{1}F_{1}(-{\tfrac {\lambda }{2}};{\tfrac {1}{2}};x^{2})$ where ${}_{1}F_{1}(a;b;z)$ 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

{\mathit {He}}_{n+1}(x)=x{\mathit {He}}_{n}(x)-{\mathit {He}}_{n}'(x).

Individual coefficients are related by the following recursion formula:

a_{n+1,k}={\begin{cases}-na_{n-1,k}&k=0,\\a_{n,k-1}-na_{n-1,k}&k>0,\end{cases}}

and $a 0,0 = 1$ , $a 1,0 = 0$ , $a 1,1 = 1$ .

For the physicist's polynomials, assuming

H_{n}(x)=\sum _{k=0}^{n}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,\\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

{\begin{aligned}{\mathit {He}}_{n}'(x)&=n{\mathit {He}}_{n-1}(x),\\H_{n}'(x)&=2nH_{n-1}(x).\end{aligned}}

Equivalently, by Taylor-expanding,

{\begin{aligned}{\mathit {He}}_{n}(x+y)&=\sum _{k=0}^{n}{\binom {n}{k}}x^{n-k}{\mathit {He}}_{k}(y)&&=2^{-{\frac {n}{2}}}\sum _{k=0}^{n}{\binom {n}{k}}{\mathit {He}}_{n-k}\left(x{\sqrt {2}}\right){\mathit {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,

{\begin{aligned}{\mathit {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 $m$ th derivatives the following relations hold:

{\begin{aligned}{\mathit {He}}_{n}^{(m)}(x)&={\frac {n!}{(n-m)!}}{\mathit {He}}_{n-m}(x)&&=m!{\binom {n}{m}}{\mathit {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

{\begin{aligned}{\mathit {He}}_{n+1}(x)&=x{\mathit {He}}_{n}(x)-n{\mathit {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 $H 0 (x)$ and $H 1 (x)$ , 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{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),\\{\mathit {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}{\mathit {He}}_{n-2i}(x).\end{aligned}}

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 $He$ have similar formulas, which may be obtained from these by replacing the power of $2 x$ with the corresponding power of $\sqrt 2 x$ and multiplying the entire sum by $2−.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}n/2$ :

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 $He$ are

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

The corresponding expressions for the physicist's Hermite polynomials $H$ follow directly by properly scaling this:[6]

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{aligned}e^{xt-{\frac {1}{2}}t^{2}}&=\sum _{n=0}^{\infty }{\mathit {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 $z \to e - z 2$ (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.

Expected values

If $X$ is a random variable with a normal distribution with standard deviation 1 and expected value $μ$ , then

\operatorname {\mathbb {E} } \left[{\mathit {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}{\mathit {He}}_{2n}(0)=(2n-1)!!,

where $(2 n - 1)!!$ is the double factorial. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments:

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

Asymptotic expansion

Asymptotically, as $n \to \infty$ , the expansion[7]

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:

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 {2\Gamma (n)}{\Gamma \left({\frac {n}{2}}\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

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.

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)^{-{\frac {1}{4}}}.

A finer approximation,[8] which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution

x={\sqrt {2n+1}}\cos(\varphi ),\quad 0<\varepsilon \leq \varphi \leq \pi -\varepsilon ,

with which one has the uniform approximation

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

x={\sqrt {2n+1}}\cosh(\varphi ),\quad 0<\varepsilon \leq \varphi \leq \omega <\infty ,

then

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

x={\sqrt {2n+1}}+t

with $t$ complex and bounded, the approximation is

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.

Special values

The physicist's Hermite polynomials evaluated at zero argument $H n (0)$ 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 $H n (0) = -2(n - 1) H n - 2 (0)$ .

In terms of the probabilist's polynomials this translates to

He_{n}(0)={\begin{cases}0&{\text{for odd }}n,\\(-1)^{\frac {n}{2}}(n-1)!!&{\text{for even }}n.\end{cases}}

Relations to other functions

Laguerre polynomials

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

{\begin{aligned}H_{2n}(x)&=(-4)^{n}n!L_{n}^{\left(-{\frac {1}{2}}\right)}(x^{2})&&=4^{n}n!\sum _{k=0}^{n}(-1)^{n-k}{\binom {n-{\frac {1}{2}}}{n-k}}{\frac {x^{2k}}{k!}},\\H_{2n+1}(x)&=2(-4)^{n}n!xL_{n}^{\left({\frac {1}{2}}\right)}(x^{2})&&=2\cdot 4^{n}n!\sum _{k=0}^{n}(-1)^{n-k}{\binom {n+{\frac {1}{2}}}{n-k}}{\frac {x^{2k+1}}{k!}}.\end{aligned}}

Relation to confluent hypergeometric functions

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

H_{n}(x)=2^{n}U\left(-{\tfrac {1}{2}}n,{\tfrac {1}{2}},x^{2}\right)

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

{\begin{aligned}H_{2n}(x)&=(-1)^{n}{\frac {(2n)!}{n!}}\,_{1}F_{1}{\big (}-n,{\tfrac {1}{2}};x^{2}{\big )},\\H_{2n+1}(x)&=(-1)^{n}{\frac {(2n+1)!}{n!}}\,2x\,_{1}F_{1}{\big (}-n,{\tfrac {3}{2}};x^{2}{\big )},\end{aligned}}

where $1 F 1 (a, b; z) = M (a, b; z)$ is Kummer's confluent hypergeometric function.

Differential-operator representation

The probabilist's Hermite polynomials satisfy the identity

{\mathit {He}}_{n}(x)=e^{-{\frac {D^{2}}{2}}}x^{n},

where $D$ represents differentiation with respect to $x$ , 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 $x n$ can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients of $H n$ that can be used to quickly compute these polynomials.

Since the formal expression for the Weierstrass transform $W$ is $e D 2$ , we see that the Weierstrass transform of $(\sqrt 2) n He n (x / \sqrt 2)$ is $x n$ . Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series.

The existence of some formal power series $g (D)$ with nonzero constant coefficient, such that $He n (x) = g (D) x n$ , 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.

Contour-integral representation

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

{\begin{aligned}{\mathit {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.

Generalizations

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

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

which has expected value 0 and variance 1.

Scaling, one may analogously speak of generalized Hermite polynomials[9]

{\mathit {He}}_{n}^{[\alpha ]}(x)

of variance $α$ , where $α$ is any positive number. These are then orthogonal with respect to the normal probability distribution whose density function is

(2\pi \alpha )^{-{\frac {1}{2}}}e^{-{\frac {x^{2}}{2\alpha }}}.

They are given by

{\mathit {He}}_{n}^{[\alpha ]}(x)=\alpha ^{\frac {n}{2}}{\mathit {He}}_{n}\left({\frac {x}{\sqrt {\alpha }}}\right)=\left({\frac {\alpha }{2}}\right)^{\frac {n}{2}}H_{n}\left({\frac {x}{\sqrt {2\alpha }}}\right)=e^{-{\frac {\alpha D^{2}}{2}}}\left(x^{n}\right).

Now, if

{\mathit {He}}_{n}^{[\alpha ]}(x)=\sum _{k=0}^{n}h_{n,k}^{[\alpha ]}x^{k},

then the polynomial sequence whose $n$ th term is

\left({\mathit {He}}_{n}^{[\alpha ]}\circ {\mathit {He}}^{[\beta ]}\right)(x)\equiv \sum _{k=0}^{n}h_{n,k}^{[\alpha ]}\,{\mathit {He}}_{k}^{[\beta ]}(x)

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

\left({\mathit {He}}_{n}^{[\alpha ]}\circ {\mathit {He}}^{[\beta ]}\right)(x)={\mathit {He}}_{n}^{[\alpha +\beta ]}(x)

and

{\mathit {He}}_{n}^{[\alpha +\beta ]}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}{\mathit {He}}_{k}^{[\alpha ]}(x){\mathit {He}}_{n-k}^{[\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 $α = β = 1 / 2$ , has already been encountered in the above section on #Recursion relations.)

"Negative variance"

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

{\mathit {He}}_{n}^{[-\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 $α > 0$ , the coefficients of ${\mathit {He}}_{n}^{[-\alpha ]}(x)$ are just the absolute values of the corresponding coefficients of ${\mathit {He}}_{n}^{[\alpha ]}(x)$ .

These arise as moments of normal probability distributions: The $n$ th moment of the normal distribution with expected value $μ$ and variance $σ 2$ is

E[X^{n}]={\mathit {He}}_{n}^{[-\sigma ^{2}]}(\mu ),

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

\sum _{k=0}^{n}{\binom {n}{k}}{\mathit {He}}_{k}^{[\alpha ]}(x){\mathit {He}}_{n-k}^{[-\alpha ]}(y)={\mathit {He}}_{n}^{[0]}(x+y)=(x+y)^{n}.

Applications

Hermite functions

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

\psi _{n}(x)=\left(2^{n}n!{\sqrt {\pi }}\right)^{-{\frac {1}{2}}}e^{-{\frac {x^{2}}{2}}}H_{n}(x)=(-1)^{n}\left(2^{n}n!{\sqrt {\pi }}\right)^{-{\frac {1}{2}}}e^{\frac {x^{2}}{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}.

Thus,

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

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

\int _{-\infty }^{\infty }\psi _{n}(x)\psi _{m}(x)\,dx=\delta _{nm},

and they form an orthonormal basis of $L 2 (R)$ . This fact is equivalent to the corresponding statement for Hermite polynomials (see above).

The Hermite functions are closely related to the Whittaker function (Whittaker & Watson 1996) $D n (z)$ :

D_{n}(z)=\left(n!{\sqrt {\pi }}\right)^{\frac {1}{2}}\psi _{n}\left({\frac {z}{\sqrt {2}}}\right)=(-1)^{n}e^{\frac {z^{2}}{4}}{\frac {d^{n}}{dz^{n}}}e^{\frac {-z^{2}}{2}}

and thereby to other parabolic cylinder functions.

The Hermite functions satisfy the differential equation

\psi _{n}''(x)+\left(2n+1-x^{2}\right)\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.

Hermite functions: 0 (blue, solid), 1 (orange, dashed), 2 (green, dot-dashed), 3 (red, dotted), 4 (purple, solid), and 5 (brown, dashed)

{\begin{aligned}\psi _{0}(x)&=\pi ^{-{\frac {1}{4}}}\,e^{-{\frac {1}{2}}x^{2}},\\\psi _{1}(x)&={\sqrt {2}}\,\pi ^{-{\frac {1}{4}}}\,x\,e^{-{\frac {1}{2}}x^{2}},\\\psi _{2}(x)&=\left({\sqrt {2}}\,\pi ^{\frac {1}{4}}\right)^{-1}\,\left(2x^{2}-1\right)\,e^{-{\frac {1}{2}}x^{2}},\\\psi _{3}(x)&=\left({\sqrt {3}}\,\pi ^{\frac {1}{4}}\right)^{-1}\,\left(2x^{3}-3x\right)\,e^{-{\frac {1}{2}}x^{2}},\\\psi _{4}(x)&=\left(2{\sqrt {6}}\,\pi ^{\frac {1}{4}}\right)^{-1}\,\left(4x^{4}-12x^{2}+3\right)\,e^{-{\frac {1}{2}}x^{2}},\\\psi _{5}(x)&=\left(2{\sqrt {15}}\,\pi ^{\frac {1}{4}}\right)^{-1}\,\left(4x^{5}-20x^{3}+15x\right)\,e^{-{\frac {1}{2}}x^{2}}.\end{aligned}}

Hermite functions: 0 (blue, solid), 2 (orange, dashed), 4 (green, dot-dashed), and 50 (red, solid)

Recursion relation

Following recursion relations of Hermite polynomials, the Hermite functions obey

\psi _{n}'(x)={\sqrt {\frac {n}{2}}}\,\psi _{n-1}(x)-{\sqrt {\frac {n+1}{2}}}\psi _{n+1}(x)

and

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

Extending the first relation to the arbitrary $m$ th derivatives for any positive integer $m$ leads to

\psi _{n}^{(m)}(x)=\sum _{k=0}^{m}{\binom {m}{k}}(-1)^{k}2^{\frac {m-k}{2}}{\sqrt {\frac {n!}{(n-m+k)!}}}\psi _{n-m+k}(x){\mathit {He}}_{k}(x).

This formula can be used in connection with the recurrence relations for $He n$ and $ψ n$ to calculate any derivative of the Hermite functions efficiently.

Cramér's inequality

For real $x$ , the Hermite functions satisfy the following bound due to Harald Cramér[10][11] and Jack Indritz:[12]

{\bigl |}\psi _{n}(x){\bigr |}\leq \pi ^{-{\frac {1}{4}}}.

Hermite functions as eigenfunctions of the Fourier transform

The Hermite functions $ψ n (x)$ are a set of eigenfunctions of the continuous Fourier transform F. To see this, take the physicist's version of the generating function and multiply by $e - 1 / 2 x 2$ . This gives

e^{-{\frac {1}{2}}x^{2}+2xt-t^{2}}=\sum _{n=0}^{\infty }e^{-{\frac {1}{2}}x^{2}}H_{n}(x){\frac {t^{n}}{n!}}.

The Fourier transform of the left side is given by

{\begin{aligned}{\mathcal {F}}\left\{e^{-{\frac {1}{2}}x^{2}+2xt-t^{2}}\right\}(k)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }e^{-ixk}e^{-{\frac {1}{2}}x^{2}+2xt-t^{2}}\,dx\\&=e^{-{\frac {1}{2}}k^{2}-2kit+t^{2}}\\&=\sum _{n=0}^{\infty }e^{-{\frac {1}{2}}k^{2}}H_{n}(k){\frac {(-it)^{n}}{n!}}.\end{aligned}}

The Fourier transform of the right side is given by

{\mathcal {F}}\left\{\sum _{n=0}^{\infty }e^{-{\frac {1}{2}}x^{2}}H_{n}(x){\frac {t^{n}}{n!}}\right\}=\sum _{n=0}^{\infty }{\mathcal {F}}\left\{e^{-{\frac {1}{2}}x^{2}}H_{n}(x)\right\}{\frac {t^{n}}{n!}}.

Equating like powers of $t$ in the transformed versions of the left and right sides finally yields

{\mathcal {F}}\left\{e^{-{\frac {1}{2}}x^{2}}H_{n}(x)\right\}=(-i)^{n}e^{-{\frac {1}{2}}k^{2}}H_{n}(k).

The Hermite functions $ψ n (x)$ are thus an orthonormal basis of $L 2 (R)$ , which diagonalizes the Fourier transform operator.[13]

Wigner distributions of Hermite functions

The Wigner distribution function of the $n$ th-order Hermite function is related to the $n$ th-order Laguerre polynomial. The Laguerre polynomials are

L_{n}(x):=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{k!}}x^{k},

leading to the oscillator Laguerre functions

l_{n}(x):=e^{-{\frac {x}{2}}}L_{n}(x).

For all natural integers $n$ , it is straightforward to see[14] that

W_{\psi _{n}}(t,f)=(-1)^{n}l_{n}{\big (}4\pi (t^{2}+f^{2}){\big )},

where the Wigner distribution of a function $x \in L 2 (R, C)$ is defined as

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

This is a fundamental result for the quantum harmonic oscillator discovered by Hip Groenewold in 1946 in his PhD thesis.[15] It is the standard paradigm of quantum mechanics in phase space.

There are further relations between the two families of polynomials.

Combinatorial interpretation of coefficients

In the Hermite polynomial $He n (x)$ of variance 1, the absolute value of the coefficient of $x k$ is the number of (unordered) partitions of an $n$ -element set into $k$ singletons and $n - k / 2$ (unordered) pairs. Equivalently, it is the number of involutions of an $n$ -element set with precisely $k$ fixed points, or in other words, the number of matchings in the complete graph on $n$ vertices that leave $k$ 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,... (sequence A000085 in the OEIS).

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

{\mathit {He}}_{n}(x)=B_{n}(x,-1,0,\ldots ,0),

where $x i = 0$ for all $i > 2$ .

These numbers may also be expressed as a special value of the Hermite polynomials:[16]

T(n)={\frac {{\mathit {He}}_{n}(i)}{i^{n}}}.

Completeness relation

The Christoffel–Darboux formula for Hermite polynomials reads

\sum _{k=0}^{n}{\frac {H_{k}(x)H_{k}(y)}{k!2^{k}}}={\frac {1}{n!2^{n+1}}}\,{\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:

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

where $δ$ is the Dirac delta function, $ψ n$ the Hermite functions, and $δ (x - y)$ represents the Lebesgue measure on the line $y = x$ in $R 2$ , normalized so that its projection on the horizontal axis is the usual Lebesgue measure.

This distributional identity follows Wiener (1958) by taking $u \to 1$ in Mehler's formula, valid when $-1 < u < 1$ :

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

which is often stated equivalently as a separable kernel,[17][18]

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

The function $(x, y) \to E (x, y; u)$ is the bivariate Gaussian probability density on $R 2$ , which is, when $u$ is close to 1, very concentrated around the line $y = x$ , and very spread out on that line. It follows that

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

when $f$ and $g$ are continuous and compactly supported.

This yields that $f$ can be expressed in Hermite functions as the sum of a series of vectors in $L 2 (R)$ , namely,

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

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

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

The Hermite polynomial is then represented as

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

With this representation for $H n (x)$ and $H n (y)$ , it is evident that

{\begin{aligned}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{aligned}}

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

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

Notes

Laplace 1810 (online).
Laplace, P.-S. (1812), Théorie analytique des probabilités [Analytic Probability Theory], vol. 2, pp. 194–203 Collected in Œuvres complètes VII.
Chebyshev, P. L. (1859). "Sur le développement des fonctions à une seule variable" [On the development of single-variable functions]. Bull. Acad. Sci. St. Petersb. 1: 193–200. Collected in Œuvres I, 501–508.
Hermite, C. (1864). "Sur un nouveau développement en série de fonctions" [On a new development in function series]. C. R. Acad. Sci. Paris. 58: 93–100. Collected in Œuvres II, 293–303.
Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al. (2010) and Abramowitz & Stegun.
"18. Orthogonal Polynomials, Classical Orthogonal Polynomials, Sums". Digital Library of Mathematical Functions. National Institute of Standards and Technology. Retrieved 30 January 2015.
Abramowitz & Stegun 1983, p. 508–510, 13.6.38 and 13.5.16.
Szegő 1955, p. 201
Roman, Steven (1984), The Umbral Calculus, Pure and Applied Mathematics, vol. 111 (1st ed.), Academic Press, pp. 87–93, ISBN 978-0-12-594380-2
Erdélyi et al. 1955, p. 207.
Szegő 1955.
Indritz, Jack (1961), "An inequality for Hermite polynomials", Proceedings of the American Mathematical Society, 12 (6): 981–983, doi:10.1090/S0002-9939-1961-0132852-2, MR 0132852
In this case, we used the unitary version of the Fourier transform, so the eigenvalues are $(- i) n$ . 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.
Folland, G. B. (1989), Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122, Princeton University Press, ISBN 978-0-691-08528-9
Groenewold, H. J. (1946). "On the Principles of elementary quantum mechanics". Physica. 12 (7): 405–460. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4.
Banderier, Cyril; Bousquet-Mélou, Mireille; Denise, Alain; Flajolet, Philippe; Gardy, Danièle; Gouyou-Beauchamps, Dominique (2002), "Generating functions for generating trees", Discrete Mathematics, 246 (1–3): 29–55, arXiv:math/0411250, doi:10.1016/S0012-365X(01)00250-3, MR 1884885
Mehler, F. G. (1866), "Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung" [On the development of a function of arbitrarily many variables according to higher-order Laplace functions], Journal für die Reine und Angewandte Mathematik (in German) (66): 161–176, ISSN 0075-4102, ERAM 066.1720cj. See p. 174, eq. (18) and p. 173, eq. (13).
Erdélyi et al. 1955, p. 194, 10.13 (22).

References

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
Courant, Richard; Hilbert, David (1989) [1953], Methods of Mathematical Physics, vol. 1, Wiley-Interscience, ISBN 978-0-471-50447-4
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher transcendental functions (PDF), vol. II, McGraw-Hill, ISBN 978-0-07-019546-2
Fedoryuk, M.V. (2001) [1994], "Hermite function", Encyclopedia of Mathematics, EMS Press
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
Laplace, P. S. (1810), "Mémoire sur les intégrales définies et leur application aux probabilités, et spécialement a la recherche du milieu qu'il faut choisir entre les résultats des observations", Mémoires de l'Académie des Sciences: 279–347 Oeuvres complètes 12, pp.357-412, English translation.
Shohat, J.A.; Hille, Einar; Walsh, Joseph L. (1940), A bibliography on orthogonal polynomials, Bulletin of the National Research Council, vol. Number 103, Washington D.C.: National Academy of Sciences - 2000 references of Bibliography on Hermite polynomials.
Suetin, P. K. (2001) [1994], "Hermite polynomials", Encyclopedia of Mathematics, EMS Press
Szegő, Gábor (1955) [1939], Orthogonal Polynomials, Colloquium Publications, vol. 23 (4th ed.), American Mathematical Society, ISBN 978-0-8218-1023-1
Temme, Nico (1996), Special Functions: An Introduction to the Classical Functions of Mathematical Physics, New York: Wiley, ISBN 978-0-471-11313-3
Wiener, Norbert (1958) [1933], The Fourier Integral and Certain of its Applications (revised ed.), New York: Dover Publications, ISBN 0-486-60272-9
Whittaker, E. T.; Watson, G. N. (1996) [1927], A Course of Modern Analysis (4th ed.), London: Cambridge University Press, ISBN 978-0-521-58807-2

External links

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Weisstein, Eric W. "Hermite Polynomial". MathWorld.
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[1] Laplace 1810 (online).

[2] Laplace, P.-S. (1812), Théorie analytique des probabilités [Analytic Probability Theory], vol. 2, pp. 194–203 Collected in Œuvres complètes VII.

[3] Chebyshev, P. L. (1859). "Sur le développement des fonctions à une seule variable" [On the development of single-variable functions]. Bull. Acad. Sci. St. Petersb. 1: 193–200. Collected in Œuvres I, 501–508.

[4] Hermite, C. (1864). "Sur un nouveau développement en série de fonctions" [On a new development in function series]. C. R. Acad. Sci. Paris. 58: 93–100. Collected in Œuvres II, 293–303.

[5] Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al. (2010) and Abramowitz & Stegun.

[6] "18. Orthogonal Polynomials, Classical Orthogonal Polynomials, Sums". Digital Library of Mathematical Functions. National Institute of Standards and Technology. Retrieved 30 January 2015.

[7] Abramowitz & Stegun 1983, p. 508–510, 13.6.38 and 13.5.16.

[8] Szegő 1955, p. 201

[9] Roman, Steven (1984), The Umbral Calculus, Pure and Applied Mathematics, vol. 111 (1st ed.), Academic Press, pp. 87–93, ISBN 978-0-12-594380-2

[10] Erdélyi et al. 1955, p. 207.

[11] Szegő 1955.

[indritz-12] Indritz, Jack (1961), "An inequality for Hermite polynomials", Proceedings of the American Mathematical Society, 12 (6): 981–983, doi:10.1090/S0002-9939-1961-0132852-2, MR 0132852

[13] In this case, we used the unitary version of the Fourier transform, so the eigenvalues are $(- i) n$ . 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.

[14] Folland, G. B. (1989), Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122, Princeton University Press, ISBN 978-0-691-08528-9

[Groenewold1946-15] Groenewold, H. J. (1946). "On the Principles of elementary quantum mechanics". Physica. 12 (7): 405–460. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4.

[gfgt-16] Banderier, Cyril; Bousquet-Mélou, Mireille; Denise, Alain; Flajolet, Philippe; Gardy, Danièle; Gouyou-Beauchamps, Dominique (2002), "Generating functions for generating trees", Discrete Mathematics, 246 (1–3): 29–55, arXiv:math/0411250, doi:10.1016/S0012-365X(01)00250-3, MR 1884885

[17] Mehler, F. G. (1866), "Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung" [On the development of a function of arbitrarily many variables according to higher-order Laplace functions], Journal für die Reine und Angewandte Mathematik (in German) (66): 161–176, ISSN 0075-4102, ERAM 066.1720cj. See p. 174, eq. (18) and p. 173, eq. (13).

[18] Erdélyi et al. 1955, p. 194, 10.13 (22).

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