# Hermite polynomials

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; in finite element methods as shape functions for beams; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator. They are also used in systems theory in connection with nonlinear operations on Gaussian noise. They were defined by Template:Harvtxt [1] though in scarcely recognizable form, and studied in detail by Chebyshev (1859).[2] Chebyshev's work was overlooked and they were named later after Charles Hermite who wrote on the polynomials in 1864 describing them as new.[3] They were consequently not new although in later 1865 papers Hermite was the first to define the multidimensional polynomials.

## Definition

There are two different ways of standardizing the Hermite polynomials:

• The "probabilists' Hermite polynomials" are given by
${\displaystyle {\mathit {He}}_{n}(x)=(-1)^{n}e^{\frac {x^{2}}{2}}{\frac {d^{n}}{dx^{n}}}e^{-{\frac {x^{2}}{2}}}=\left(x-{\frac {d}{dx}}\right)^{n}\cdot 1}$,
• while the "physicists' Hermite polynomials" are given by
${\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}=\left(2x-{\frac {d}{dx}}\right)^{n}\cdot 1}$.

These two definitions are not exactly identical; each one is a rescaling of the other,

${\displaystyle H_{n}(x)=2^{\tfrac {n}{2}}{\mathit {He}}_{n}({\sqrt {2}}\,x),\qquad {\mathit {He}}_{n}(x)=2^{-{\tfrac {n}{2}}}H_{n}\left({\frac {x}{\sqrt {2}}}\right).}$

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 Template:Harvs and Abramowitz & Stegun. The polynomials Hen are sometimes denoted by Hn, 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 probabilists' Hermite polynomials Hen(x).

The first eleven probabilists' Hermite polynomials are:

${\displaystyle {\mathit {He}}_{0}(x)=1\,}$
${\displaystyle {\mathit {He}}_{1}(x)=x\,}$
${\displaystyle {\mathit {He}}_{2}(x)=x^{2}-1\,}$
${\displaystyle {\mathit {He}}_{3}(x)=x^{3}-3x\,}$
${\displaystyle {\mathit {He}}_{4}(x)=x^{4}-6x^{2}+3\,}$
${\displaystyle {\mathit {He}}_{5}(x)=x^{5}-10x^{3}+15x\,}$
${\displaystyle {\mathit {He}}_{6}(x)=x^{6}-15x^{4}+45x^{2}-15\,}$
${\displaystyle {\mathit {He}}_{7}(x)=x^{7}-21x^{5}+105x^{3}-105x\,}$
${\displaystyle {\mathit {He}}_{8}(x)=x^{8}-28x^{6}+210x^{4}-420x^{2}+105\,}$
${\displaystyle {\mathit {He}}_{9}(x)=x^{9}-36x^{7}+378x^{5}-1260x^{3}+945x\,}$
${\displaystyle {\mathit {He}}_{10}(x)=x^{10}-45x^{8}+630x^{6}-3150x^{4}+4725x^{2}-945\,}$
The first six (physicists') Hermite polynomials Hn(x).

and the first eleven physicists' Hermite polynomials are:

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

## Properties

Hn is a polynomial of degree n. The probabilists' version He has leading coefficient 1, while the physicists' version H has leading coefficient 2n.

### Orthogonality

Hn(x) and Hen(x) are nth-degree polynomials for n = 0, 1, 2, 3, .... These polynomials are orthogonal with respect to the weight function (measure)

${\displaystyle w(x)=e^{-{\frac {x^{2}}{2}}}}$      (He)

or

${\displaystyle w(x)=e^{-x^{2}}}$      (H)

i.e., we have

${\displaystyle \int _{-\infty }^{\infty }H_{m}(x)H_{n}(x)\,w(x)\,\mathrm {d} x=0,\qquad m\neq n.}$

Furthermore,

${\displaystyle \int _{-\infty }^{\infty }{\mathit {He}}_{m}(x){\mathit {He}}_{n}(x)\,e^{-{\frac {x^{2}}{2}}}\,\mathrm {d} x={\sqrt {2\pi }}n!\delta _{nm}}$     (probabilists')

or

${\displaystyle \int _{-\infty }^{\infty }H_{m}(x)H_{n}(x)e^{-x^{2}}\,\mathrm {d} x={\sqrt {\pi }}2^{n}n!\delta _{nm}}$      (physicists').

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

### Completeness

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

${\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,w(x)\,\mathrm {d} x<\infty ~,}$

in which the inner product is given by the integral including the Gaussian weight function w(x) defined in the preceding section,

${\displaystyle \langle f,g\rangle =\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,w(x)\,\mathrm {d} x~.}$

An orthogonal basis for L2(Rw(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 ƒ ∈ L2(Rw(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 Template:Mvar satisfies

${\displaystyle \int _{-\infty }^{\infty }f(x)x^{n}e^{-x^{2}}\,\mathrm {d} x=0}$

for every Template:Mvar ≥ 0, then Template:Mvar = 0.

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

${\displaystyle F(z)=\int _{-\infty }^{\infty }f(x)e^{zx-x^{2}}\,\mathrm {d} x=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\int f(x)x^{n}e^{-x^{2}}\,\mathrm {d} x=0}$

vanishes identically. The fact then that F(it) = 0 for every Template:Mvar real means that the Fourier transform of f(x) exp(−x2) is 0, hence Template:Mvar 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(Rw(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 probabilists' Hermite polynomials are solutions of the differential equation

${\displaystyle \left(e^{-{\frac {x^{2}}{2}}}u'\right)'+\lambda e^{-{\frac {1}{2}}x^{2}}u=0}$

where Template:Mvar is a constant, with the boundary conditions that Template:Mvar should be polynomially bounded at infinity. With these boundary conditions, the equation has solutions only if λ is a non-negative integer, and up to an overall scaling, the solution is uniquely given by u(x) = Heλ(x).

Rewriting the differential equation as an eigenvalue problem

${\displaystyle L[u]=u''-xu'=-\lambda u~,}$

solutions are the eigenfunctions of the differential operator Template:Mvar. 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 solutions are the physicists' Hermite polynomials.

With more general boundary conditions, the Hermite polynomials can be generalized to obtain more general analytic functions Heλ(z) for Template:Mvar a complex index. An explicit formula can be given in terms of a contour integral Template:Harv.

### Recursion relation

The sequence of Hermite polynomials also satisfies the recursion

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

Individual coefficients are related by the following recursion formula:

${\displaystyle a_{n+1,k}=a_{n,k-1}-na_{n-1,k}\ \ k>0}$
${\displaystyle a_{n+1,k}=-na_{n-1,k}\ \ k=0}$

and Template:Mvar[0,0]=1, Template:Mvar[1,0]=0, Template:Mvar[1,1]=1.

${\displaystyle H_{n+1}(x)=2xH_{n}(x)-H_{n}'(x).}$   (physicists')

Individual coefficients are related by the following recursion formula:

${\displaystyle a_{n+1,k}=2a_{n,k-1}-2na_{n-1,k}\ \ k>0}$
${\displaystyle a_{n+1,k}=-2na_{n-1,k}\ \ k=0}$

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

${\displaystyle {\mathit {He}}_{n}'(x)=n{\mathit {He}}_{n-1}(x),\,\!}$    (probabilists')
${\displaystyle H_{n}'(x)=2nH_{n-1}(x),\,\!}$    (physicists')

or, equivalently, by Taylor expanding,

${\displaystyle {\mathit {He}}_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}x^{n-k}{\mathit {He}}_{k}(y)}$    (probabilists')
${\displaystyle H_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}H_{k}(x)(2y)^{(n-k)}=2^{-{\frac {n}{2}}}\cdot \sum _{k=0}^{n}{n \choose k}H_{n-k}\left(x{\sqrt {2}}\right)H_{k}\left(y{\sqrt {2}}\right).}$    (physicists')

In consequence, for the m-th derivatives the following relations hold:

${\displaystyle {\mathit {He}}_{n}^{(m)}(x)={\frac {n!}{(n-m)!}}\cdot {\mathit {He}}_{n-m}(x)=m!\cdot {n \choose m}\cdot {\mathit {He}}_{n-m}(x),\,\!}$    (probabilists')
${\displaystyle {\mathit {H}}_{n}^{(m)}(x)=2^{m}\cdot {\frac {n!}{(n-m)!}}\cdot {\mathit {H}}_{n-m}(x)=2^{m}\cdot m!\cdot {n \choose m}\cdot {\mathit {H}}_{n-m}(x).\,\!}$    (physicists')

It follows that the Hermite polynomials also satisfy the recurrence relation

${\displaystyle {\mathit {He}}_{n+1}(x)=x{\mathit {He}}_{n}(x)-n{\mathit {He}}_{n-1}(x),}$   (probabilists')
${\displaystyle H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x).\,\!}$    (physicists')

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

${\displaystyle {\mathit {He}}_{n}(x)^{2}-{\mathit {He}}_{n-1}(x){\mathit {He}}_{n+1}(x)=(n-1)!\cdot \sum _{i=0}^{n-1}{\frac {2^{n-i}}{i!}}{\mathit {He}}_{i}(x)^{2}>0.}$

Moreover, the following multiplication theorem holds:

${\displaystyle {\mathit {H}}_{n}(\gamma x)=\sum _{i=0}^{\lfloor {\tfrac {n}{2}}\rfloor }\gamma ^{n-2i}(\gamma ^{2}-1)^{i}{n \choose 2i}{\frac {(2i)!}{i!}}{\mathit {H}}_{n-2i}(x).}$

### Explicit expression

The physicists' Hermite polynomials can be written explicitly as

${\displaystyle H_{n}(x)=n!\sum _{\ell =0}^{\tfrac {n}{2}}{\frac {(-1)^{{\tfrac {n}{2}}-\ell }}{(2\ell )!\left({\tfrac {n}{2}}-\ell \right)!}}(2x)^{2\ell }}$

for even value of Template:Mvar and

${\displaystyle H_{n}(x)=n!\sum _{\ell =0}^{\frac {n-1}{2}}{\frac {(-1)^{{\frac {n-1}{2}}-\ell }}{(2\ell +1)!\left({\frac {n-1}{2}}-\ell \right)!}}(2x)^{2\ell +1}}$

for odd values of Template:Mvar.

These two equations may be combined into one using the floor function,

${\displaystyle H_{n}(x)=n!\sum _{m=0}^{\lfloor {\tfrac {n}{2}}\rfloor }{\frac {(-1)^{m}}{m!(n-2m)!}}(2x)^{n-2m}.}$

The probabilists' Hermite polynomials He have similar formulas, which may be obtained from these by replacing the power of 2Template:Mvar with the corresponding power of (√2)x, and multiplying the entire sum by 2n/2.

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

### Generating function

The Hermite polynomials are given by the exponential generating function

${\displaystyle \exp(xt-{\frac {t^{2}}{2}})=\sum _{n=0}^{\infty }{\mathit {He}}_{n}(x){\frac {t^{n}}{n!}}}$    (probabilists')

${\displaystyle \exp(2xt-t^{2})=\sum _{n=0}^{\infty }H_{n}(x){\frac {t^{n}}{n!}}\,\!}$    (physicists').

This equality is valid for all x, t complex, and can be obtained by writing the Taylor expansion at x of the entire function z → exp(−z2) (in physicist's case). One can also derive the (physicist's) generating function by using Cauchy's Integral Formula to write the Hermite polynomials as

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

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.

### Expected values

If X is a random variable with a normal distribution with standard deviation 1 and expected value Template:Mvar, then

${\displaystyle E({\mathit {He}}_{n}(X))=\mu ^{n}.}$    (probabilists')

The moments of the standard normal may be read off directly from the relation for even indices

${\displaystyle E(X^{2n})=(-1)^{n}{\mathit {He}}_{2n}(0)=(2n-1)!!}$

where ${\displaystyle (2n-1)!!}$ is the double factorial. Note that the above expression is a special case of the representation of the probabilists' Hermite polynomials as moments

${\displaystyle {\mathit {He}}_{n}(x)=\int _{-\infty }^{\infty }(x+iy)^{n}e^{-{\frac {y^{2}}{2}}}\,\mathrm {d} y/{\sqrt {2\pi }}.}$

### Asymptotic expansion

Asymptotically, as n → ∞, the expansion

${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim {\frac {2^{n}}{\sqrt {\pi }}}\Gamma \left({\frac {n+1}{2}}\right)\cos \left(x{\sqrt {2n}}-n{\frac {\pi }{2}}\right)}$   (physicist[4])

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

${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim {\frac {2^{n}}{\sqrt {\pi }}}\Gamma \left({\frac {n+1}{2}}\right)\cos \left(x{\sqrt {2n}}-n{\frac {\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n}}\right)^{-{\frac {1}{4}}}={\frac {2\Gamma \left(n\right)}{\Gamma \left({\frac {n}{2}}\right)}}\cos \left(x{\sqrt {2n}}-n{\frac {\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n}}\right)^{-{\frac {1}{4}}}}$

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

${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim \left({\frac {2n}{e}}\right)^{\frac {n}{2}}{\sqrt {2}}\cos \left(x{\sqrt {2n}}-n{\frac {\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n}}\right)^{-{\frac {1}{4}}}}$

This expansion is needed to resolve the wave-function of a quantum harmonic oscillator such that it agrees with the classical approximation in the limit of the correspondence principle.

A finer approximation,[5] which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution

${\displaystyle x={\sqrt {2n+1}}\cos(\phi ),\qquad 0<\epsilon \leq \phi \leq \pi -\epsilon ~,}$

with which one has the uniform approximation

${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=2^{{\tfrac {n}{2}}+{\frac {1}{4}}}{\sqrt {n!}}(\pi n)^{-{\frac {1}{4}}}(\sin \phi )^{-{\tfrac {1}{2}}}\cdot \left[\sin \left(\left({\frac {n}{2}}+{\frac {1}{4}}\right)\left(\sin(2\phi )-2\phi \right)+{\frac {3\pi }{4}}\right)+O(n^{-1})\right]~.}$

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

${\displaystyle x={\sqrt {2n+1}}\cosh(\phi ),\qquad 0<\epsilon \leq \phi \leq \omega <\infty ~,}$

then

${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=2^{{\tfrac {n}{2}}-{\frac {3}{4}}}{\sqrt {n!}}(\pi n)^{-{\frac {1}{4}}}(\sinh \phi )^{-{\frac {1}{2}}}\cdot \exp \left(\left({\frac {n}{2}}+{\frac {1}{4}}\right)\left(2\phi -\sinh(2\phi )\right)\right)\left[1+O(n^{-1})\right],}$

while for

${\displaystyle x={\sqrt {2n+1}}-2^{-{\frac {1}{2}}}3^{-1/3}n^{-1/6}t}$

with Template:Mvar complex and bounded, then

${\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=\pi ^{\frac {1}{4}}2^{{\tfrac {n}{2}}+{\frac {1}{4}}}{\sqrt {n!}}n^{-1/12}\left[\mathrm {Ai} (-3^{-1/3}t)+O(n^{-2/3})\right]}$

where Ai(·) is the Airy function of the first kind.

### Special Values

The Hermite polynomials evaluated at zero argument ${\displaystyle H_{n}(0)}$ are called Hermite numbers.

${\displaystyle H_{n}(0)={\begin{cases}0,&{\mbox{if }}n{\mbox{ is odd}}\\(-1)^{\tfrac {n}{2}}2^{\tfrac {n}{2}}(n-1)!!,&{\mbox{if }}n{\mbox{ is even}}\end{cases}}}$

which satisfy the recursion relation ${\displaystyle H_{n}(0)=-2(n-1)H_{n-2}(0)}$. In terms of the probabilist's polynomials this translates to

${\displaystyle He_{n}(0)={\begin{cases}0,&{\mbox{if }}n{\mbox{ is odd}}\\(-1)^{\tfrac {n}{2}}(n-1)!!&{\mbox{if }}n{\mbox{ is even}}.\end{cases}}}$

## Relations to other functions

### Laguerre polynomials

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

${\displaystyle H_{2n}(x)=(-4)^{n}\,n!\,L_{n}^{(-{\frac {1}{2}})}(x^{2})=4^{n}\,n!\sum _{i=0}^{n}(-1)^{n-i}{n-{\frac {1}{2}} \choose n-i}{\frac {x^{2i}}{i!}}}$    (physicists')
${\displaystyle H_{2n+1}(x)=2(-4)^{n}\,n!\,x\,L_{n}^{({\frac {1}{2}})}(x^{2})=2\cdot 4^{n}\,n!\sum _{i=0}^{n}(-1)^{n-i}{n+{\frac {1}{2}} \choose n-i}{\frac {x^{2i+1}}{i!}}}$    (physicists')

### Relation to confluent hypergeometric functions

The Hermite polynomials can be expressed as a special case of the parabolic cylinder functions.

${\displaystyle H_{n}(x)=2^{n}\,U\left(-{\frac {n}{2}},{\frac {1}{2}},x^{2}\right)}$    (physicists')
${\displaystyle H_{2n}(x)=(-1)^{n}\,{\frac {(2n)!}{n!}}\,_{1}F_{1}\left(-n,{\frac {1}{2}};x^{2}\right)}$    (physicists')
${\displaystyle H_{2n+1}(x)=(-1)^{n}\,{\frac {(2n+1)!}{n!}}\,2x\,_{1}F_{1}\left(-n,{\frac {3}{2}};x^{2}\right)}$    (physicists')

## Differential operator representation

The probabilists' Hermite polynomials satisfy the identity

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

where Template:Mvar represents differentiation with respect to Template:Mvar, 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 xn can be written down explicitly, this differential operator representation gives rise to a concrete formula for the coefficients of Hn that can be used to quickly compute these polynomials.

Since the formal expression for the Weierstrass transform Template:Mvar is eD2, we see that the Weierstrass transform of (√2)nHen(x/√2) is xn. 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 Hen(x) = g(D) xn, is another equivalent to the statement that these polynomials form an Appell sequence−−cf. [[Weierstrass transform#The inverse|Template:Mvar]]. 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

${\displaystyle {\mathit {He}}_{n}(x)={\frac {n!}{2\pi i}}\oint {\frac {e^{tx-{\frac {t^{2}}{2}}}}{t^{n+1}}}\,dt}$    (probabilists')
${\displaystyle H_{n}(x)={\frac {n!}{2\pi i}}\oint {\frac {e^{2tx-t^{2}}}{t^{n+1}}}\,dt}$    (physicists')

with the contour encircling the origin.

## Generalizations

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

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

which has expected value 0 and variance 1.

Scaling, one may analogously speak of generalized Hermite polynomials

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

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

${\displaystyle (2\pi \alpha )^{-{\frac {1}{2}}}e^{-x^{2}/(2\alpha )}~.}$

They are given by

${\displaystyle {\mathit {He}}_{n}^{[\alpha ]}(x)=\alpha ^{\tfrac {n}{2}}H\!e_{n}^{[1]}\left({\frac {x}{\sqrt {\alpha }}}\right)=\left({\frac {\alpha }{2}}\right)^{\tfrac {n}{2}}H_{n}\left({\frac {x}{\sqrt {2\alpha }}}\right)=e^{-\alpha D^{2}/2}x^{n}.}$

In particular, the physicists' Hermite polynomials are thus

${\displaystyle H_{n}(x)=2^{n}{\mathit {He}}_{n}^{[{\frac {1}{2}}]}(x).}$

Now, if

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

then the polynomial sequence whose Template:Mvarth term is

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

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

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

and

${\displaystyle {\mathit {He}}_{n}^{[\alpha +\beta ]}(x+y)=\sum _{k=0}^{n}{n \choose k}{\mathit {He}}_{k}^{[\alpha ]}(x){\mathit {He}}_{n-k}^{[\beta ]}(y)~.}$

The last identity is expressed by saying that this parameterized family of polynomial sequences is a cross-sequence. (See the above section on Appel sequences and on the #Differential operator representation, which leads to a ready derivation of it. This binomial type identity, for Template:Mvar = Template:Mvar = 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

${\displaystyle {\mathit {He}}_{n}^{[-\alpha ]}(x)\,\!}$

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 Template:Mvar > 0, the coefficients of Hen[−α](x) are just the absolute values of the corresponding coefficients of Hen[α](x).

These arise as moments of normal probability distributions: The n-th moment of the normal distribution with expected value Template:Mvar and variance Template:Mvar2 is

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

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

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

## Applications

### Hermite functions

One can define the Hermite functions from the physicists' polynomials:

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

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

${\displaystyle \int _{-\infty }^{\infty }\psi _{n}(x)\psi _{m}(x)\,{\mathrm {d} }x=\delta _{n\,m}\,}$

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

The Hermite functions are closely related to the Whittaker function (Whittaker and Watson, 1962) Dn(z),

${\displaystyle D_{n}(z)=(n!{\sqrt {\pi }})^{\frac {1}{2}}\psi _{n}(z/{\sqrt {2}})=\pi ^{-{\frac {1}{4}}}{\sqrt {2}}e^{z^{2}/4}{\frac {d^{n}}{dz^{n}}}e^{-z^{2}}~,}$

and thereby to other parabolic cylinder functions.

The Hermite functions satisfy the differential equation,

${\displaystyle \psi _{n}''(x)+(2n+1-x^{2})\psi _{n}(x)=0.}$

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

Hermite functions 0 (black), 1 (red), 2 (blue), 3 (yellow), 4 (green), and 5 (magenta).
${\displaystyle \psi _{0}(x)=\pi ^{-{\frac {1}{4}}}\,e^{-{\frac {1}{2}}x^{2}}}$
${\displaystyle \psi _{1}(x)={\sqrt {2}}\,\pi ^{-{\frac {1}{4}}}\,x\,e^{-{\frac {1}{2}}x^{2}}}$
${\displaystyle \psi _{2}(x)=({\sqrt {2}}\,\pi ^{\frac {1}{4}})^{-1}\,(2x^{2}-1)\,e^{-{\frac {1}{2}}x^{2}}}$
${\displaystyle \psi _{3}(x)=({\sqrt {3}}\,\pi ^{\frac {1}{4}})^{-1}\,(2x^{3}-3x)\,e^{-{\frac {1}{2}}x^{2}}}$
${\displaystyle \psi _{4}(x)=(2{\sqrt {6}}\,\pi ^{\frac {1}{4}})^{-1}\,(4x^{4}-12x^{2}+3)\,e^{-{\frac {1}{2}}x^{2}}}$
${\displaystyle \psi _{5}(x)=(2{\sqrt {15}}\,\pi ^{\frac {1}{4}})^{-1}\,(4x^{5}-20x^{3}+15x)\,e^{-{\frac {1}{2}}x^{2}}}$
Hermite functions 0 (black), 2 (blue), 4 (green), and 50 (magenta).

### Recursion relation

Following recursion relations of Hermite polynomials, the Hermite functions obey

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

as well as

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

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

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

This formula can be used in connection with the recurrence relations for Hen and ψn to calculate any derivative of the Hermite functions efficiently.

### Cramér's inequality

The Hermite functions satisfy the following bound due to Harald Cramér[6][7]

${\displaystyle |\psi _{n}(x)|\leq K\pi ^{-{\frac {1}{4}}}}$

for Template:Mvar real, where the constant Template:Mvar is less than 1.086435.

### Hermite functions as eigenfunctions of the Fourier transform

The Hermite functions ψn(x) are a set of eigenfunctions of the continuous Fourier transform. To see this, take the physicist's version of the generating function and multiply by exp(−x2/2). This gives