Hermite polynomials
In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced "air MEET"), compose a polynomial sequence defined either by

or sometimes by

which is
not equivalent. These are Hermite polynomial sequences of different variances; see the material on variances below.
Below, we follow the first convention. That convention is sometimes preferred by probabilists because

is the
probability density function for the
normal distribution with
expected value 0 and
standard deviation 1. The other convention is often followed by physicists.
The first several Hermite polynomials are:






Orthogonality
The nth function in this list is an nthdegree polynomial for n = 0, 1, 2, 3, ....
These polynomials are orthogonal with respect to the weight

i.e., we have

They form an orthogonal basis of the
Hilbert space of functions satisfying

in which the inner product is given by
Various properties
The nth Hermite polynomial satisfies Hermite's differential equation:

The sequence of Hermite polynomials also satisfies the recursion

The Hermite polynomials constitute an
Appell sequence, i.e., they are a polynomial sequence satisfying the identity

or equivalently,

(the equivalence of these last two identities may not be obvious, but its proof is a routine exercise).
The Hermite polynomials satisfy the identity

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. Indeed, the existence of some formal power series
g(
D), with nonzero constant coefficient, such that
H_{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.
If X is a random variable with a normal distribution with standard deviation 1 and expected value μ then
Generalization
The Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution

which has expected value 0 and variance 1. One may speak of Hermite polynomials

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

They are given by

If

then the polynomial sequence whose nth term is

is the umbral composition of the two polynomial sequences, and it can be shown to satisfy the identities

and

The last identity is expressed by saying that this parametrized family of polynomial sequences is a crosssequence.
Since polynomial sequences form a group under the operation of umbral composition, one may denote by

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
H_{n}^{[−α]}(
x) are just the absolute values of the corresponding coefficients of
H_{n}^{[α]}(
x).
These arise as moments of normal probability distributions: The nth moment of the normal distribution with expected value μ and variance σ^{2} is

where
X is a random variable with the specified normal distribution. A special case of the crosssequence identity then says that
Eigenfunctions of the Fourier transform
The functions
are eigenfunctions of the Fourier transform, with eigenvalues
−
i^{n}.
In the Hermite polynomial H_{n}(x) of variance 1, the absolute value of the coefficient of x^{k} is the number of (unordered) partitions of an nmember set into k singletons and (n − k)/2 (unordered) pairs.