Hermite polynomials
Definition
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 nth-degree 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 Hn(x) = g(D)xn 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 cross-sequence.
"Negative variance"
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 Hn[−α](x) are just the absolute values of the corresponding coefficients of Hn[α](x).
These arise as moments of normal probability distributions: The nth moment of the normal distribution with expected value μ and variance σ2 is
where X is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that
Eigenfunctions of the Fourier transform
The functions
are eigenfunctions of the Fourier transform, with eigenvalues
−in.
Combinatorial interpretation of the coefficients
In the Hermite polynomial Hn(x) of variance 1, the absolute value of the coefficient of xk is the number of (unordered) partitions of an n-member set into k singletons and (n − k)/2 (unordered) pairs.
Referenced By
Charles Hermite | Hermite | Hermite Spline | Ladder operator | List of mathematical topics (G-I) | List of mathematical topics (G-Z) | List of polynomial topics | Monic polynomial | Orthogonal functions | Orthogonal polynomials | Polynomial | Polynomial ring | Polynomial sequence | Quantum harmonic oscillator | Sheffer sequence
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