Table of contents |
2 Orthogonality 3 Various properties 4 Generalization 5 "Negative variance" 6 Eigenfunctions of the Fourier transform 7 Combinatorial interpretation of the coefficients |
In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced "air MEET"), compose a polynomial sequence defined either by
Below, we follow the first convention. That convention is sometimes preferred by probabilists because
Definition
or sometimes by
which is not equivalent. These are Hermite polynomial sequences of different variances; see the material on variances below.
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:
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
The nth Hermite polynomial satisfies Hermite's differential equation:
If X is a random variable with a normal distribution with standard deviation 1 and expected value μ then
The Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution
Since polynomial sequences form a group under the operation of umbral composition, one may denote by
These arise as moments of normal probability distributions: The nth moment of the normal distribution with expected value μ and variance σ2 is
The functions
"Negative variance"
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).
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
are eigenfunctions of the Fourier transform, with eigenvalues
−in.