Hermite Polynomials Orthogonality Various Properties

(the "physicists' Hermite polynomials"). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. These are Hermite polynomial sequences of different variances; see the material on variances below.

Below, we follow the first convention. That convention is often preferred by probabilists because

1 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 measure

in which the inner product is given by the integral including a gaussian function

2 Various properties

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

(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. The existence of some formal power series g(D), with nonzero constant coefficient, such that H_n(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.

3 Generalization

The Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution

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

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

The last identity is expressed by saying that this parametrized family of polynomial sequences is a cross-sequence.

4 "Negative variance"

Since polynomial sequences form a groupIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G 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 σ² is

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

5 Eigenfunctions of the Fourier transform

The functions

are eigenfunctions of the Fourier transformThe Fourier transform named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes")., with eigenvalueIn linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value of a linear mapping A if there exists a nonzero vector x such that Ax &lambda x''. The vector x is called an eigenvector. In matrix theory, an elemens −iⁿ.

6 Combinatorial interpretation of the coefficients

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 n-member set into k singletons and (n − k)/2 (unordered) pairs.

7 Edgeworth series

Hermite polynomials arise in the theory of Edgeworth seriesThe Edgeworth series or Gram-Charlier A series named in honor of Francis Ysidro Edgeworth, are series that approximate a probability distribution in terms of its cumulants. Gram-Charlier A series The key idea of these expansions is to write the characteri.

<Hide>

Home > Hermite polynomials