Hermite polynomials

Like the other classical orthogonal polynomials, the Hermite polynomials can be defined from several different starting points. Noting from the outset that there are two different standardizations in common use, one convenient method is as follows:

These equations have the form of a Rodrigues' formula and can also be written as,

The two definitions are not exactly identical; each is a rescaling of the other:

These are Hermite polynomial sequences of different variances; see the material on variances below.

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

In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see section on the Completeness relation below).

The probabilist's Hermite polynomials are solutions of the differential equation

The general solutions to the above second-order differential equations are in fact linear combinations of both Hermite polynomials and confluent hypergeometric functions of the first kind. For example, for the physicist's Hermite equation

The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation

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

It follows that the Hermite polynomials also satisfy the recurrence relation

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

The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials He are

The Hermite polynomials are given by the exponential generating function

The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices:

A better approximation, which accounts for the variation in frequency, is given by

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

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

The physicist's Hermite polynomials can be expressed as a special case of the parabolic cylinder functions:

From the generating-function representation above, we see that the Hermite polynomials have a representation in terms of a contour integral, as

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

One can define the Hermite functions (often called Hermite-Gaussian functions) from the physicist's polynomials:

Hermite functions: 0 (blue, solid), 1 (orange, dashed), 2 (green, dot-dashed), 3 (red, dotted), 4 (purple, solid), and 5 (brown, dashed)
Hermite functions: 0 (blue, solid), 2 (orange, dashed), 4 (green, dot-dashed), and 50 (red, solid)

Following recursion relations of Hermite polynomials, the Hermite functions obey

Equating like powers of t in the transformed versions of the left and right sides finally yields

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496,... (sequence in the OEIS).

This combinatorial interpretation can be related to complete exponential Bell polynomials as