Hermite Polynomials - Generalizations

Generalizations

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

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 whose density function is

They are given by

In particular, the physicists' Hermite polynomials are

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 parameterized family of polynomial sequences is a cross-sequence.