Definition
There are two different ways of standardizing the Hermite polynomials:
(the "probabilists' Hermite polynomials"), and
(the "physicists' Hermite polynomials"). These two definitions are not exactly equivalent; either is a rescaling of the other, to wit
These are Hermite polynomial sequences of different variances; see the material on variances below.
The notation He and H is that used in the standard references Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al. (2010) and Abramowitz & Stegun. The polynomials Hen are sometimes denoted by Hn, especially in probability theory, because
is the probability density function for the normal distribution with expected value 0 and standard deviation 1.
The first eleven probabilists' Hermite polynomials are:
and the first eleven physicists' Hermite polynomials are:
Read more about this topic: Hermite Polynomials
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“... we all know the wags definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)