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
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