Hermite Polynomials - Definition

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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