Hermite polynomial
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
Hermite polynomials were defined by Laplace (1810) though in scarcely recognizable form, and studied in detail by Chebyshev (1859). Chebyshev's work was overlooked and they were named later after Charles Hermite who wrote on the polynomials in 1864 describing them as new. They were consequently not new although in later 1865 papers Hermite was the first to define the multidimensional polynomials.
There are two different ways of standardizing the Hermite polynomials:
These two definitions are not exactly identical; each one is a rescaling of the other,
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:
Hn is a polynomial of degree n. The probabilists' version He has leading coefficient 1, while the physicists' version H has leading coefficient 2n.
Hn(x) and Hen(x) are nth-degree polynomials for n = 0, 1, 2, 3, .... These polynomials are orthogonal with respect to the weight function (measure)
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