Hermite differential equation

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.

The polynomials arise in:

Hermite polynomials were defined by Pierre-Simon Laplace in 1810 though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 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 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. The polynomials $He n$ are sometimes denoted by $H n$ , especially in probability theory, because

is the probability density function for the normal distribution with expected value 0 and standard deviation 1.

The $n$ th-order Hermite polynomial is a polynomial of degree $n$ . The probabilists' version $He n$ has leading coefficient 1, while the physicists' version $H n$ has leading coefficient $2 n$ .

$H n (x)$ and $He n (x)$ are $n$ th-degree polynomials for $n = 0, 1, 2, 3,...$ . These polynomials are orthogonal with respect to the weight function (measure)

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