# Hermite polynomials

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Chebyshev–Hermite polynomials

Polynomials orthogonal on with the weight function . The standardized Hermite polynomials are defined by the Rodrigues formula

The most commonly used formulas are:

The first few Hermite polynomials are:

The polynomial satisfies the differential equation

The orthonormal Hermite polynomials are defined by

The Hermite polynomials with leading coefficient one have the form

Fourier series in Hermite polynomials in the interior of behave analogous to trigonometric Fourier series.

In mathematical statistics and probability theory one uses the Hermite polynomials corresponding to the weight function

The definition of Hermite polynomials is encountered in P. Laplace [1]. A detailed study of them was published by P.L. Chebyshev in 1859 (see [2]). Later, these polynomials were studied by Ch. Hermite . V.A. Steklov [4] proved that the set of them is dense in the space of square-summable functions with the weight on the whole real line.

#### References

 [1] P.S. Laplace, Mém. Cl. Sci. Math. Phys. Inst. France , 58 (1810) pp. 279–347 [2] P.L. Chebyshev, , Collected works , 2 , Moscow-Leningrad (1947) pp. 335–341 (In Russian) [3a] Ch. Hermite, C.R. Acad. Sci. Paris , 58 (1864) pp. 93–100 [3b] Ch. Hermite, C.R. Acad. Sci. Paris , 58 (1864) pp. 266–273 [4] V.A. Steklov, Izv. Akad. Nauk , 10 (1956) pp. 403–416 [5] P.K. Suetin, "Classical orthogonal polynomials" , Moscow (1979) (In Russian)