# Hermite polynomials

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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

How to Cite This Entry:
Hermite polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_polynomials&oldid=14720
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article