This article is about the family of orthogonal polynomials on the real line. For polynomial interpolation on a segment using derivatives, see Hermite interpolation. For integral transform of Hermite polynomials, see Hermite transform.
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
signal processing as Hermitian wavelets for wavelet transform analysis
probability, such as the Edgeworth series, as well as in connection with Brownian motion;
combinatorics, as an example of an Appell sequence, obeying the umbral calculus;
numerical analysis as Gaussian quadrature;
physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term is present);
systems theory in connection with nonlinear operations on Gaussian noise.
random matrix theory in Gaussian ensembles.
Hermite polynomials were defined by Pierre-Simon Laplace in 1810,[1][2] though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859.[3] Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new.[4] They were consequently not new, although Hermite was the first to define the multidimensional polynomials.
^Laplace (1811). "Mémoire sur les intégrales définies et leur application aux probabilités, et spécialement a la recherche du milieu qu'il faut choisir entre les resultats des observations" [Memoire on definite integrals and their application to probabilities, and especially to the search for the mean which must be chosen among the results of observations]. Mémoires de la Classe des Sciences Mathématiques et Physiques de l'Institut Impérial de France (in French). 11: 297–347.
^Laplace, P.-S. (1812), Théorie analytique des probabilités [Analytic Probability Theory], vol. 2, pp. 194–203 Collected in Œuvres complètesVII.
^Tchébychef, P. (1860). "Sur le développement des fonctions à une seule variable" [On the development of single-variable functions]. Bulletin de l'Académie impériale des sciences de St.-Pétersbourg (in French). 1: 193–200. Collected in ŒuvresI, 501–508.
^Hermite, C. (1864). "Sur un nouveau développement en série de fonctions" [On a new development in function series]. C. R. Acad. Sci. Paris (in French). 58: 93–100, 266–273. Collected in ŒuvresII, 293–308.
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