View source for Hermite polynomials

''Chebyshev–Hermite polynomials''

Polynomials orthogonal on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470101.png" /> with the weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470102.png" />. The standardized Hermite polynomials are defined by the [[Rodrigues formula|Rodrigues formula]]

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The most commonly used formulas are:

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The first few Hermite polynomials are:

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The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701011.png" /> satisfies the differential equation

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The orthonormal Hermite polynomials are defined by

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The Hermite polynomials with leading coefficient one have the form

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Fourier series in Hermite polynomials in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701015.png" /> behave analogous to trigonometric Fourier series.

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

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The definition of Hermite polynomials is encountered in P. Laplace [[#References|[1]]]. A detailed study of them was published by P.L. Chebyshev in 1859 (see [[#References|[2]]]). Later, these polynomials were studied by Ch. Hermite . V.A. Steklov [[#References|[4]]] proved that the set of them is dense in the space of square-summable functions with the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701017.png" /> on the whole real line.

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



====Comments====
The result by Steklov mentioned in the last sentence of the main article goes back at least to H. Weyl (1908), cf. the references in [[#References|[a3]]], Sect. 5.7.

One possible way to prove the Plancherel formula for the [[Fourier transform|Fourier transform]] is by use of Hermite polynomials, cf. [[#References|[a4]]]. Hermite polynomials occur in solutions of the heat and Schrödinger equations and in the so-called heat polynomials, cf. [[#References|[a1]]]. A canonical orthonormal basis of the representation space for the Schrödinger representation of the Heisenberg group is given in terms of Hermite polynomials, cf. [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Miller jr.,   "Symmetry and separation of variables" , Addison-Wesley  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Schempp,   "Harmonic analysis on the Heisenberg nilpotent Lie group" , Longman  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Szegö,   "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.C. Titchmarsh,   "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR></table>