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 x u x {\displaystyle {\begin{aligned}xu_{x}\end{aligned}}} 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, 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 not new, although Hermite was the first to define the multidimensional polynomials.
Definition Like the other classical orthogonal polynomials, the Hermite polynomials can be defined from several different starting points. Noting from the outset that there are two different standardizations in common use, one convenient method is as follows:
The "probabilist's Hermite polynomials" are given by He n ( x ) = ( − 1 ) n e x 2 2 d n d x n e − x 2 2 , {\displaystyle \operatorname {He} _{n}(x)=(-1)^{n}e^{\frac {x^{2}}{2}}{\frac {d^{n}}{dx^{n}}}e^{-{\frac {x^{2}}{2}}},}
while the "physicist's Hermite polynomials" are given by H n ( x ) = ( − 1 ) n e x 2 d n d x n e − x 2 . {\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}.}
These equations have the form of a Rodrigues' formula and can also be written as,
He n ( x ) = ( x − d d x ) n ⋅ 1 , H n ( x ) = ( 2 x − d d x ) n ⋅ 1. {\displaystyle \operatorname {He} _{n}(x)=\left(x-{\frac {d}{dx}}\right)^{n}\cdot 1,\quad H_{n}(x)=\left(2x-{\frac {d}{dx}}\right)^{n}\cdot 1.}
The two definitions are not exactly identical; each is a rescaling of the other:
H n ( x ) = 2 n 2 He n ( 2 x ) , He n ( x ) = 2 − n 2 H n ( x 2 ) . {\displaystyle H_{n}(x)=2^{\frac {n}{2}}\operatorname {He} _{n}\left({\sqrt {2}}\,x\right),\quad \operatorname {He} _{n}(x)=2^{-{\frac {n}{2}}}H_{n}\left({\frac {x}{\sqrt {2}}}\right).}
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