Hermite polynomials

In mathematics, the Hermite polynomials, named in honor of Charles Hermite (Hermite is pronounced "air MIT"), are a polynomial sequence defined either by

Related Topics:
Mathematics - Charles Hermite - Polynomial sequence

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:H_n(x)=(-1)^n e^{x^2/2}rac{d^n}{dx^n}e^{-x^2/2}

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(the "probabilists' Hermite polynomials"), or sometimes by

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:H_n(x)=(-1)^n e^{x^2}rac{d^n}{dx^n}e^{-x^2}

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(the "physicists' Hermite polynomials"). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. These are Hermite polynomial sequences of different variances; see the material on variances below.

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Below, we follow the first convention. That convention is often preferred by probabilists because

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:rac{1}{sqrt{2pi}}e^{-x^2/2}

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is the probability density function for the normal distribution with expected value 0 and standard deviation 1.

Related Topics:
Probability density function - Normal distribution - Expected value - Standard deviation

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The first several Hermite polynomials are:

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:H_0(x)=1

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:H_1(x)=x

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:H_2(x)=x^2-1

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:H_3(x)=x^3-3x

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:H_4(x)=x^4-6x^2+3

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:H_5(x)=x^5-10x^3+15x

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:H_6(x)=x^6-15x^4+45x^2-15

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